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<!DOCTYPE html>
<html lang="en">
<head>
<title>Bayesian Statistics in R: Intuition First</title>
<meta charset="utf-8">
<meta name="Description" content="Bayesian statistics in R from scratch: simulate the prior-likelihood-posterior update, watch a wrong prior get washed out by data, then bridge to MCMC.">
<meta name="Keywords" content="Bayesian statistics R, prior likelihood posterior, Beta-Binomial conjugate, posterior simulation, credible interval, Bayesian inference, prior distribution, posterior distribution, grid approximation, MCMC introduction">
<meta name="Distribution" content="Global">
<meta name="Author" content="Selva Prabhakaran">
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.sidebar-chevron{display:inline-block;font-size:10px;margin-right:4px;color:#757575}
.sidebar-section-items{display:none;padding-left:0;margin:2px 0 6px}
.sidebar-section.expanded .sidebar-section-items{display:block}
.sidebar-section-items li{line-height:1.5}
.sidebar-section-items li a{display:block;padding:4px 10px 4px 32px;font-size:14px;color:#495057;text-decoration:none;border-left:2px solid transparent;border-radius:0 4px 4px 0}
.sidebar-divider{margin:12px 0 2px;padding:0 12px 4px 24px;font-size:11px;font-weight:700;letter-spacing:0.06em;text-transform:uppercase;color:#9ca3af;border-bottom:1px solid #f0f2f5;list-style:none}
.sidebar-divider:first-child{margin-top:4px}
.progress-dot{display:inline-block;width:10px;height:10px;border-radius:50%;border:1.5px solid #d1d5db;margin-right:7px;vertical-align:middle;flex-shrink:0;position:relative}
/* Empty progress-dot (no status class) reads as an unchecked todo /
disabled radio next to every sidebar item — visual noise that doesn't
belong in a nav. Reserve the layout slot but hide the circle until
the item actually has progress. visibility:hidden (not display:none)
keeps the width stable when state flips. */
.progress-dot:not(.started):not(.visited):not(.completed){visibility:hidden}
/* CLS prevention: TOC sidebar matches main.css final state */
#toc-wrapper{position:sticky;top:20px;max-height:calc(100vh - 40px);overflow-y:auto;padding-left:14px;border-left:1px solid #e9ecef}
.toc-title{font-size:11px;font-weight:600;text-transform:uppercase;letter-spacing:0.8px;color:#6c757d;margin-bottom:10px;padding-bottom:0}
#toc{margin:0;padding:0}
#toc li{line-height:1.45;margin-bottom:1px}
#toc li a{font-size:13.5px;color:#6c757d;text-decoration:none;display:block;padding:3px 0;border-left:2px solid transparent;padding-left:8px;margin-left:-15px}
#toc li a.toc-active{color:#1d3158;font-weight:600;border-left-color:#1d3158}
/* CLS prevention: engagement strip layout (matches engagement.css) */
.engagement-header{display:flex;flex-wrap:wrap;align-items:center;gap:6px 10px;font-family:'Inter',-apple-system,BlinkMacSystemFont,'Segoe UI',sans-serif;font-size:12px;color:#64748b;letter-spacing:0.01em;margin:0.4em 0 1em;min-height:22px}
.engagement-header:empty{display:none}
/* CLS prevention: progress bar space (matches engagement.css) */
.engagement-progress{min-height:32px;margin:1.2em 0 1.6em}
/* CLS prevention: WebR editor min-height so the static→textarea swap doesn't shift layout */
.webr-editor{min-height:60px}
/* CLS prevention: images in content always reserve aspect ratio */
#content img{max-width:100%;height:auto}
/* =====================================================
LAYOUT V2 (opt-in via body.layout-v2). Retry 2026-05-29.
Differences from the first attempt:
1. Sidebar columns use clamp(180px, 16vw, 260px) so they
SHRINK on narrow viewports (first attempt used minmax
which pinned both sides at 260px, squeezing content on
typical laptops to ~460px).
2. Sidebar links get overflow-wrap so long tutorial titles
don't escape the narrower column.
3. min-width:0 on every grid child so they can shrink.
4. Right TOC breakpoint raised from 1100px -> 1280px so it
only shows when the viewport can actually afford it.
5. The inline-code background restyle is dropped - kept
current syntax-highlighter behavior.
All rules scoped under body.layout-v2 so pages without it
render identically.
===================================================== */
body.layout-v2{
--lv2-bg:#f8f9fb;--lv2-card:#fff;--lv2-ink:#0d1117;
--lv2-mut:#4b5260;--lv2-faint:#5b6573;
--lv2-navy:#1c2c4f;--lv2-navy-h:#2d4173;--lv2-navy-soft:#eef1f7;
--lv2-accent:#2056d2;--lv2-accent-soft:#e9efff;
--lv2-line:#e4e7ee;--lv2-border:#d4d9e3;
--lv2-green:#15803d;
background:var(--lv2-bg);color:var(--lv2-ink);
}
html.dark body.layout-v2{
--lv2-bg:#0a0f1c;--lv2-card:#0f1525;--lv2-ink:#eef2fa;
--lv2-mut:#a5afc4;--lv2-faint:#9aa6c0;
--lv2-navy:#5170b3;--lv2-navy-h:#6587c4;--lv2-navy-soft:#192240;
--lv2-accent:#7faaff;--lv2-accent-soft:#1a2748;
--lv2-line:#1f2a48;--lv2-border:#2c3a62;
}
/* Wider container; clamp-based grid that scales with viewport.
width:auto is REQUIRED because the base critical CSS has
@media(min-width:1200px){.container{width:1170px}} which would
otherwise pin width to 1170 and ignore max-width:1340. */
body.layout-v2 .container{max-width:1340px;width:auto;padding:0 22px}
body.layout-v2 .row{
display:grid;
grid-template-columns:
clamp(180px, 16vw, 260px)
minmax(0, 1fr)
clamp(180px, 16vw, 260px);
gap:0;margin-left:0;margin-right:0;
}
/* Bootstrap inserts .row::before AND .row::after pseudo-elements
(content:" "; display:table) as a clearfix. Under display:grid
these pseudos count as grid items. ::before grabs col 1, pushing
every real child one column to the right and wrapping the last
child to a new row. The 2026-05-29 first attempt only disabled
::after, resulting in nav in col 2, content in col 3, and toc
wrapping to a new row at col 1 (the bug the user reported as
"sidebar wide, content narrow"). BOTH pseudos must be killed. */
body.layout-v2 .row::before,
body.layout-v2 .row::after{display:none;content:none}
/* min-width:0 lets grid children shrink below their content's
intrinsic min-width; without this, long unbreakable text in any
child forces the column wider than the grid template intended. */
/* Drop Bootstrap's float and width on the legacy .col-sm-* hooks
so the parent .row's CSS grid owns sizing. DO NOT zero horizontal
padding here — per-column padding lives in the dedicated rules
below (sidebar, #content, #toc-sidebar) and gives the article
the 44px breathing room the mock has. */
body.layout-v2 #nav.col-sm-3,
body.layout-v2 #content.col-sm-7,
body.layout-v2 #toc-sidebar.col-sm-2{
width:auto!important;float:none!important;
min-width:0;min-height:0;
}
body.layout-v2 #nav.col-sm-3{
padding:14px 10px 40px;border-right:1px solid var(--lv2-line);
position:sticky;top:60px;height:calc(100vh - 60px);overflow-y:auto;
}
/* Inner #sidebar-nav had its own sticky + overflow from the legacy critical
CSS; under layout-v2 the outer #nav owns scrolling, so flatten the inner
one to avoid a double scrollbar. */
body.layout-v2 #sidebar-nav{
position:static;max-height:none;overflow:visible;padding:0;
}
/* Wrap long tutorial titles in the sidebar so they never overflow */
body.layout-v2 #nav a,
body.layout-v2 #nav span,
body.layout-v2 #sidebar-nav a,
body.layout-v2 #sidebar-nav li{
overflow-wrap:break-word;word-break:break-word;
}
body.layout-v2 #content{padding:24px 44px 60px}
/* S3 — Code-block bottom spacing.
Default .webr-container margin is 22/22; the dark block visually
carries a lot of weight, so the next paragraph reads as cramped.
Lift bottom to 30px under layout-v2. */
body.layout-v2 #content .webr-container{margin-bottom:30px}
/* S7 — Further Reading: convert auto-injected <ul> to a 2-column
card grid that matches the .rt-grid pattern used elsewhere.
No HTML/auto_link.py changes needed — pure CSS. */
body.layout-v2 #content #auto-further-reading .further-reading-list{
list-style:none;padding:0;margin:14px 0 0;
display:grid;grid-template-columns:repeat(2,minmax(0,1fr));gap:12px;
}
body.layout-v2 #content #auto-further-reading .further-reading-list li{margin:0}
body.layout-v2 #content #auto-further-reading .further-reading-list li a{
display:block;padding:14px 16px;
background:var(--lv2-card);border:1px solid var(--lv2-border);
border-radius:8px;color:var(--lv2-ink);
font-weight:500;font-size:15px;line-height:1.4;
text-decoration:none;
transition:border-color .15s,background .15s,transform .15s;
}
body.layout-v2 #content #auto-further-reading .further-reading-list li a:hover{
border-color:var(--lv2-accent);background:var(--lv2-accent-soft);
text-decoration:none;transform:translateY(-1px);
}
@media(max-width:820px){
body.layout-v2 #content #auto-further-reading .further-reading-list{
grid-template-columns:1fr;
}
}
body.layout-v2 #toc-sidebar{padding:24px 18px;position:sticky;top:74px;align-self:start}
@media(max-width:1280px){
body.layout-v2 .row{grid-template-columns:clamp(180px, 18vw, 260px) minmax(0,1fr)}
body.layout-v2 #toc-sidebar{display:none}
}
@media(max-width:820px){
body.layout-v2 .row{grid-template-columns:1fr}
body.layout-v2 #nav{display:none}
body.layout-v2 #content{padding:20px}
}
/* Typography refresh - sized to mock spec */
body.layout-v2 #content{
font-size:16.5px;line-height:1.7;color:var(--lv2-ink);
font-feature-settings:"tnum","ss01","cv11";text-rendering:optimizeLegibility;
}
body.layout-v2 #content p{line-height:1.7;margin:13px 0;color:var(--lv2-ink);text-wrap:pretty}
body.layout-v2 #content h1{
font-size:38px;font-weight:700;line-height:1.2;letter-spacing:-0.018em;
margin:10px 0 13px;text-wrap:balance;color:var(--lv2-ink);
}
/* H2: hairline divider + breathing room above. Mock breaks each
section with a top rule and ~48px space; staging used 34px and no
divider, leaving sections to blur together. :first-of-type keeps
the very first H2 flush with the actionbar (no double divider). */
body.layout-v2 #content h2{
font-size:25px;font-weight:600;line-height:1.25;letter-spacing:-0.018em;
margin:48px 0 12px;padding-top:32px;border-top:1px solid var(--lv2-line);
text-wrap:balance;color:var(--lv2-ink);
}
body.layout-v2 #content h2:first-of-type{
margin-top:24px;padding-top:0;border-top:none;
}
body.layout-v2 #content h3,body.layout-v2 #content h4{
font-size:18px;font-weight:600;line-height:1.3;margin:24px 0 8px;text-wrap:balance;color:var(--lv2-ink);
}
/* Specificity bump: `body.layout-v2 #content p` (1 id + 1 class
+ 2 elements) was beating `body.layout-v2 p.lead` (2 classes +
2 elements) and forcing the dek to var(--lv2-ink) instead of
var(--lv2-mut). Adding #content here matches that specificity. */
body.layout-v2 #content p.lead{
font-size:18.5px;line-height:1.6;color:var(--lv2-mut);
border-left:3px solid var(--lv2-navy);padding-left:17px;margin:-2px 0 20px;font-weight:400;
}
/* Body link color: legacy a{color:#1d3158} and Bootstrap .text-muted
leave inline links almost indistinguishable from body text. Use
the mock accent so links are obviously clickable. Excludes
buttons, cards, prev/next nav, and TOC items which have their
own palette. */
body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]){
color:var(--lv2-accent);
}
body.layout-v2 #content a:not(.btn):not(.rt-card):not(.pn-link):not(.lv2-anchor):not([class*="toc"]):hover{
text-decoration:underline;
}
/* Inline <code> palette: match the mock's navy-soft tint instead
of the legacy purple-blue (#eef vs #eef1f7). */
body.layout-v2 #content :not(pre) > code{
background:var(--lv2-navy-soft);color:var(--lv2-ink);
border-radius:4px;padding:2px 5px;font-size:0.92em;
}
/* TOC right rail */
body.layout-v2 #toc-sidebar .toc-title{font-size:10.5px;font-weight:700;text-transform:uppercase;letter-spacing:.08em;color:var(--lv2-faint);margin-bottom:9px}
body.layout-v2 #toc{font-size:13px}
body.layout-v2 #toc li{margin:0;line-height:1.45}
body.layout-v2 #toc li a{
display:block;padding:5px 0 5px 11px;
border-left:2px solid var(--lv2-line);
color:var(--lv2-faint);font-size:13px;
transition:color .15s,border-color .15s;
}
body.layout-v2 #toc li a:hover{color:var(--lv2-ink)}
body.layout-v2 #toc li a.toc-active{
border-left-color:var(--lv2-navy);color:var(--lv2-ink);font-weight:600;
}
/* Reading progress bar - uses transform:scaleX so we don't fight any
containing-block / overflow quirks that can pin a width-% bar to 0. */
.lv2-progress{
position:fixed;top:0;left:0;width:100vw;height:3px;
background:var(--lv2-accent,#2056d2);z-index:90;
transform:scaleX(0);transform-origin:left center;
transition:transform .1s ease-out;pointer-events:none;
}
/* Heading anchor links (hover-reveal) */
body.layout-v2 #content h2,
body.layout-v2 #content h3,
body.layout-v2 #content h4{position:relative}
body.layout-v2 .lv2-anchor{
position:absolute;left:-26px;top:50%;transform:translateY(-50%);
font-size:.55em;color:var(--lv2-faint);text-decoration:none;
opacity:0;transition:opacity .15s,color .15s;
cursor:pointer;padding:4px 6px;line-height:1;font-weight:400;
}
body.layout-v2 #content h2:hover .lv2-anchor,
body.layout-v2 #content h3:hover .lv2-anchor,
body.layout-v2 #content h4:hover .lv2-anchor,
body.layout-v2 .lv2-anchor:focus-visible{opacity:1}
body.layout-v2 .lv2-anchor:hover{color:var(--lv2-accent)}
@media(max-width:820px){body.layout-v2 .lv2-anchor{left:0;top:-22px;transform:none}}
/* Focus ring refresh */
body.layout-v2 a:focus-visible,
body.layout-v2 button:focus-visible,
body.layout-v2 input:focus-visible,
body.layout-v2 textarea:focus-visible{
outline:none;
box-shadow:0 0 0 3px rgba(32,86,210,.22);
border-radius:4px;
}
html.dark body.layout-v2 a:focus-visible,
html.dark body.layout-v2 button:focus-visible,
html.dark body.layout-v2 input:focus-visible{box-shadow:0 0 0 3px rgba(127,170,255,.22)}
/* Reduced motion: shut everything down for users who request it */
@media (prefers-reduced-motion: reduce){
body.layout-v2 *,body.layout-v2 *::before,body.layout-v2 *::after{
animation-duration:0.01ms!important;
animation-iteration-count:1!important;
transition-duration:0.01ms!important;
scroll-behavior:auto!important;
}
}
</style>
<!-- Full Bootstrap deferred (non-render-blocking) -->
<link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet" media="print" onload="this.media='all'">
<noscript><link href="www/bootstrap.min.css?h=dae0a568" rel="stylesheet"></noscript>
<!-- Roadmap-jel: load code-highlight CSS normally (not deferred) so code blocks paint COLORED on first load, not after interaction. Small file; negligible render cost. -->
<link href="www/highlight.min.css?h=f103014a" rel="stylesheet">
<link href="css/main.min.css?h=a2bb70ae" rel="stylesheet" media="print" onload="this.media='all'">
<noscript><link href="css/main.min.css?h=a2bb70ae" rel="stylesheet"></noscript>
<!-- Sitewide navbar (canonical roadmap .nav design). Loaded sync: the nav
is above the fold and must never flash unstyled. -->
<link href="/www/site-nav.css?v=7" rel="stylesheet">
<style>
a{color:#1d3158}a:hover{text-decoration:underline}
li{line-height:1.65}
.MathJax_Display{margin:0}
blockquote p{line-height:1.75;color:#717171}
/* engagement.js renders a meta strip into .engagement-header that
duplicates the byline (difficulty + time). Hide it; saved-posts-
button.js reads the data-* attrs directly for the byline. */
.engagement-header{display:none!important}
/* Byline (saved-posts widget rebuilds this after H1+lead, mock-style) */
.byline{display:flex;flex-wrap:wrap;align-items:center;gap:8px;
font-size:13.5px;color:#4b5260;margin:6px 0 16px;line-height:1.5}
/* P1 — bump avatar 26→30 to match mock visual weight. */
.byline .av-sm{width:30px;height:30px;border-radius:50%;
background:linear-gradient(135deg,#1c2c4f,#2d4173);color:#fff;
display:inline-flex;align-items:center;justify-content:center;
font-weight:700;font-size:12px;flex:none;letter-spacing:.02em}
.byline .who{color:#0d1117;font-weight:600}
.byline .dot{color:#c0c5cf}
.byline .rev{color:#137a3e;font-weight:600;display:inline-flex;align-items:center;gap:4px}
.byline .time{display:inline-flex;align-items:center;gap:4px;color:#6b7280}
.byline .diff{font-size:10.5px;font-weight:700;text-transform:uppercase;
letter-spacing:.05em;background:#eef1f7;color:#4b5260;
padding:3px 8px;border-radius:999px;margin-left:2px}
/* Action bar (saved-posts widget injects this after byline) */
.actionbar{display:flex;align-items:center;gap:9px;margin:10px 0 26px;padding:12px 0;
border-top:1px solid #e4e7ee;border-bottom:1px solid #e4e7ee;flex-wrap:wrap}
.actionbar .act{display:inline-flex;align-items:center;gap:6px;border:1px solid #d4d9e3;
background:#fff;border-radius:9px;padding:7px 12px;font-size:13px;font-weight:600;
color:#4b5260;cursor:pointer;font-family:inherit;
transition:border-color .15s,color .15s,background .15s}
.actionbar .act:hover{border-color:#0a0d14;color:#0a0d14}
.actionbar .act.bookmark-btn[data-saved="1"]{background:#fef3c7;border-color:#f59e0b;color:#92400e}
.actionbar .act.bookmark-btn[data-saved="1"] svg{fill:#f59e0b}
.actionbar .act:disabled{opacity:.55;cursor:wait}
.actionbar-sync{margin-left:auto;font-size:12px;color:#6b7280}
.actionbar-sync a{color:#2056d2;font-weight:600;text-decoration:none}
.actionbar-sync a:hover{text-decoration:underline}
/* Dark-mode overrides for byline + actionbar. The hard-coded #0d1117
on .who renders invisible on the dark background; the action-bar
borders and surfaces also need to flip. Tokens (--lv2-ink, --lv2-mut,
--lv2-line, --lv2-card) already swap under html.dark body.layout-v2;
we just need to point the hard-coded rules at them under dark. */
html.dark body.layout-v2 .byline{color:var(--lv2-mut)}
html.dark body.layout-v2 .byline .who{color:var(--lv2-ink)}
html.dark body.layout-v2 .byline .dot{color:var(--lv2-faint)}
html.dark body.layout-v2 .byline .time{color:var(--lv2-mut)}
html.dark body.layout-v2 .byline .diff{background:var(--lv2-navy-soft);color:var(--lv2-mut)}
html.dark body.layout-v2 .actionbar{border-top-color:var(--lv2-line);border-bottom-color:var(--lv2-line)}
html.dark body.layout-v2 .actionbar .act{background:var(--lv2-card);border-color:var(--lv2-border);color:var(--lv2-mut)}
html.dark body.layout-v2 .actionbar .act:hover{border-color:var(--lv2-ink);color:var(--lv2-ink)}
html.dark body.layout-v2 .actionbar-sync{color:var(--lv2-mut)}
/* Toast (Share button "Link copied" feedback) */
.rs-toast{position:fixed;bottom:24px;left:50%;transform:translateX(-50%) translateY(8px);
background:#0a0d14;color:#fff;padding:9px 18px;border-radius:8px;font-size:13px;
font-family:'IBM Plex Sans',sans-serif;z-index:1000;opacity:0;pointer-events:none;
transition:opacity .2s,transform .2s;box-shadow:0 8px 24px rgba(10,13,20,.25)}
.rs-toast.show{opacity:1;transform:translateX(-50%) translateY(0)}
/* Auth-state visibility helpers (used by masthead variants) */
.auth-anon,.auth-user{display:none}
body.state-anon .auth-anon{display:inline-flex;align-items:center}
body.state-pro .auth-user{display:inline-flex;align-items:center;gap:8px}
.auth-signout.masthead-icon-btn{background:none;border:1px solid #d8dce2;border-radius:6px;padding:4px 9px;font-size:14px;line-height:1;color:#4a5160;cursor:pointer;transition:border-color .15s,color .15s}
.auth-signout.masthead-icon-btn:hover{border-color:#9a1f1f;color:#9a1f1f}
html.dark .auth-signout.masthead-icon-btn{border-color:#262a31;color:#c3cad9}
html.dark .auth-signout.masthead-icon-btn:hover{border-color:#f08989;color:#f08989}
@media(min-width:1200px){.container{width:1170px}}
#nav,#content,#toc-sidebar{box-sizing:border-box}
#content{padding-left:15px;padding-right:15px;overflow-wrap:break-word;word-wrap:break-word;overflow-x:clip}
@media(max-width:767px){#nav{display:none}}
.mobile-sidebar-overlay{display:none;position:fixed;top:0;left:0;right:0;bottom:0;z-index:9999;background:rgba(0,0,0,0.4)}
.mobile-sidebar-overlay.open{display:block}
.mobile-sidebar-panel{position:fixed;top:0;left:0;bottom:0;width:280px;background:#fff;z-index:10000;overflow-y:auto;padding:16px;box-shadow:2px 0 16px rgba(0,0,0,0.15);transform:translateX(-100%);transition:transform 0.25s ease}
.mobile-sidebar-overlay.open .mobile-sidebar-panel{transform:translateX(0)}
.mobile-sidebar-close{position:absolute;top:10px;right:12px;background:none;border:none;font-size:22px;cursor:pointer;color:#666}
/* Mobile top-nav drawer section */
.mnav-section{margin-bottom:18px;padding-bottom:14px;border-bottom:1px solid #e5e7eb}
.mnav-section:last-child{border-bottom:none}
.mnav-title{font-family:'IBM Plex Sans',sans-serif;font-size:11px;font-weight:600;color:#8a8f99;text-transform:uppercase;letter-spacing:0.06em;margin:0 0 8px}
.mnav-link{display:block;padding:9px 10px;color:#3a3f4a;text-decoration:none;font-size:15px;border-radius:6px;font-family:'IBM Plex Sans',sans-serif}
.mnav-link:hover,.mnav-link:active{background:#f1f3f6;color:#0d1117;text-decoration:none}
.mnav-accordion{padding:0}
.mnav-accordion>summary{padding:9px 10px;cursor:pointer;font-size:15px;font-weight:500;color:#3a3f4a;list-style:none;border-radius:6px;font-family:'IBM Plex Sans',sans-serif;display:flex;align-items:center;justify-content:space-between}
.mnav-accordion>summary::-webkit-details-marker{display:none}
.mnav-accordion>summary::after{content:'\25BE';font-size:11px;color:#8a8f99;transition:transform 0.15s}
.mnav-accordion[open]>summary::after{transform:rotate(180deg)}
.mnav-accordion>summary:hover{background:#f1f3f6}
.mnav-accordion-body{padding:4px 6px 8px 10px}
.mnav-accordion-body .mnav-link{padding:7px 8px;font-size:14px;color:#4a5160}
.mnav-accordion-body h6{margin:10px 0 4px 8px;font-family:'IBM Plex Sans',sans-serif;font-size:10px;font-weight:600;color:#8a8f99;text-transform:uppercase;letter-spacing:0.06em}
html.dark .mobile-sidebar-panel{background:#1e293b}
html.dark .mobile-sidebar-overlay{background:rgba(0,0,0,0.6)}
html.dark .mobile-sidebar-close{color:#94a3b8}
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title="R Control Flow"><span class="progress-dot"></span>R Control Flow</a></li><li data-subkey="sec0sub2"><a href="/R-Special-Values.html" title="R Special Values"><span class="progress-dot"></span>R Special Values</a></li><li data-subkey="sec0sub2"><a href="/R-Type-Coercion.html" title="R Type Coercion"><span class="progress-dot"></span>R Type Coercion</a></li><li data-subkey="sec0sub2"><a href="/R-Functions.html" title="Writing R Functions"><span class="progress-dot"></span>Writing R Functions</a></li><li data-subkey="sec0sub2" class="is-quiz"><a href="/R-Beginner-Exercises-quiz.html" title="R Fundamentals Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Working Effectively</li><li 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Users</a></li><li data-subkey="sec0sub4"><a href="/R-Interview-Questions.html" title="R Interview Questions"><span class="progress-dot"></span>R Interview Questions</a></li><li data-subkey="sec0sub4" class="is-quiz"><a href="/R-Interview-Questions-quiz.html" title="R Interview Readiness Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec0sub4"><a href="/R-Cheat-Sheet.html" title="R Cheat Sheet"><span class="progress-dot"></span>R Cheat Sheet</a></li><li data-subkey="sec0sub4"><a href="/Bias-in-Data-and-Models.html" title="Bias in Data and Models"><span class="progress-dot"></span>Bias in Data and Models</a></li><li data-subkey="sec0sub4"><a href="/Communicating-Uncertainty.html" title="Communicating Uncertainty"><span class="progress-dot"></span>Communicating Uncertainty</a></li><li data-subkey="sec0sub4"><a href="/Data-Ethics-in-R.html" title="Data Ethics"><span class="progress-dot"></span>Data Ethics</a></li><li data-subkey="sec0sub4"><a href="/Data-Privacy-in-R.html" title="Data Privacy in R"><span class="progress-dot"></span>Data Privacy in R</a></li><li data-subkey="sec0sub4"><a href="/R-Foundations-Basics-Course.html" title="R Foundations: The Basics (Course)"><span class="progress-dot"></span>R Foundations: The Basics (Course)</a></li><li data-subkey="sec0sub4"><a href="/R-Foundations-Debugging-Course.html" title="R Foundations: Debugging (Course)"><span class="progress-dot"></span>R Foundations: Debugging (Course)</a></li><li data-subkey="sec0sub4"><a href="/R-Foundations-Iteration-Course.html" title="R Foundations: Iteration (Course)"><span class="progress-dot"></span>R Foundations: Iteration (Course)</a></li><li data-subkey="sec0sub4"><a href="/R-Foundations-Structures-Course.html" title="R Foundations: Data Structures (Course)"><span class="progress-dot"></span>R Foundations: Data Structures (Course)</a></li><li data-subkey="sec0sub4"><a href="/R-Foundations-Workflow-Course.html" title="R Foundations: Reproducible Workflow (Course)"><span class="progress-dot"></span>R Foundations: Reproducible Workflow (Course)</a></li><li data-subkey="sec0sub4"><a href="/Reproducibility-Crisis.html" title="Reproducibility Crisis"><span class="progress-dot"></span>Reproducibility Crisis</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">2.</span> <span class="sec-title-t">Data Wrangling</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Import & Setup</li><li data-subkey="sec1sub1"><a href="/Importing-Data-in-R.html" title="Importing Data"><span class="progress-dot"></span>Importing Data</a></li><li data-subkey="sec1sub1"><a href="/R-Pipe-Operator.html" title="Pipe Operator"><span class="progress-dot"></span>Pipe Operator</a></li><li data-subkey="sec1sub1"><a href="/Tidy-Data-in-R.html" title="Tidy Data"><span class="progress-dot"></span>Tidy Data</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> dplyr Essentials</li><li data-subkey="sec1sub2"><a href="/dplyr-filter-select.html" title="dplyr filter & select"><span class="progress-dot"></span>dplyr filter & select</a></li><li data-subkey="sec1sub2"><a href="/dplyr-mutate-rename.html" title="dplyr mutate & rename"><span class="progress-dot"></span>dplyr mutate & rename</a></li><li data-subkey="sec1sub2"><a href="/dplyr-group-by-summarise.html" title="dplyr group_by & summarise"><span class="progress-dot"></span>dplyr group_by & summarise</a></li><li data-subkey="sec1sub2"><a href="/dplyr-arrange-slice.html" title="dplyr arrange & slice"><span class="progress-dot"></span>dplyr arrange & slice</a></li><li data-subkey="sec1sub2"><a href="/dplyr-across.html" title="dplyr across()"><span class="progress-dot"></span>dplyr across()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-case-when.html" title="dplyr case_when()"><span class="progress-dot"></span>dplyr case_when()</a></li><li data-subkey="sec1sub2" class="is-quiz"><a href="/dplyr-Exercises-in-R-quiz.html" title="dplyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Join & Reshape</li><li data-subkey="sec1sub3"><a href="/R-Joins.html" title="R Joins"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html" title="pivot_longer & pivot_wider"><span class="progress-dot"></span>pivot_longer & pivot_wider</a></li><li data-subkey="sec1sub3"><a href="/tidyr-separate-unite-Split-Combine-Columns-in-R.html" title="separate() & unite()"><span class="progress-dot"></span>separate() & unite()</a></li><li data-subkey="sec1sub3" class="is-quiz"><a href="/tidyr-Exercises-in-R-quiz.html" title="tidyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Clean & Quality</li><li data-subkey="sec1sub4"><a href="/Missing-Values-in-R-Detect-Count-Remove-Impute-NA.html" title="Missing Values (NA)"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html" title="Data Quality Checking"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html" title="janitor Package"><span class="progress-dot"></span>janitor Package</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Strings & Dates</li><li data-subkey="sec1sub5"><a href="/stringr-in-R.html" title="stringr"><span class="progress-dot"></span>stringr</a></li><li data-subkey="sec1sub5"><a href="/R-Regex-stringr-Pattern-Matching.html" title="Regex Patterns"><span class="progress-dot"></span>Regex Patterns</a></li><li data-subkey="sec1sub5"><a href="/lubridate-in-R.html" title="lubridate"><span class="progress-dot"></span>lubridate</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec1sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Scale & Connect</li><li data-subkey="sec1sub6"><a href="/DBI-in-R.html" title="DBI & Databases"><span class="progress-dot"></span>DBI & Databases</a></li><li data-subkey="sec1sub6"><a href="/DuckDB-in-R.html" title="DuckDB & duckplyr"><span class="progress-dot"></span>DuckDB & duckplyr</a></li><li data-subkey="sec1sub6"><a href="/Web-Scraping-in-R-with-rvest.html" title="Web Scraping (rvest)"><span class="progress-dot"></span>Web Scraping (rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html" title="REST APIs (httr2)"><span class="progress-dot"></span>REST APIs (httr2)</a></li><li data-subkey="sec1sub6"><a href="/Data-Wrangling-dplyr-Course.html" title="Data Wrangling with dplyr (Course)"><span class="progress-dot"></span>Data Wrangling with dplyr (Course)</a></li><li data-subkey="sec1sub6"><a href="/Join-Reshape-Course.html" title="Join & Reshape (Course)"><span class="progress-dot"></span>Join & Reshape (Course)</a></li><li data-subkey="sec1sub6"><a href="/data-table-Course.html" title="data.table (Course)"><span class="progress-dot"></span>data.table (Course)</a></li><li data-subkey="sec1sub6"><a href="/Report-Tables-Course.html" title="Report-Ready Tables (Course)"><span class="progress-dot"></span>Report-Ready Tables (Course)</a></li><li data-subkey="sec1sub6"><a href="/Communicate-Automate-Course.html" title="Communicate & Automate (Course)"><span class="progress-dot"></span>Communicate & Automate (Course)</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">3.</span> <span class="sec-title-t">Statistics</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> EDA & Data Quality</li><li data-subkey="sec2sub1"><a href="/Automated-EDA-in-R.html" title="Automated EDA"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec2sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html" title="Missing Data Viz (naniar)"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec2sub1"><a href="/Outlier-Detection-in-R.html" title="Outlier Detection"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Probability</li><li data-subkey="sec2sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html" title="Probability Axioms"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec2sub2"><a href="/Conditional-Probability-in-R.html" title="Conditional Probability"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec2sub2"><a href="/Random-Variables-in-R.html" title="Random Variables"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec2sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html" title="Binomial vs Poisson"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec2sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html" title="Normal, t, F, Chi-Squared"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec2sub2"><a href="/Central-Limit-Theorem-in-R.html" title="Central Limit Theorem"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec2sub2"><a href="/Sampling-Distributions-in-R.html" title="Sampling Distributions"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec2sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html" title="LLN vs CLT"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec2sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html" title="Probability (Simulation-First)"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec2sub2"><a href="/Expected-Value-and-Variance-in-R.html" title="Expected Value and Variance"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Inference & Estimation</li><li data-subkey="sec2sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html" title="Maximum Likelihood Estimation"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec2sub3"><a href="/Hypothesis-Testing-in-R.html" title="Hypothesis Testing"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec2sub3"><a href="/Sample-Size-Planning-in-R.html" title="Sample Size Planning"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec2sub3"><a href="/Which-Statistical-Test-in-R.html" title="Choosing the Right Test"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec2sub3"><a href="/Statistical-Tests-in-R.html" title="Statistical Tests"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec2sub3"><a href="/Measures-of-Association-in-R.html" title="Measures of Association"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec2sub3"><a href="/Point-Estimation-in-R.html" title="Point Estimation"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec2sub3"><a href="/Confidence-Intervals-in-R.html" title="Confidence Intervals"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec2sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html" title="Type I and II Errors"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec2sub3"><a href="/Statistical-Power-Analysis-in-R.html" title="Power Analysis"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec2sub3"><a href="/Effect-Size-in-R.html" title="Effect Size"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec2sub3"><a href="/t-Tests-in-R.html" title="t-Tests"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec2sub3"><a href="/Proportion-Tests-in-R.html" title="Proportion Tests"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec2sub3"><a href="/Normality-and-Variance-Tests-in-R.html" title="Normality & Variance Tests"><span class="progress-dot"></span>Normality & Variance Tests</a></li><li data-subkey="sec2sub3"><a href="/Chi-Square-Tests-in-R.html" title="Chi-Square Tests"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec2sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html" title="Wilcoxon, Mann-Whitney & Kruskal-Wallis"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney & Kruskal-Wallis</a></li><li data-subkey="sec2sub3"><a href="/Multiple-Comparisons-in-R.html" title="Multiple Testing Correction"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec2sub3" class="is-quiz"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html" title="Hypothesis Testing Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression</li><li data-subkey="sec2sub4"><a href="/Linear-Regression.html" title="Linear Regression"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec2sub4"><a href="/Logistic-Regression-With-R.html" title="Logistic Regression"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec2sub4"><a href="/Variable-Selection-and-Importance-With-R.html" title="Feature Selection"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec2sub4"><a href="/Model-Selection-in-R.html" title="Model Selection"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec2sub4"><a href="/Missing-Value-Treatment-With-R.html" title="Missing Value Treatment"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec2sub4"><a href="/Outlier-Treatment-With-R.html" title="Outlier Analysis"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec2sub4"><a href="/adv-regression-models.html" title="Advanced Regression Models"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec2sub4" class="is-quiz"><a href="/Linear-Regression-Exercises-in-R-quiz.html" title="Linear Regression Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Reporting & Communication</li><li data-subkey="sec2sub5"><a href="/Statistical-Consulting-in-R.html" title="Statistical Consulting"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec2sub5"><a href="/Statistical-Report-Writing-in-R.html" title="Statistical Report Writing"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec2sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html" title="Bootstrap Confidence Intervals"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec2sub5"><a href="/Reporting-Statistics-in-R.html" title="Reporting Statistics"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec2sub5"><a href="/Regression-Tables-in-R.html" title="Regression Tables (3 packages)"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression in Practice</li><li data-subkey="sec2sub6"><a href="/Simple-Linear-Regression-in-R.html" title="Simple Linear Regression"><span class="progress-dot"></span>Simple Linear Regression</a></li><li data-subkey="sec2sub6"><a href="/Multiple-Regression-in-R.html" title="Multiple Regression"><span class="progress-dot"></span>Multiple Regression</a></li><li data-subkey="sec2sub6"><a href="/Correlation-in-R.html" title="Correlation (Pearson, Spearman, Kendall)"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec2sub6"><a href="/Linear-Regression-Assumptions-in-R.html" title="Linear Regression Assumptions"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec2sub6"><a href="/Dummy-Variables-in-R.html" title="Dummy Variables in R"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec2sub6"><a href="/Interaction-Effects-in-R.html" title="Interaction Effects"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec2sub6"><a href="/Regression-Diagnostics-in-R.html" title="Regression Diagnostics"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec2sub6"><a href="/Variable-Selection-in-R.html" title="Variable Selection"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec2sub6"><a href="/Polynomial-and-Spline-Regression-in-R.html" title="Polynomial & Splines"><span class="progress-dot"></span>Polynomial & Splines</a></li><li data-subkey="sec2sub6"><a href="/Ridge-and-Lasso-Regression-in-R.html" title="Ridge & Lasso Regression"><span class="progress-dot"></span>Ridge & Lasso Regression</a></li><li data-subkey="sec2sub6"><a href="/Robust-Regression-in-R.html" title="Robust Regression (rlm)"><span class="progress-dot"></span>Robust Regression (rlm)</a></li><li data-subkey="sec2sub6"><a href="/Quantile-Regression-in-R-2.html" title="Quantile Regression"><span class="progress-dot"></span>Quantile Regression</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> ANOVA & Experiments</li><li data-subkey="sec2sub7"><a href="/One-Way-ANOVA-in-R.html" title="One-Way ANOVA"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Post-Hoc-Tests-After-ANOVA.html" title="Post-Hoc Tests After ANOVA"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Two-Way-ANOVA-in-R.html" title="Two-Way ANOVA"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec2sub7"><a href="/Repeated-Measures-ANOVA-in-R.html" title="Repeated Measures ANOVA"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec2sub7"><a href="/ANCOVA-in-R.html" title="ANCOVA"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec2sub7"><a href="/Experimental-Design-Principles-in-R.html" title="Experimental Design in R"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec2sub7"><a href="/Factorial-Experiments-in-R.html" title="Factorial Designs (2^k)"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec2sub7"><a href="/AB-Testing-in-R.html" title="A/B Testing"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec2sub7"><a href="/MANOVA-in-R.html" title="MANOVA"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec2sub7"><a href="/Mixed-ANOVA-in-R.html" title="Mixed ANOVA"><span class="progress-dot"></span>Mixed ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> GLMs & Categorical Data</li><li data-subkey="sec2sub8"><a href="/Categorical-Data-in-R.html" title="Categorical Data (Tables & Mosaic)"><span class="progress-dot"></span>Categorical Data (Tables & Mosaic)</a></li><li data-subkey="sec2sub8"><a href="/Chi-Square-Test-of-Independence-in-R.html" title="Chi-Square Test of Independence"><span class="progress-dot"></span>Chi-Square Test of Independence</a></li><li data-subkey="sec2sub8"><a href="/Chi-Square-Goodness-of-Fit-Test-in-R.html" title="Chi-Square Goodness-of-Fit"><span class="progress-dot"></span>Chi-Square Goodness-of-Fit</a></li><li data-subkey="sec2sub8"><a href="/Fishers-Exact-Test-in-R.html" title="Fisher's Exact Test"><span class="progress-dot"></span>Fisher's Exact Test</a></li><li data-subkey="sec2sub8"><a href="/Odds-Ratios-and-Relative-Risk-in-R.html" title="Odds Ratios & Relative Risk"><span class="progress-dot"></span>Odds Ratios & Relative Risk</a></li><li data-subkey="sec2sub8"><a href="/Logistic-Regression-in-R.html" title="Logistic Regression (glm + ROC)"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec2sub8"><a href="/Logistic-Regression-in-R-2.html" title="Logistic Regression (Diagnostics)"><span class="progress-dot"></span>Logistic Regression (Diagnostics)</a></li><li data-subkey="sec2sub8"><a href="/Poisson-Regression-in-R.html" title="Poisson Regression"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec2sub8"><a href="/Poisson-and-Negative-Binomial-Regression.html" title="Poisson & Negative Binomial Regression"><span class="progress-dot"></span>Poisson & Negative Binomial Regression</a></li><li data-subkey="sec2sub8"><a href="/Multinomial-and-Ordinal-Logistic-Regression-in-R.html" title="Multinomial & Ordinal Logistic Regression"><span class="progress-dot"></span>Multinomial & Ordinal Logistic Regression</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub9" data-collapsed="false"><span class="subsec-chevron">▼</span> Multivariate Methods</li><li data-subkey="sec2sub9"><a href="/Multivariate-Statistics-in-R.html" title="Multivariate Distances & Hotelling's T²"><span class="progress-dot"></span>Multivariate Distances & Hotelling's T²</a></li><li data-subkey="sec2sub9"><a href="/PCA-in-R.html" title="PCA with prcomp()"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec2sub9"><a href="/Interpreting-PCA-Results-in-R.html" title="Interpreting PCA Output"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec2sub9"><a href="/factoextra-and-FactoMineR.html" title="factoextra (PCA + Clusters)"><span class="progress-dot"></span>factoextra (PCA + Clusters)</a></li><li data-subkey="sec2sub9"><a href="/Exploratory-Factor-Analysis-in-R.html" title="Exploratory Factor Analysis"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li data-subkey="sec2sub9"><a href="/CFA-and-Structural-Equation-Modeling-in-R.html" title="SEM and CFA (lavaan)"><span class="progress-dot"></span>SEM and CFA (lavaan)</a></li><li data-subkey="sec2sub9"><a href="/Linear-Discriminant-Analysis-in-R.html" title="LDA (Linear Discriminant Analysis)"><span class="progress-dot"></span>LDA (Linear Discriminant Analysis)</a></li><li data-subkey="sec2sub9"><a href="/Cluster-Analysis-in-R.html" title="Clustering (k-Means / HC / DBSCAN)"><span class="progress-dot"></span>Clustering (k-Means / HC / DBSCAN)</a></li><li data-subkey="sec2sub9"><a href="/Correspondence-Analysis-in-R.html" title="Correspondence Analysis"><span class="progress-dot"></span>Correspondence Analysis</a></li><li data-subkey="sec2sub9"><a href="/t-SNE-and-UMAP-in-R.html" title="t-SNE and UMAP"><span class="progress-dot"></span>t-SNE and UMAP</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub10" data-collapsed="false"><span class="subsec-chevron">▼</span> Nonparametric & Resampling</li><li data-subkey="sec2sub10"><a href="/When-to-Use-Nonparametric-Tests-in-R.html" title="When to Use Nonparametric Tests"><span class="progress-dot"></span>When to Use Nonparametric Tests</a></li><li data-subkey="sec2sub10"><a href="/Wilcoxon-Signed-Rank-Test-in-R.html" title="Wilcoxon Signed-Rank Test"><span class="progress-dot"></span>Wilcoxon Signed-Rank Test</a></li><li data-subkey="sec2sub10"><a href="/Mann-Whitney-U-Test-in-R.html" title="Mann-Whitney U Test"><span class="progress-dot"></span>Mann-Whitney U Test</a></li><li data-subkey="sec2sub10"><a href="/Kruskal-Wallis-Test-in-R-2.html" title="Kruskal-Wallis Test"><span class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec2sub10"><a href="/Friedman-Test-in-R.html" title="Friedman Test"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec2sub10"><a href="/Spearman-and-Kendall-Correlation-in-R.html" title="Spearman & Kendall Correlation"><span class="progress-dot"></span>Spearman & Kendall Correlation</a></li><li data-subkey="sec2sub10"><a href="/Bootstrap-in-R.html" title="Bootstrap (boot package)"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub11" data-collapsed="false"><span class="subsec-chevron">▼</span> Linear Algebra for Statistics</li><li data-subkey="sec2sub11"><a href="/Matrix-Operations-in-R.html" title="Matrix Operations in R"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec2sub11"><a href="/Solving-Linear-Systems-in-R.html" title="Solving Linear Systems in R"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec2sub11"><a href="/Eigenvalues-and-Eigenvectors-in-R.html" title="Eigenvalues & Eigenvectors in R"><span class="progress-dot"></span>Eigenvalues & Eigenvectors in R</a></li><li data-subkey="sec2sub11"><a href="/Singular-Value-Decomposition-in-R.html" title="Singular Value Decomposition in R"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec2sub11"><a href="/Projections-and-the-Hat-Matrix-in-R.html" title="Projections & the Hat Matrix"><span class="progress-dot"></span>Projections & the Hat Matrix</a></li><li data-subkey="sec2sub11"><a href="/QR-Decomposition-in-R.html" title="QR Decomposition in R"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec2sub11"><a href="/Quadratic-Forms-in-R.html" title="Quadratic Forms"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec2sub11"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html" title="Matrix Derivatives & Hessian"><span class="progress-dot"></span>Matrix Derivatives & Hessian</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub12" data-collapsed="false"><span class="subsec-chevron">▼</span> Statistical Theory</li><li data-subkey="sec2sub12"><a href="/Exponential-Family-Distributions-in-R.html" title="Exponential Family Distributions"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec2sub12"><a href="/Sufficient-Statistics-in-R.html" title="Sufficient Statistics"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec2sub12"><a href="/Complete-and-Ancillary-Statistics-in-R.html" title="Complete & Ancillary Statistics"><span class="progress-dot"></span>Complete & Ancillary Statistics</a></li><li data-subkey="sec2sub12"><a href="/UMVUE-in-R-2.html" title="UMVUE (Rao-Blackwell & Lehmann-Scheffé)"><span class="progress-dot"></span>UMVUE (Rao-Blackwell & Lehmann-Scheffé)</a></li><li data-subkey="sec2sub12"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html" title="Cramér-Rao Lower Bound"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec2sub12"><a href="/Asymptotic-Theory-in-R-2.html" title="Asymptotic Theory"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec2sub12"><a href="/Neyman-Pearson-Lemma-in-R-2.html" title="Neyman-Pearson Lemma"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec2sub12"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html" title="Likelihood Ratio & Pivotal Methods"><span class="progress-dot"></span>Likelihood Ratio & Pivotal Methods</a></li><li data-subkey="sec2sub12"><a href="/Decision-Theory-in-R.html" title="Decision Theory"><span class="progress-dot"></span>Decision Theory</a></li><li data-subkey="sec2sub12"><a href="/Asymptotic-Relative-Efficiency-in-R.html" title="Asymptotic Relative Efficiency"><span class="progress-dot"></span>Asymptotic Relative Efficiency</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub13" data-collapsed="false"><span class="subsec-chevron">▼</span> Bayesian Foundations</li><li data-subkey="sec2sub13"><a href="/Bayes-Theorem-in-R.html" title="Bayes' Theorem"><span class="progress-dot"></span>Bayes' Theorem</a></li><li data-subkey="sec2sub13"><a href="/Bayesian-Statistics-in-R.html" class="active" title="Bayesian Statistics"><span class="progress-dot"></span>Bayesian Statistics</a></li><li data-subkey="sec2sub13"><a href="/Conjugate-Priors-in-R.html" title="Conjugate Priors"><span class="progress-dot"></span>Conjugate Priors</a></li><li data-subkey="sec2sub13"><a href="/Grid-Approximation-in-R.html" title="Grid Approximation"><span class="progress-dot"></span>Grid Approximation</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub14" data-collapsed="false"><span class="subsec-chevron">▼</span> MCMC & Stan</li><li data-subkey="sec2sub14"><a href="/MCMC-in-R.html" title="MCMC in R"><span class="progress-dot"></span>MCMC in R</a></li><li data-subkey="sec2sub14"><a href="/Gibbs-Sampling-in-R.html" title="Gibbs Sampling"><span class="progress-dot"></span>Gibbs Sampling</a></li><li data-subkey="sec2sub14"><a href="/Hamiltonian-Monte-Carlo-in-R.html" title="Hamiltonian Monte Carlo"><span class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec2sub14"><a href="/Stan-in-R.html" title="Stan"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec2sub14"><a href="/brms-in-R.html" title="brms"><span class="progress-dot"></span>brms</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub15" data-collapsed="false"><span class="subsec-chevron">▼</span> Bayesian Modeling</li><li data-subkey="sec2sub15"><a href="/Choosing-Priors-in-R.html" title="Choosing Priors"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec2sub15"><a href="/Prior-Predictive-Checks-in-R.html" title="Prior Predictive Checks"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec2sub15"><a href="/Compare-Bayesian-Models-in-R.html" title="Compare Bayesian Models"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec2sub15"><a href="/Posterior-Predictive-Checks-in-R.html" title="Posterior Predictive Checks"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Linear-Regression-in-R.html" title="Bayesian Linear Regression"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Logistic-Regression-in-R.html" title="Bayesian Logistic Regression"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-Hierarchical-Models-in-R.html" title="Bayesian Hierarchical Models"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec2sub15"><a href="/Multilevel-Models-in-R.html" title="Multilevel Models"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec2sub15"><a href="/Bayesian-ANOVA-in-R.html" title="Bayesian ANOVA"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub16" data-collapsed="false"><span class="subsec-chevron">▼</span> Machine Learning</li><li data-subkey="sec2sub16"><a href="/Random-Forest-Course.html" title="Random Forests (Course)"><span class="progress-dot"></span>Random Forests (Course)</a></li><li data-subkey="sec2sub16"><a href="/R-Gradient-Boosting-Course.html" title="Gradient Boosting (Course)"><span class="progress-dot"></span>Gradient Boosting (Course)</a></li><li data-subkey="sec2sub16"><a href="/R-tidymodels-Course.html" title="tidymodels (Course)"><span class="progress-dot"></span>tidymodels (Course)</a></li><li data-subkey="sec2sub16" class="is-quiz"><a href="/Machine-Learning-Exercises-in-R-quiz.html" title="Machine Learning Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec2sub16"><a href="/T-Test-Course.html" title="The t-test (Lesson)"><span class="progress-dot"></span>The t-test (Lesson)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">4.</span> <span class="sec-title-t">Visualization</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> ggplot2 Foundations</li><li data-subkey="sec3sub1"><a href="/ggplot2-Grammar-of-Graphics.html" title="Grammar of Graphics"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Getting-Started.html" title="ggplot2 Getting Started"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html" title="ggplot2 Aesthetics (aes)"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Colours.html" title="ggplot2 Colours"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Scales.html" title="ggplot2 Scales"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Themes-in-R.html" title="ggplot2 Themes"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Labels-and-Annotations.html" title="Labels & Annotations"><span class="progress-dot"></span>Labels & Annotations</a></li><li data-subkey="sec3sub1"><a href="/ggplot2-Facets.html" title="ggplot2 Facets"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec3sub1" class="is-quiz"><a href="/ggplot2-Exercises-in-R-quiz.html" title="ggplot2 Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Core Charts</li><li data-subkey="sec3sub2"><a href="/ggplot2-Scatter-Plots.html" title="Scatter Plots"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Line-Charts.html" title="Line Charts"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Bar-Charts.html" title="Bar Charts"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec3sub2"><a href="/ggplot2-Distribution-Charts.html" title="Distribution Charts"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec3sub2"><a href="/Error-Bars-in-R.html" title="Error Bars"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec3sub2"><a href="/geom_smooth-in-R.html" title="geom_smooth()"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Distributions & Groups</li><li data-subkey="sec3sub3"><a href="/Violin-Plot-in-R.html" title="Violin Plot"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec3sub3"><a href="/Ridgeline-Plot-in-R.html" title="Ridgeline Plot"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec3sub3"><a href="/Lollipop-Chart-in-R.html" title="Lollipop Chart"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Relationships</li><li data-subkey="sec3sub4"><a href="/Bubble-Chart-in-R.html" title="Bubble Chart"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec3sub4"><a href="/Heatmap-in-R.html" title="Heatmap in R"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec3sub4"><a href="/Correlation-Matrix-Plot-in-R.html" title="Correlation Matrix"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Advanced Charts</li><li data-subkey="sec3sub5"><a href="/Pie-Donut-Chart-in-R.html" title="Pie & Donut Chart"><span class="progress-dot"></span>Pie & Donut Chart</a></li><li data-subkey="sec3sub5"><a href="/Treemap-in-R.html" title="Treemap"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec3sub5"><a href="/Waffle-Chart-in-R.html" title="Waffle Chart"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Exploratory Analysis</li><li data-subkey="sec3sub6"><a href="/Exploratory-Data-Analysis-in-R.html" title="EDA (7-Step Framework)"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec3sub6"><a href="/Univariate-EDA-in-R.html" title="Univariate EDA"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec3sub6"><a href="/Bivariate-EDA-in-R.html" title="Bivariate EDA"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec3sub6"><a href="/Descriptive-Statistics-in-R.html" title="Descriptive Statistics"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec3sub6"><a href="/Correlation-Analysis-in-R.html" title="Correlation Analysis"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> Interactive & Maps</li><li data-subkey="sec3sub7"><a href="/Combining-ggplot2-with-plotly.html" title="ggplot2 + plotly Interactive"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec3sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html" title="Leaflet Interactive Maps"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec3sub7"><a href="/Spatial-Data-in-R-with-sf.html" title="Spatial Data (sf)"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec3sub7"><a href="/Choropleth-Maps-in-R.html" title="Choropleth Maps (sf)"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> Customization & Reference</li><li data-subkey="sec3sub8"><a href="/ggplot2-Legends-in-R.html" title="ggplot2 Legends"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Secondary-Axis.html" title="Secondary Axis"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Log-Scale.html" title="Log Scale"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec3sub8"><a href="/patchwork-Package.html" title="patchwork (Combine Plots)"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec3sub8"><a href="/Publication-Quality-Figures-in-R.html" title="Publication-Ready Figures"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-cheatsheet.html" title="ggplot2 Quickref"><span class="progress-dot"></span>ggplot2 Quickref</a></li><li data-subkey="sec3sub8"><a href="/Advanced-ggplot2-Course.html" title="Advanced ggplot2 (Course)"><span class="progress-dot"></span>Advanced ggplot2 (Course)</a></li><li data-subkey="sec3sub8"><a href="/ggplot2-Course.html" title="ggplot2 (Course)"><span class="progress-dot"></span>ggplot2 (Course)</a></li><li data-subkey="sec3sub8"><a href="/Dashboards-Course.html" title="Interactive Dashboards (Course)"><span class="progress-dot"></span>Interactive Dashboards (Course)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">5.</span> <span class="sec-title-t">Time Series</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec4sub0"><a href="/Time-Series-Analysis-With-R.html" title="Time Series Analysis"><span class="progress-dot"></span>Time Series Analysis</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R.html" title="Time Series Forecasting"><span class="progress-dot"></span>Time Series Forecasting</a></li><li data-subkey="sec4sub0"><a href="/Time-Series-Forecasting-With-R-part2.html" title="More Time Series Forecasting"><span class="progress-dot"></span>More Time Series Forecasting</a></li><li data-subkey="sec4sub0" class="is-quiz"><a href="/Time-Series-Exercises-in-R-quiz.html" title="Time Series Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">6.</span> <span class="sec-title-t">Advanced R</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Functional Programming</li><li data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html" title="Functional Programming"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1" class="is-quiz"><a href="/R-Functional-Programming-Exercises-quiz.html" title="Functional Programming Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html" title="purrr map() Variants"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html" title="R Anonymous Functions"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html" title="R Function Factories"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html" title="R Function Operators"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html" title="Reduce, Filter, Map"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html" title="Memoization in R"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html" title="Composable R Code"><span class="progress-dot"></span>Composable R Code</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> OOP in R</li><li data-subkey="sec5sub2"><a href="/OOP-in-R.html" title="OOP in R: S3/S4/R6"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html" title="S3 Classes"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html" title="S3 Method Dispatch"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html" title="S4 Classes"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html" title="S4 Methods & Dispatch"><span class="progress-dot"></span>S4 Methods & Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html" title="R6 Classes"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html" title="R6 Advanced"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html" title="Operator Overloading"><span class="progress-dot"></span>Operator Overloading</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> How R Works</li><li data-subkey="sec5sub3"><a href="/R-Names-and-Values.html" title="R Names & Values"><span class="progress-dot"></span>R Names & Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html" title="R Assignment Deep Dive"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html" title="R Memory & lobstr"><span class="progress-dot"></span>R Memory & lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html" title="R Environments"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html" title="Lexical Scoping"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html" title="R Closures"><span class="progress-dot"></span>R Closures</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec5sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Debugging & Performance</li><li data-subkey="sec5sub4"><a href="/R-Conditions-System.html" title="Conditions System"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html" title="Debugging R Code"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html" title="50 Common R Errors"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html" title="Parallel Computing"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html" title="Speedup R Code"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4" class="is-quiz"><a href="/Shiny-Exercises-in-R-quiz.html" title="Shiny Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">7.</span> <span class="sec-title-t">Classic Tutorials</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li data-subkey="sec6sub0"><a href="/R-Tutorial.html" title="R Tutorial (Classic)"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html" title="ggplot2 Short Tutorial"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html" title="ggplot2 Tutorial 1 - Intro"><span class="progress-dot"></span>ggplot2 Tutorial 1 - Intro</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part2-Customizing-Theme-With-R-Code.html" title="ggplot2 Tutorial 2 - Theme"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html" title="ggplot2 Tutorial 3 - Masterlist"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html" title="Association Mining"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html" title="Multi Dimensional Scaling"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html" title="Optimization"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html" title="InformationValue Package"><span class="progress-dot"></span>InformationValue Package</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span><span class="sec-num">8.</span> <span class="sec-title-t">Practice Exercises</span><span class="section-meta" data-section-meta></span></div><ul class="sidebar-section-items list-unstyled"><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub1" data-collapsed="false"><span class="subsec-chevron">▼</span> Featured Problem Sets</li><li data-subkey="sec7sub1"><a href="/R-Interview-Questions.html" title="R Interview Questions (Top 50)"><span class="progress-dot"></span>R Interview Questions (Top 50)</a></li><li data-subkey="sec7sub1"><a href="/Statistics-Interview-Questions.html" title="Statistics Interview Questions (25)"><span class="progress-dot"></span>Statistics Interview Questions (25)</a></li><li data-subkey="sec7sub1"><a href="/ML-Interview-Questions-in-R.html" title="ML Interview Questions (25)"><span class="progress-dot"></span>ML Interview Questions (25)</a></li><li data-subkey="sec7sub1"><a href="/Top-20-Bayesian-Problems-in-R.html" title="Top 20 Bayesian Problems"><span class="progress-dot"></span>Top 20 Bayesian Problems</a></li><li data-subkey="sec7sub1"><a href="/Probability-Puzzles-for-Interviews.html" title="Probability Puzzles (Top 20)"><span class="progress-dot"></span>Probability Puzzles (Top 20)</a></li><li data-subkey="sec7sub1"><a href="/AB-Testing-Interview-Cases.html" title="A/B Testing Interview Cases (20)"><span class="progress-dot"></span>A/B Testing Interview Cases (20)</a></li><li data-subkey="sec7sub1"><a href="/SQL-to-dplyr-Translations.html" title="SQL to dplyr (30 Translations)"><span class="progress-dot"></span>SQL to dplyr (30 Translations)</a></li><li data-subkey="sec7sub1"><a href="/Top-25-Regression-Problems-in-R.html" title="Top 25 Regression Problems"><span class="progress-dot"></span>Top 25 Regression Problems</a></li><li data-subkey="sec7sub1"><a href="/Top-20-Time-Series-Problems-in-R.html" title="Top 20 Time Series Problems"><span class="progress-dot"></span>Top 20 Time Series Problems</a></li><li data-subkey="sec7sub1"><a href="/Resampling-Problems-in-R.html" title="The Resampling 15"><span class="progress-dot"></span>The Resampling 15</a></li><li data-subkey="sec7sub1"><a href="/Data-Cleaning-Gauntlet.html" title="Data Cleaning Gauntlet (25)"><span class="progress-dot"></span>Data Cleaning Gauntlet (25)</a></li><li data-subkey="sec7sub1"><a href="/Error-Triage-Drills-in-R.html" title="Error Triage Drills (20)"><span class="progress-dot"></span>Error Triage Drills (20)</a></li><li data-subkey="sec7sub1"><a href="/Regex-Drills-in-R.html" title="The Regex 20"><span class="progress-dot"></span>The Regex 20</a></li><li data-subkey="sec7sub1"><a href="/Dates-and-Times-Drills-in-R.html" title="Dates and Times Drills"><span class="progress-dot"></span>Dates and Times Drills</a></li><li data-subkey="sec7sub1"><a href="/Base-R-Speed-Round.html" title="Base R Speed Round"><span class="progress-dot"></span>Base R Speed Round</a></li><li data-subkey="sec7sub1"><a href="/ggplot2-Recreation-Challenge.html" title="ggplot2 Recreation Challenge"><span class="progress-dot"></span>ggplot2 Recreation Challenge</a></li><li data-subkey="sec7sub1"><a href="/Take-Home-Assignment-Simulator.html" title="Take-Home Simulator"><span class="progress-dot"></span>Take-Home Simulator</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Mastery Quizzes (Certificate)</li><li data-subkey="sec7sub2" class="is-quiz"><a href="/R-Beginner-Exercises-quiz.html" title="R Fundamentals Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/dplyr-Exercises-in-R-quiz.html" title="dplyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/ggplot2-Exercises-in-R-quiz.html" title="ggplot2 Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html" title="Hypothesis Testing Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/Linear-Regression-Exercises-in-R-quiz.html" title="Linear Regression Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/Machine-Learning-Exercises-in-R-quiz.html" title="Machine Learning Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/tidyr-Exercises-in-R-quiz.html" title="tidyr Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/Time-Series-Exercises-in-R-quiz.html" title="Time Series Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/Shiny-Exercises-in-R-quiz.html" title="Shiny Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/R-Interview-Questions-quiz.html" title="R Interview Readiness Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li data-subkey="sec7sub2" class="is-quiz"><a href="/R-Functional-Programming-Exercises-quiz.html" title="Functional Programming Quiz"><span class="quiz-marker" aria-hidden="true"><svg viewBox="0 0 24 24" fill="currentColor" aria-hidden="true"><path d="M12 2 4 5.1v5.7c0 4.9 3.4 8.4 8 9.9 4.6-1.5 8-5 8-9.9V5.1L12 2z"/></svg></span>Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> R Fundamentals</li><li data-subkey="sec7sub3"><a href="/R-Basics-Exercises.html" title="R Basics (15 problems)"><span class="progress-dot"></span>R Basics (15 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Vectors-Exercises.html" title="R Vectors (12 problems)"><span class="progress-dot"></span>R Vectors (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Data-Frames-Exercises.html" title="R Data Frames (15 problems)"><span class="progress-dot"></span>R Data Frames (15 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Lists-Exercises.html" title="R Lists (10 problems)"><span class="progress-dot"></span>R Lists (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Control-Flow-Exercises.html" title="R Control Flow (12 problems)"><span class="progress-dot"></span>R Control Flow (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Functions-Exercises.html" title="R Functions (10 problems)"><span class="progress-dot"></span>R Functions (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-String-Exercises.html" title="R Strings (10 problems)"><span class="progress-dot"></span>R Strings (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Date-Time-Exercises.html" title="R Date & Time (10 problems)"><span class="progress-dot"></span>R Date & Time (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Apply-Exercises.html" title="R apply Family (12 problems)"><span class="progress-dot"></span>R apply Family (12 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Subsetting-Exercises.html" title="R Subsetting (10 problems)"><span class="progress-dot"></span>R Subsetting (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-Functional-Programming-Exercises.html" title="Functional Programming (10 problems)"><span class="progress-dot"></span>Functional Programming (10 problems)</a></li><li data-subkey="sec7sub3"><a href="/R-OOP-Exercises.html" title="OOP in R (8 problems)"><span class="progress-dot"></span>OOP in R (8 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Data Wrangling</li><li data-subkey="sec7sub4"><a href="/R-Data-Import-Exercises.html" title="Data Import (10 problems)"><span class="progress-dot"></span>Data Import (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/dplyr-Exercises.html" title="dplyr (15 problems)"><span class="progress-dot"></span>dplyr (15 problems)</a></li><li data-subkey="sec7sub4"><a href="/dplyr-filter-select-Exercises.html" title="dplyr filter() & select() (12 problems)"><span class="progress-dot"></span>dplyr filter() & select() (12 problems)</a></li><li data-subkey="sec7sub4"><a href="/dplyr-group-by-summarise-Exercises.html" title="dplyr group_by() & summarise() (10 problems)"><span class="progress-dot"></span>dplyr group_by() & summarise() (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/dplyr-Join-Exercises.html" title="dplyr Joins (10 problems)"><span class="progress-dot"></span>dplyr Joins (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/data-table-Exercises.html" title="data.table (12 problems)"><span class="progress-dot"></span>data.table (12 problems)</a></li><li data-subkey="sec7sub4"><a href="/purrr-Exercises.html" title="purrr (10 problems)"><span class="progress-dot"></span>purrr (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/tidyr-Reshaping-Exercises.html" title="tidyr Reshaping (10 problems)"><span class="progress-dot"></span>tidyr Reshaping (10 problems)</a></li><li data-subkey="sec7sub4"><a href="/Missing-Data-in-R-Exercises.html" title="Missing Data in R (10 problems)"><span class="progress-dot"></span>Missing Data in R (10 problems)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec7sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Visualization</li><li data-subkey="sec7sub5"><a href="/ggplot2-Exercises.html" title="ggplot2 (15 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width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.3" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 8H4.5"/><path d="M2 6.2V9.8"/><rect x="4.5" y="5" width="6" height="6" rx="1"/><path d="M7.3 5V11"/><path d="M10.5 8H12"/><path d="M12 6.2V9.8"/><circle cx="14" cy="8" r="1" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">IQR & Outliers</span></a></li><li><a href="/tools/box-plot-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M4.4 2.5V4.6M4.4 11.4V13.5"/><rect x="2.7" y="4.6" width="3.4" height="6.8" rx="0.7"/><path d="M2.7 8.2H6.1"/><path d="M11.4 4V6M11.4 12V14"/><rect x="9.7" y="6" width="3.4" height="6" rx="0.7"/><path d="M9.7 9.4H13.1"/></svg></span><span class="tool-label">Box Plot</span></a></li><li><a href="/tools/chi-square-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M4 3c4 3.5 4 6.5 8 10"/><path d="M12 3c-4 3.5-4 6.5-8 10"/></svg></span><span class="tool-label">Chi-Square Test</span></a></li><li><a href="/tools/confidence-interval-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M3.5 4v8M3.5 4H5M3.5 12H5"/><circle cx="8" cy="8" r="1.4" fill="currentColor" stroke="none"/><path d="M12.5 4v8M12.5 4H11M12.5 12H11"/></svg></span><span class="tool-label">Confidence Interval</span></a></li><li><a href="/tools/margin-of-error-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 8H13.5"/><path d="M2.5 8l2.2-2M2.5 8l2.2 2M13.5 8l-2.2-2M13.5 8l-2.2 2"/><circle cx="8" cy="8" r="1.5" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Margin of Error</span></a></li><li><a href="/tools/bootstrap-ci-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M13 8a5 5 0 1 1-1.4-3.5"/><path d="M13 2.5V5h-2.5"/></svg></span><span class="tool-label">Bootstrap CI</span></a></li><li><a href="/tools/effect-size-converter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 5.5h11M11 3l2.5 2.5L11 8"/><path d="M13.5 10.5h-11M5 8l-2.5 2.5L5 13"/></svg></span><span class="tool-label">Effect Size Converter</span></a></li><li><a href="/tools/power-analysis.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M9 2L4 9h3.5L7 14l5-7H8.5z"/></svg></span><span class="tool-label">Power Analysis</span></a></li><li><a href="/tools/survival-power-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 3.5h3.5V7H9v3.5h3.5V13H14"/></svg></span><span class="tool-label">Survival Power</span></a></li><li><a href="/tools/type-i-ii-error-visualizer.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 12c1.5 0 2-2 3.5-2S7 12 8 12s2-8 3.5-8S13 12 14.5 12"/><line x1="8" y1="2" x2="8" y2="14" stroke-dasharray="2 2"/></svg></span><span class="tool-label">Type I / II Error</span></a></li><li><a href="/tools/z-score-percentile.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13c2 0 3-1 4-3s1.5-6 2-6 .5 6 2 6 2.5 0 4 0"/><path d="M2 13.5h12"/></svg></span><span class="tool-label">Z-Score & Percentile</span></a></li><li><a href="/tools/normal-distribution-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M1.5 13c2 0 3-1 4-3s1.5-6 2.5-6 1.5 6 2.5 6 2-1 3.5-1"/><path d="M1.5 13.5h13"/><path d="M5.5 13C6.3 11 7 7 8 4C9 7 9.7 11 10.5 13Z" fill="currentColor" stroke="none" opacity="0.4"/></svg></span><span class="tool-label">Normal Distribution</span></a></li><li><a href="/tools/empirical-rule-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.3" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M1.5 12.5h13"/><path d="M2 12.5C5 12.5 5.2 4 8 4C10.8 4 11 12.5 14 12.5"/><path d="M5.6 12.5C6.5 10.8 7.2 7 8 4C8.8 7 9.5 10.8 10.4 12.5Z" fill="currentColor" stroke="none" opacity="0.32"/><path d="M5.6 12.5V10.2M10.4 12.5V10.2" stroke-dasharray="1.5 1.4"/><path d="M3.4 12.5V11M12.6 12.5V11" stroke-dasharray="1.5 1.4"/></svg></span><span class="tool-label">Empirical Rule</span></a></li><li><a href="/tools/equivalence-noninferiority-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M3 3.5v9M13 3.5v9"/><path d="M5.5 8h5"/><circle cx="8" cy="8" r="1.3" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Equivalence / NI</span></a></li><li><a href="/tools/outlier-detection-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><circle cx="5" cy="10" r="1" fill="currentColor" stroke="none"/><circle cx="7.5" cy="8.5" r="1" fill="currentColor" stroke="none"/><circle cx="5.5" cy="12.5" r="1" fill="currentColor" stroke="none"/><circle cx="12" cy="4" r="2.4"/></svg></span><span class="tool-label">Outlier Detection</span></a></li><li><a href="/tools/roc-auc-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 13.5L13.5 2.5" stroke-dasharray="2 2"/><path d="M2.5 13.5C2.5 7 7 2.5 13.5 2.5"/></svg></span><span class="tool-label">ROC / AUC</span></a></li><li><a href="/tools/cronbachs-alpha-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 4h5M2 8h5M2 12h5"/><path d="M7 4c3.5 0 3.5 4 6 4M7 12c3.5 0 3.5-4 6-4"/><circle cx="13" cy="8" r="1.2" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Cronbach's Alpha</span></a></li><li><a href="/tools/cohens-kappa-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="2.5" width="11" height="11" rx="1"/><path d="M6.17 2.5v11M9.83 2.5v11M2.5 6.17h11M2.5 9.83h11"/><circle cx="4.33" cy="4.33" r="1" fill="currentColor" stroke="none"/><circle cx="8" cy="8" r="1" fill="currentColor" stroke="none"/><circle cx="11.67" cy="11.67" r="1" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Cohen's Kappa</span></a></li><li><a href="/tools/correlation-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 2v11.5h11"/><path d="M4 12.5 13 4"/><circle cx="5" cy="10.5" r=".9" fill="currentColor" stroke="none"/><circle cx="7.5" cy="8.5" r=".9" fill="currentColor" stroke="none"/><circle cx="9.5" cy="7" r=".9" fill="currentColor" stroke="none"/><circle cx="11.5" cy="5" r=".9" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Correlation</span></a></li><li><a href="/tools/linear-regression-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 2v11.5h11"/><path d="M3.8 12.2 13.2 4.2"/><circle cx="5.2" cy="11.4" r=".9" fill="currentColor" stroke="none"/><circle cx="7.4" cy="8.6" r=".9" fill="currentColor" stroke="none"/><circle cx="9.4" cy="7.4" r=".9" fill="currentColor" stroke="none"/><circle cx="11.4" cy="4.8" r=".9" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Linear Regression</span></a></li><li><a href="/tools/anova-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M4 4v4M8 6v4M12 3v4"/><circle cx="4" cy="6" r="1.1" fill="currentColor" stroke="none"/><circle cx="8" cy="8" r="1.1" fill="currentColor" stroke="none"/><circle cx="12" cy="5" r="1.1" fill="currentColor" stroke="none"/><path d="M2 13.5h12"/></svg></span><span class="tool-label">ANOVA Calculator</span></a></li><li><a href="/tools/odds-ratio-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="2.5" width="11" height="11" rx="1"/><path d="M8 2.5v11M2.5 8h11"/><circle cx="5.25" cy="5.25" r="1.05" fill="currentColor" stroke="none"/><circle cx="10.75" cy="10.75" r="1.05" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Odds Ratio & RR</span></a></li><li><a href="/tools/fisher-exact-test-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2.5 2v11.5h11"/><rect x="4" y="8" width="1.7" height="3.5" fill="currentColor" stroke="none" opacity=".5"/><rect x="6.5" y="4.8" width="1.7" height="6.7" fill="currentColor" stroke="none"/><rect x="9" y="7" width="1.7" height="4.5" fill="currentColor" stroke="none"/><rect x="11.5" y="9.5" width="1.7" height="2" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Fisher's Exact Test</span></a></li><li><a href="/tools/binomial-probability-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.3" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M1.5 14h13"/><rect x="2.2" y="10.5" width="1.7" height="3.5" fill="currentColor" stroke="none" opacity="0.45"/><rect x="4.9" y="7.5" width="1.7" height="6.5" fill="currentColor" stroke="none" opacity="0.45"/><rect x="7.6" y="5" width="1.7" height="9" fill="currentColor" stroke="none"/><rect x="10.3" y="7.5" width="1.7" height="6.5" fill="currentColor" stroke="none" opacity="0.45"/><rect x="13" y="10.5" width="1.2" height="3.5" fill="currentColor" stroke="none" opacity="0.45"/></svg></span><span class="tool-label">Binomial Probability</span></a></li><li><a href="/tools/poisson-distribution-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.3" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M1.5 14h13"/><rect x="2" y="9" width="1.6" height="5" fill="currentColor" stroke="none" opacity="0.45"/><rect x="4.3" y="5" width="1.6" height="9" fill="currentColor" stroke="none"/><rect x="6.6" y="6.5" width="1.6" height="7.5" fill="currentColor" stroke="none" opacity="0.7"/><rect x="8.9" y="9" width="1.6" height="5" fill="currentColor" stroke="none" opacity="0.55"/><rect x="11.2" y="11" width="1.6" height="3" fill="currentColor" stroke="none" opacity="0.45"/><rect x="13.5" y="12.5" width="1.2" height="1.5" fill="currentColor" stroke="none" opacity="0.4"/></svg></span><span class="tool-label">Poisson Distribution</span></a></li><li><a href="/tools/proportion-test-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.3" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M1.5 13c2 0 3-1 4-3s1.5-6 2.5-6 1.5 6 2.5 6 2-1 3.5-1"/><path d="M1.5 13.5h13"/><path d="M11.3 13 L14.2 13 L14.2 10.4 Z" fill="currentColor" stroke="none" opacity="0.4"/><path d="M9.6 5.4V13" stroke-dasharray="1.8 1.6"/></svg></span><span class="tool-label">Proportion Test</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">▼</span> Reference tables</li><li><a href="/tools/t-table.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="3" width="11" height="10" rx="1"/><path d="M2.5 6.5h11M6 3v10"/></svg></span><span class="tool-label">t Table</span></a></li><li><a href="/tools/z-table.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 12c2 0 3-1 4-3s1.5-6 2-6 .5 6 2 6 2.5 0 4 0"/><path d="M2 12.5h12"/><path d="M10.5 4v8" stroke-dasharray="2 2"/></svg></span><span class="tool-label">z Table</span></a></li><li><a href="/tools/chi-square-table.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 12c1-5 2.5-7 4-7s3 5 8 6.5"/><path d="M2 12.5h12"/><path d="M10 5v7.5" stroke-dasharray="2 2"/></svg></span><span class="tool-label">Chi-Square Table</span></a></li><li><a href="/tools/f-table.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="2.5" y="3" width="11" height="10" rx="1"/><path d="M2.5 6.5h11M6 3v10"/><path d="M8.5 9.5h3.5" stroke-width="2"/></svg></span><span class="tool-label">F Distribution Table</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">▼</span> Bayesian</li><li><a href="/tools/bayes-theorem-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M3 8h4M7 8l4-4.5M7 8l4 4.5"/><circle cx="3" cy="8" r="1.2" fill="currentColor" stroke="none"/><circle cx="12" cy="3.5" r="1.2" fill="currentColor" stroke="none"/><circle cx="12" cy="12.5" r="1.2" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">Bayes Theorem</span></a></li><li><a href="/tools/bayes-factor-calculator.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M8 3v9.5M4 4.5h8"/><path d="M4 4.5L2.5 8h3zM12 4.5L10.5 8h3z"/><path d="M5.5 12.5h5"/></svg></span><span class="tool-label">Bayes Factor</span></a></li><li class="sidebar-divider"><span class="subsec-chevron">▼</span> Interpreters</li><li><a href="/tools/lm-output-interpreter.html"><span class="tool-icon"><svg viewBox="0 0 16 16" width="14" height="14" fill="none" stroke="currentColor" stroke-width="1.4" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><path d="M2 13.5L14 3"/><circle cx="4" cy="11.5" r="1" fill="currentColor" stroke="none"/><circle cx="7.5" cy="9" r="1" fill="currentColor" stroke="none"/><circle cx="11" cy="6.5" r="1" fill="currentColor" stroke="none"/></svg></span><span class="tool-label">lm() Output</span></a></li><li><a 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<h1>Bayesian Statistics in R: Intuition First</h1>
<p class="lead">Bayesian statistics in R updates a prior belief about an unknown parameter with observed data, producing a <a class="auto-link" href="Bayesian-Statistics-Exercises-in-R.html" title="Bayesian Statistics Exercises in R: 20 Practice Problems">posterior distribution</a> you can plot, integrate, and reason about. Unlike frequentist methods that return a single point estimate plus a confidence interval whose interpretation trips up most students, the Bayesian workflow gives you the full probability curve over plausible parameter values, ready for decision-making.</p>
<div class="post-byline" style="color:#6b7280;font-size:14px;margin:2px 0 18px 0;line-height:1.5;">By <strong>Selva Prabhakaran</strong> · Published July 16, 2026 · Last updated July 16, 2026</div>
<div class="engagement-header" data-difficulty="Intermediate" data-time="35" data-exercises="10" data-xp="150"></div>
<h2>How does Bayes' theorem turn data into a posterior?</h2>
<p>Frequentist tools answer "what is the parameter?" with a point estimate and a confidence interval whose interpretation bends most readers' minds. <a class="auto-link" href="Bayesian-Linear-Regression-in-R.html" title="Bayesian Linear Regression in R: Get Uncertainty Estimates lm() Cannot Give You">Bayesian inference</a> flips the question. You start with a prior belief about the parameter, observe data, and end with a posterior distribution: a probability curve over every plausible value. This section shows that update happen in a single line of base R using the Beta-Binomial pair, the simplest example of an analytic posterior.</p>
<p>The math behind every Bayesian update is one line:</p>
<p>$$ p(\theta \mid \text{data}) \;\propto\; p(\text{data} \mid \theta) \cdot p(\theta) $$</p>
<p>Where: $p(\theta)$ is the prior, $p(\text{data} \mid \theta)$ is the likelihood, and $p(\theta \mid \text{data})$ is the posterior. The proportional sign hides a normalizing constant that does not affect the shape of the curve.</p>
<p>Suppose you flip a possibly-biased coin 100 times and see 65 heads. You want to estimate the unknown success probability theta. A Beta(2, 2) prior is mildly skeptical of extreme values, gently centered at 0.5. The posterior comes out in closed form because the Beta family is conjugate to the Binomial likelihood, meaning the posterior stays in the Beta family.</p>
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Beta-Binomial posterior from 100 flips</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">n <span class="o"><-</span> <span class="m">100</span> <span class="c1"># total flips</span></span>
<span class="cl">k <span class="o"><-</span> <span class="m">65</span> <span class="c1"># heads observed</span></span>
<span class="cl">alpha_prior <span class="o"><-</span> <span class="m">2</span> <span class="c1"># prior shape parameters</span></span>
<span class="cl">beta_prior <span class="o"><-</span> <span class="m">2</span> <span class="c1"># ... encoding "fair-ish, but uncertain"</span></span>
<span class="cl"></span>
<span class="cl">alpha_post <span class="o"><-</span> alpha_prior <span class="o">+</span> k <span class="c1"># closed-form posterior shape</span></span>
<span class="cl">beta_post <span class="o"><-</span> beta_prior <span class="o">+</span> n <span class="o">-</span> k <span class="c1"># ... thanks to Beta-Binomial conjugacy</span></span>
<span class="cl"></span>
<span class="cl">post_mean <span class="o"><-</span> alpha_post <span class="o">/</span> (alpha_post <span class="o">+</span> beta_post)</span>
<span class="cl">cri <span class="o"><-</span> <span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.025</span>, <span class="m">0.975</span>), alpha_post, beta_post)</span>
<span class="cl"></span>
<span class="cl">post_mean</span>
<span class="cl"><span class="c1">#> [1] 0.6442308</span></span>
<span class="cl">cri</span>
<span class="cl"><span class="c1">#> [1] 0.5497073 0.7321076</span></span></div>
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<p>The posterior is Beta(67, 37). Its mean of 0.644 sits between the prior mean of 0.5 and the data proportion of 0.65, gently pulled toward 0.5 by the prior's weight. The 95% credible interval [0.55, 0.73] is the range that contains 95% of the <a class="auto-link" href="Bayes-Theorem-in-R.html" title="Bayes' Theorem in R: Why Medical Tests Mislead You, A Simulation That Shows Why">posterior probability</a> mass. That is the interpretation people incorrectly give a frequentist confidence interval.</p>
<p><img src="screenshots/Bayesian-Statistics-in-R-workflow.webp" alt="The Bayesian update workflow" class="img-responsive img-zoomable" loading="lazy" width="2248" height="380" /></p>
<p><em>Figure 1: The Bayesian update workflow. Prior plus data give a posterior, which you then <a class="auto-link" href="dplyr-group-by-summarise.html" title="dplyr group_by() + summarise(): The Combination That Answers Most Business Questions">summarize</a>.</em></p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>The posterior is the prior reweighted by the likelihood.</strong> No exotic computation, just multiplication and renormalization. The Beta-Binomial pair gives you the answer in closed form because the Beta family is conjugate to the Binomial, meaning the posterior stays in the Beta family.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Repeat the calculation with a much tighter prior, Beta(20, 20). What does the posterior mean become and why?</p>
<div class="webr-container" data-block-title="Your turn: tighten the prior">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: tighten the prior</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_alpha <span class="o"><-</span> <span class="m">20</span> <span class="c1"># try a tight prior</span></span>
<span class="cl">ex_beta <span class="o"><-</span> <span class="m">20</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># compute the posterior shape parameters and posterior mean here</span></span>
<span class="cl"><span class="c1"># ex_alpha_post <- ?</span></span>
<span class="cl"><span class="c1"># ex_beta_post <- ?</span></span>
<span class="cl"><span class="c1"># ex_post_mean <- ?</span></span>
<span class="cl"><span class="c1">#> Expected: posterior mean closer to 0.5 than 0.65</span></span></div>
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<details><summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Tighter prior solution">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Tighter prior solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_alpha_post <span class="o"><-</span> ex_alpha <span class="o">+</span> k</span>
<span class="cl">ex_beta_post <span class="o"><-</span> ex_beta <span class="o">+</span> n <span class="o">-</span> k</span>
<span class="cl">ex_post_mean <span class="o"><-</span> ex_alpha_post <span class="o">/</span> (ex_alpha_post <span class="o">+</span> ex_beta_post)</span>
<span class="cl">ex_post_mean</span>
<span class="cl"><span class="c1">#> [1] 0.6071429</span></span></div>
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<p>A Beta(20, 20) prior is equivalent to having seen 38 prior flips with 19 heads. Adding the new 100 flips gives a posterior that is pulled noticeably back toward 0.5. A stronger prior carries more weight against the same data, that is the lesson here.</p>
</details>
</section>
<h2>What does a prior actually encode?</h2>
<p>A prior is just a probability distribution over the parameter. Anything you can put on a curve, you can use as a prior. The Beta family is convenient for proportions because it lives on [0, 1] and supports two intuitive shape parameters that act like pseudo-counts of prior successes and failures. Three Beta priors illustrate the spectrum from ignorance to strong belief.</p>
<div class="webr-container" data-block-title="Three priors plotted on one panel">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Three priors plotted on one panel</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">theta_grid <span class="o"><-</span> <span class="nf">seq</span>(<span class="m">0</span>, <span class="m">1</span>, length.out <span class="o">=</span> <span class="m">200</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, <span class="m">1</span>, <span class="m">1</span>), type <span class="o">=</span> <span class="s">"l"</span>, lwd <span class="o">=</span> <span class="m">2</span>,</span>
<span class="cl"> ylim <span class="o">=</span> <span class="nf">c</span>(<span class="m">0</span>, <span class="m">6</span>), xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">"density"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Three priors over a proportion"</span>)</span>
<span class="cl"><span class="nf">lines</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, <span class="m">20</span>, <span class="m">20</span>), lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">"steelblue"</span>)</span>
<span class="cl"><span class="nf">lines</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, <span class="m">2</span>, <span class="m">5</span>), lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">"tomato"</span>)</span>
<span class="cl"><span class="nf">legend</span>(<span class="s">"topright"</span>, lwd <span class="o">=</span> <span class="m">2</span>,</span>
<span class="cl"> col <span class="o">=</span> <span class="nf">c</span>(<span class="s">"black"</span>, <span class="s">"steelblue"</span>, <span class="s">"tomato"</span>),</span>
<span class="cl"> legend <span class="o">=</span> <span class="nf">c</span>(<span class="s">"Beta(1,1) uniform"</span>, <span class="s">"Beta(20,20) tight at 0.5"</span>, <span class="s">"Beta(2,5) skewed low"</span>))</span></div>
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<p>Beta(1, 1) is flat: every value of theta is equally plausible before seeing data. Beta(20, 20) is tight around 0.5: a strong belief that the coin is fair. Beta(2, 5) is skewed low: a belief that small values of theta are more likely. Each shape encodes a different domain assumption, and each will pull the posterior in a different direction.</p>
<p>You often have a substantive belief such as "I think theta is between 0.4 and 0.6 with about 90% probability." That language has a direct Beta translation. Search for shape parameters whose 5th and 95th percentiles match the beliefs. A symmetric, moderately tight Beta(45, 45) lands close.</p>
<div class="webr-container" data-block-title="Quantile-based prior elicitation">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Quantile-based prior elicitation</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.05</span>, <span class="m">0.95</span>), <span class="m">45</span>, <span class="m">45</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.4133497 0.5866503</span></span></div>
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<p>A Beta(45, 45) prior places 90% of its mass between 0.41 and 0.59, a near-perfect match for the stated belief. If you wanted a less tight prior, lower both shape parameters; for a more confident prior, raise them. This is how to translate qualitative belief into a quantitative prior without throwing darts.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Pick a prior whose shape matches your prior belief, document the choice, and plan to report sensitivity.</strong> Beta(2, 2) is mildly skeptical of extreme values; Beta(1, 1) is informationless; Beta(50, 50) is hard to budge. Match the prior to the belief, then check how much the answer changes if you nudge the prior.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Encode the belief "I think theta is around 0.7 with mild uncertainty" as Beta shape parameters. A good answer keeps most mass in the [0.6, 0.8] range.</p>
<div class="webr-container" data-block-title="Your turn: encode a belief at 0.7">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Your turn: encode a belief at 0.7</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_alpha2 <span class="o"><-</span> <span class="m">0</span> <span class="c1"># replace</span></span>
<span class="cl">ex_beta2 <span class="o"><-</span> <span class="m">0</span> <span class="c1"># replace</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Verify with: qbeta(c(0.05, 0.95), ex_alpha2, ex_beta2)</span></span>
<span class="cl"><span class="c1">#> Expected: 90% interval roughly [0.6, 0.8]</span></span></div>
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<details><summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Belief-at-0.7 solution">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Belief-at-0.7 solution</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">ex_alpha2 <span class="o"><-</span> <span class="m">35</span></span>
<span class="cl">ex_beta2 <span class="o"><-</span> <span class="m">15</span></span>
<span class="cl"><span class="nf">qbeta</span>(<span class="nf">c</span>(<span class="m">0.05</span>, <span class="m">0.95</span>), ex_alpha2, ex_beta2)</span>
<span class="cl"><span class="c1">#> [1] 0.5901019 0.8003617</span></span></div>
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<p>Beta(35, 15) has mean 35/50 = 0.7 with 90% of its mass between 0.59 and 0.80. The ratio alpha/(alpha+beta) controls the center; the sum alpha+beta controls how tight the curve is. Combine those two levers to encode any belief on [0,1].</p>
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</section>
<h2>How do prior and likelihood combine into a posterior?</h2>
<p>The likelihood is a function of the parameter, given fixed data. It is not itself a probability distribution over theta, just a curve showing which theta values are most consistent with what you saw. Multiply the likelihood curve by the prior curve, normalize so the area is 1, and you have the posterior. Plotting all three on the same axes makes the arithmetic visual.</p>
<div class="webr-container" data-block-title="Prior, likelihood, posterior on one panel">
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<div class="webr-header"><div class="webr-header-left"><span class="webr-header-badge">R</span><span class="webr-header-label">Prior, likelihood, posterior on one panel</span></div><div class="webr-header-right"><button type="button" class="webr-copy-btn" aria-label="Copy code" title="Copy code"><svg width="14" height="14" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" aria-hidden="true"><rect x="9" y="9" width="13" height="13" rx="2" ry="2"/><path d="M5 15H4a2 2 0 0 1-2-2V4a2 2 0 0 1 2-2h9a2 2 0 0 1 2 2v1"/></svg></button><button class="btn btn-sm btn-primary webr-run-btn" onclick="runWebR(this)">▶ Run <span class="webr-run-shortcut">Ctrl+Enter</span></button></div></div>
<div class="webr-editor" data-language="r"><span class="cl">likelihood_vals <span class="o"><-</span> <span class="nf">dbinom</span>(k, size <span class="o">=</span> n, prob <span class="o">=</span> theta_grid)</span>
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<span class="cl"><span class="nf">par</span>(mfrow <span class="o">=</span> <span class="nf">c</span>(<span class="m">1</span>, <span class="m">3</span>), mar <span class="o">=</span> <span class="nf">c</span>(<span class="m">4</span>, <span class="m">4</span>, <span class="m">3</span>, <span class="m">1</span>))</span>
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<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, alpha_prior, beta_prior),</span>
<span class="cl"> type <span class="o">=</span> <span class="s">"l"</span>, lwd <span class="o">=</span> <span class="m">2</span>, main <span class="o">=</span> <span class="s">"Prior Beta(2, 2)"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">"density"</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, likelihood_vals, type <span class="o">=</span> <span class="s">"l"</span>, lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">"tomato"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Likelihood Binomial(n=100, k=65)"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">"L(theta)"</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">plot</span>(theta_grid, <span class="nf">dbeta</span>(theta_grid, alpha_post, beta_post),</span>
<span class="cl"> type <span class="o">=</span> <span class="s">"l"</span>, lwd <span class="o">=</span> <span class="m">2</span>, col <span class="o">=</span> <span class="s">"steelblue"</span>,</span>
<span class="cl"> main <span class="o">=</span> <span class="s">"Posterior Beta(67, 37)"</span>,</span>
<span class="cl"> xlab <span class="o">=</span> <span class="nf">expression</span>(theta), ylab <span class="o">=</span> <span class="s">"density"</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="nf">par</span>(mfrow <span class="o">=</span> <span class="nf">c</span>(<span class="m">1</span>, <span class="m">1</span>))</span></div>
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