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<!DOCTYPE html>
<html lang="en">
<head>
<title>A/B Testing in R: Plan Your Sample Size, Analyse Correctly, and Know When to Stop</title>
<meta charset="utf-8">
<meta name="Description" content="Plan A/B tests in R: pick a sample size with pwr, analyse proportions and means correctly, and stop without peeking. End-to-end workflow with runnable code.">
<meta name="Keywords" content="a/b testing in r, ab test sample size, power analysis for a/b test, pwr package, pwr.2p.test, prop.test a/b, sequential testing r, peeking problem, power.prop.test, conversion rate test">
<meta name="Distribution" content="Global">
<meta name="Author" content="Selva Prabhakaran">
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<div id="sidebar-nav"><div class="continue-chip" data-continue-chip><span class="chip-label">Continue reading</span><a href="#" data-continue-link></a></div><div class="sidebar-tabs" role="tablist"><button class="sidebar-tab active" data-tab="posts" type="button" role="tab" onclick="var n=this.dataset.tab;document.querySelectorAll('.sidebar-tab').forEach(function(x){x.classList.toggle('active',x.dataset.tab===n)});document.querySelectorAll('.sidebar-panel').forEach(function(p){p.classList.toggle('active',p.dataset.panel===n)});try{localStorage.setItem('rstat_sidebar_tab',n)}catch(e){}">Posts</button><button class="sidebar-tab" data-tab="tools" type="button" role="tab" onclick="var 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data-subkey="sec0sub4"><a href="/R-Cheat-Sheet.html"><span class="progress-dot"></span>R Cheat Sheet</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec0sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Professional R</li><li data-subkey="sec0sub5"><a href="/Data-Ethics-in-R.html"><span class="progress-dot"></span>Data Ethics</a></li><li data-subkey="sec0sub5"><a href="/Bias-in-Data-and-Models.html"><span class="progress-dot"></span>Bias in Data & Models</a></li><li data-subkey="sec0sub5"><a href="/Reproducibility-Crisis.html"><span class="progress-dot"></span>Reproducibility</a></li><li data-subkey="sec0sub5"><a href="/Data-Privacy-in-R.html"><span class="progress-dot"></span>Data Privacy</a></li><li data-subkey="sec0sub5"><a href="/Communicating-Uncertainty.html"><span class="progress-dot"></span>Communicating Uncertainty</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Data Wrangling<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"><span class="progress-dot"></span>Importing Data</a></li><li data-subkey="sec1sub1"><a href="/R-Pipe-Operator.html"><span class="progress-dot"></span>Pipe Operator</a></li><li data-subkey="sec1sub1"><a href="/Tidy-Data-in-R.html"><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"><span class="progress-dot"></span>dplyr filter & select</a></li><li data-subkey="sec1sub2"><a href="/dplyr-mutate-rename.html"><span class="progress-dot"></span>dplyr mutate & rename</a></li><li data-subkey="sec1sub2"><a href="/dplyr-group-by-summarise.html"><span class="progress-dot"></span>dplyr group_by & summarise</a></li><li data-subkey="sec1sub2"><a href="/dplyr-arrange-slice.html"><span class="progress-dot"></span>dplyr arrange & slice</a></li><li data-subkey="sec1sub2"><a href="/dplyr-across.html"><span class="progress-dot"></span>dplyr across()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-case-when.html"><span class="progress-dot"></span>dplyr case_when()</a></li><li data-subkey="sec1sub2"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr 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"><span class="progress-dot"></span>R Joins</a></li><li data-subkey="sec1sub3"><a href="/pivot_longer-pivot_wider-Reshape-Data-in-R.html"><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"><span class="progress-dot"></span>separate() & unite()</a></li><li data-subkey="sec1sub3"><a href="/tidyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>tidyr 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"><span class="progress-dot"></span>Missing Values (NA)</a></li><li data-subkey="sec1sub4"><a href="/Data-Quality-Checking-in-R.html"><span class="progress-dot"></span>Data Quality Checking</a></li><li data-subkey="sec1sub4"><a href="/janitor-Package-in-R.html"><span class="progress-dot"></span>janitor Package</a></li><li class="sidebar-divider 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(rvest)</a></li><li data-subkey="sec1sub6"><a href="/REST-APIs-in-R-with-httr2.html"><span class="progress-dot"></span>REST APIs (httr2)</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Visualization<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> ggplot2 Foundations</li><li data-subkey="sec2sub1"><a href="/ggplot2-Grammar-of-Graphics.html"><span class="progress-dot"></span>Grammar of Graphics</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Getting-Started.html"><span class="progress-dot"></span>ggplot2 Getting Started</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Aesthetics-aes-Map-Data.html"><span class="progress-dot"></span>ggplot2 Aesthetics (aes)</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Colours.html"><span class="progress-dot"></span>ggplot2 Colours</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Scales.html"><span class="progress-dot"></span>ggplot2 Scales</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Themes-in-R.html"><span class="progress-dot"></span>ggplot2 Themes</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Labels-and-Annotations.html"><span class="progress-dot"></span>Labels & Annotations</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Facets.html"><span class="progress-dot"></span>ggplot2 Facets</a></li><li data-subkey="sec2sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Core Charts</li><li data-subkey="sec2sub2"><a href="/ggplot2-Scatter-Plots.html"><span class="progress-dot"></span>Scatter Plots</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Line-Charts.html"><span class="progress-dot"></span>Line Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Bar-Charts.html"><span class="progress-dot"></span>Bar Charts</a></li><li data-subkey="sec2sub2"><a href="/ggplot2-Distribution-Charts.html"><span class="progress-dot"></span>Distribution Charts</a></li><li data-subkey="sec2sub2"><a href="/Error-Bars-in-R.html"><span class="progress-dot"></span>Error Bars</a></li><li data-subkey="sec2sub2"><a href="/geom_smooth-in-R.html"><span class="progress-dot"></span>geom_smooth()</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Distributions & Groups</li><li data-subkey="sec2sub3"><a href="/Violin-Plot-in-R.html"><span class="progress-dot"></span>Violin Plot</a></li><li data-subkey="sec2sub3"><a href="/Ridgeline-Plot-in-R.html"><span class="progress-dot"></span>Ridgeline Plot</a></li><li data-subkey="sec2sub3"><a href="/Lollipop-Chart-in-R.html"><span class="progress-dot"></span>Lollipop Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Relationships</li><li data-subkey="sec2sub4"><a href="/Bubble-Chart-in-R.html"><span class="progress-dot"></span>Bubble Chart</a></li><li data-subkey="sec2sub4"><a href="/Heatmap-in-R.html"><span class="progress-dot"></span>Heatmap in R</a></li><li data-subkey="sec2sub4"><a href="/Correlation-Matrix-Plot-in-R.html"><span class="progress-dot"></span>Correlation Matrix</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Advanced Charts</li><li data-subkey="sec2sub5"><a href="/Pie-Donut-Chart-in-R.html"><span class="progress-dot"></span>Pie & Donut Chart</a></li><li data-subkey="sec2sub5"><a href="/Treemap-in-R.html"><span class="progress-dot"></span>Treemap</a></li><li data-subkey="sec2sub5"><a href="/Waffle-Chart-in-R.html"><span class="progress-dot"></span>Waffle Chart</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Exploratory Analysis</li><li data-subkey="sec2sub6"><a href="/Exploratory-Data-Analysis-in-R.html"><span class="progress-dot"></span>EDA (7-Step Framework)</a></li><li data-subkey="sec2sub6"><a href="/Univariate-EDA-in-R.html"><span class="progress-dot"></span>Univariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Bivariate-EDA-in-R.html"><span class="progress-dot"></span>Bivariate EDA</a></li><li data-subkey="sec2sub6"><a href="/Descriptive-Statistics-in-R.html"><span class="progress-dot"></span>Descriptive Statistics</a></li><li data-subkey="sec2sub6"><a href="/Correlation-Analysis-in-R.html"><span class="progress-dot"></span>Correlation Analysis</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub7" data-collapsed="false"><span class="subsec-chevron">▼</span> Interactive & Maps</li><li data-subkey="sec2sub7"><a href="/Combining-ggplot2-with-plotly.html"><span class="progress-dot"></span>ggplot2 + plotly Interactive</a></li><li data-subkey="sec2sub7"><a href="/Interactive-Maps-in-R-with-leaflet.html"><span class="progress-dot"></span>Leaflet Interactive Maps</a></li><li data-subkey="sec2sub7"><a href="/Spatial-Data-in-R-with-sf.html"><span class="progress-dot"></span>Spatial Data (sf)</a></li><li data-subkey="sec2sub7"><a href="/Choropleth-Maps-in-R.html"><span class="progress-dot"></span>Choropleth Maps (sf)</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec2sub8" data-collapsed="false"><span class="subsec-chevron">▼</span> Customization & Reference</li><li data-subkey="sec2sub8"><a href="/ggplot2-Legends-in-R.html"><span class="progress-dot"></span>ggplot2 Legends</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Secondary-Axis.html"><span class="progress-dot"></span>Secondary Axis</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-Log-Scale.html"><span class="progress-dot"></span>Log Scale</a></li><li data-subkey="sec2sub8"><a href="/patchwork-Package.html"><span class="progress-dot"></span>patchwork (Combine Plots)</a></li><li data-subkey="sec2sub8"><a href="/Publication-Quality-Figures-in-R.html"><span class="progress-dot"></span>Publication-Ready Figures</a></li><li data-subkey="sec2sub8"><a href="/ggplot2-cheatsheet.html"><span class="progress-dot"></span>ggplot2 Quickref</a></li></ul></li><li class="sidebar-section expanded"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Statistics<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> EDA & Data Quality</li><li data-subkey="sec3sub1"><a href="/Automated-EDA-in-R.html"><span class="progress-dot"></span>Automated EDA</a></li><li data-subkey="sec3sub1"><a href="/Missing-Data-Visualization-in-R-naniar.html"><span class="progress-dot"></span>Missing Data Viz (naniar)</a></li><li data-subkey="sec3sub1"><a href="/Outlier-Detection-in-R.html"><span class="progress-dot"></span>Outlier Detection</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub2" data-collapsed="false"><span class="subsec-chevron">▼</span> Probability</li><li data-subkey="sec3sub2"><a href="/Sample-Spaces-Events-and-Probability-Axioms-in-R-With-Monte-Carlo-Proof.html"><span class="progress-dot"></span>Probability Axioms</a></li><li data-subkey="sec3sub2"><a href="/Conditional-Probability-in-R.html"><span class="progress-dot"></span>Conditional Probability</a></li><li data-subkey="sec3sub2"><a href="/Random-Variables-in-R.html"><span class="progress-dot"></span>Random Variables</a></li><li data-subkey="sec3sub2"><a href="/Binomial-and-Poisson-Distributions-in-R.html"><span class="progress-dot"></span>Binomial vs Poisson</a></li><li data-subkey="sec3sub2"><a href="/Normal-t-F-and-Chi-Squared-Distributions-in-R.html"><span class="progress-dot"></span>Normal, t, F, Chi-Squared</a></li><li data-subkey="sec3sub2"><a href="/Central-Limit-Theorem-in-R.html"><span class="progress-dot"></span>Central Limit Theorem</a></li><li data-subkey="sec3sub2"><a href="/Sampling-Distributions-in-R.html"><span class="progress-dot"></span>Sampling Distributions</a></li><li data-subkey="sec3sub2"><a href="/Law-of-Large-Numbers-vs-CLT-in-R.html"><span class="progress-dot"></span>LLN vs CLT</a></li><li data-subkey="sec3sub2"><a href="/What-Is-Probability-Simulation-First-Intuition-in-R-Before-the-Formulas.html"><span class="progress-dot"></span>Probability (Simulation-First)</a></li><li data-subkey="sec3sub2"><a href="/Expected-Value-and-Variance-in-R.html"><span class="progress-dot"></span>Expected Value and Variance</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub3" data-collapsed="false"><span class="subsec-chevron">▼</span> Inference & Estimation</li><li data-subkey="sec3sub3"><a href="/Maximum-Likelihood-Estimation-in-R.html"><span class="progress-dot"></span>Maximum Likelihood Estimation</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-in-R.html"><span class="progress-dot"></span>Hypothesis Testing</a></li><li data-subkey="sec3sub3"><a href="/Sample-Size-Planning-in-R.html"><span class="progress-dot"></span>Sample Size Planning</a></li><li data-subkey="sec3sub3"><a href="/Which-Statistical-Test-in-R.html"><span class="progress-dot"></span>Choosing the Right Test</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Tests-in-R.html"><span class="progress-dot"></span>Statistical Tests</a></li><li data-subkey="sec3sub3"><a href="/Measures-of-Association-in-R.html"><span class="progress-dot"></span>Measures of Association</a></li><li data-subkey="sec3sub3"><a href="/Point-Estimation-in-R.html"><span class="progress-dot"></span>Point Estimation</a></li><li data-subkey="sec3sub3"><a href="/Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Confidence Intervals</a></li><li data-subkey="sec3sub3"><a href="/Type-I-and-Type-II-Errors-in-R.html"><span class="progress-dot"></span>Type I and II Errors</a></li><li data-subkey="sec3sub3"><a href="/Statistical-Power-Analysis-in-R.html"><span class="progress-dot"></span>Power Analysis</a></li><li data-subkey="sec3sub3"><a href="/Effect-Size-in-R.html"><span class="progress-dot"></span>Effect Size</a></li><li data-subkey="sec3sub3"><a href="/t-Tests-in-R.html"><span class="progress-dot"></span>t-Tests</a></li><li data-subkey="sec3sub3"><a href="/Proportion-Tests-in-R.html"><span class="progress-dot"></span>Proportion Tests</a></li><li data-subkey="sec3sub3"><a href="/Normality-and-Variance-Tests-in-R.html"><span class="progress-dot"></span>Normality & Variance Tests</a></li><li data-subkey="sec3sub3"><a href="/Chi-Square-Tests-in-R.html"><span class="progress-dot"></span>Chi-Square Tests</a></li><li data-subkey="sec3sub3"><a href="/Wilcoxon-Mann-Whitney-and-Kruskal-Wallis-in-R.html"><span class="progress-dot"></span>Wilcoxon, Mann-Whitney & Kruskal-Wallis</a></li><li data-subkey="sec3sub3"><a href="/Multiple-Comparisons-in-R.html"><span class="progress-dot"></span>Multiple Testing Correction</a></li><li data-subkey="sec3sub3"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub4" data-collapsed="false"><span class="subsec-chevron">▼</span> Regression</li><li data-subkey="sec3sub4"><a href="/Linear-Regression.html"><span class="progress-dot"></span>Linear Regression</a></li><li data-subkey="sec3sub4"><a href="/Logistic-Regression-With-R.html"><span class="progress-dot"></span>Logistic Regression</a></li><li data-subkey="sec3sub4"><a href="/Variable-Selection-and-Importance-With-R.html"><span class="progress-dot"></span>Feature Selection</a></li><li data-subkey="sec3sub4"><a href="/Model-Selection-in-R.html"><span class="progress-dot"></span>Model Selection</a></li><li data-subkey="sec3sub4"><a href="/Missing-Value-Treatment-With-R.html"><span class="progress-dot"></span>Missing Value Treatment</a></li><li data-subkey="sec3sub4"><a href="/Outlier-Treatment-With-R.html"><span class="progress-dot"></span>Outlier Analysis</a></li><li data-subkey="sec3sub4"><a href="/adv-regression-models.html"><span class="progress-dot"></span>Advanced Regression Models</a></li><li data-subkey="sec3sub4"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub5" data-collapsed="false"><span class="subsec-chevron">▼</span> Reporting</li><li data-subkey="sec3sub5"><a href="/Statistical-Consulting-in-R.html"><span class="progress-dot"></span>Statistical Consulting</a></li><li data-subkey="sec3sub5"><a href="/Statistical-Report-Writing-in-R.html"><span class="progress-dot"></span>Statistical Report Writing</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-Confidence-Intervals-in-R.html"><span class="progress-dot"></span>Bootstrap Confidence Intervals</a></li><li data-subkey="sec3sub5"><a href="/Reporting-Statistics-in-R.html"><span class="progress-dot"></span>Reporting Statistics</a></li><li data-subkey="sec3sub5"><a href="/Correlation-in-R.html"><span class="progress-dot"></span>Correlation (Pearson, Spearman, Kendall)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Regression-Assumptions-in-R.html"><span class="progress-dot"></span>Linear Regression Assumptions</a></li><li data-subkey="sec3sub5"><a href="/Dummy-Variables-in-R.html"><span class="progress-dot"></span>Dummy Variables in R</a></li><li data-subkey="sec3sub5"><a href="/Interaction-Effects-in-R.html"><span class="progress-dot"></span>Interaction Effects</a></li><li data-subkey="sec3sub5"><a href="/Regression-Diagnostics-in-R.html"><span class="progress-dot"></span>Regression Diagnostics</a></li><li data-subkey="sec3sub5"><a href="/Logistic-Regression-in-R.html"><span class="progress-dot"></span>Logistic Regression (glm + ROC)</a></li><li data-subkey="sec3sub5"><a href="/Variable-Selection-in-R.html"><span class="progress-dot"></span>Variable Selection</a></li><li data-subkey="sec3sub5"><a href="/Poisson-Regression-in-R.html"><span class="progress-dot"></span>Poisson Regression</a></li><li data-subkey="sec3sub5"><a href="/Ridge-and-Lasso-Regression-in-R.html"><span class="progress-dot"></span>Ridge & Lasso Regression</a></li><li data-subkey="sec3sub5"><a href="/Polynomial-and-Spline-Regression-in-R.html"><span class="progress-dot"></span>Polynomial & Splines</a></li><li data-subkey="sec3sub5"><a href="/Regression-Tables-in-R.html"><span class="progress-dot"></span>Regression Tables (3 packages)</a></li><li data-subkey="sec3sub5"><a href="/One-Way-ANOVA-in-R.html"><span class="progress-dot"></span>One-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Post-Hoc-Tests-After-ANOVA.html"><span class="progress-dot"></span>Post-Hoc Tests After ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Two-Way-ANOVA-in-R.html"><span class="progress-dot"></span>Two-Way ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Repeated-Measures-ANOVA-in-R.html"><span class="progress-dot"></span>Repeated Measures ANOVA</a></li><li data-subkey="sec3sub5"><a href="/ANCOVA-in-R.html"><span class="progress-dot"></span>ANCOVA</a></li><li data-subkey="sec3sub5"><a href="/Experimental-Design-Principles-in-R.html"><span class="progress-dot"></span>Experimental Design in R</a></li><li data-subkey="sec3sub5"><a href="/Factorial-Experiments-in-R.html"><span class="progress-dot"></span>Factorial Designs (2^k)</a></li><li data-subkey="sec3sub5"><a href="/AB-Testing-in-R.html" class="active"><span class="progress-dot"></span>A/B Testing</a></li><li data-subkey="sec3sub5"><a href="/MANOVA-in-R.html"><span class="progress-dot"></span>MANOVA</a></li><li data-subkey="sec3sub5"><a href="/Mixed-ANOVA-in-R.html"><span class="progress-dot"></span>Mixed ANOVA</a></li><li data-subkey="sec3sub5"><a href="/Multivariate-Statistics-in-R.html"><span class="progress-dot"></span>Multivariate Distances & Hotelling's T²</a></li><li data-subkey="sec3sub5"><a href="/PCA-in-R.html"><span class="progress-dot"></span>PCA with prcomp()</a></li><li data-subkey="sec3sub5"><a href="/Interpreting-PCA-Results-in-R.html"><span class="progress-dot"></span>Interpreting PCA Output</a></li><li data-subkey="sec3sub5"><a href="/Exploratory-Factor-Analysis-in-R.html"><span class="progress-dot"></span>Exploratory Factor Analysis</a></li><li data-subkey="sec3sub5"><a href="/CFA-and-Structural-Equation-Modeling-in-R.html"><span class="progress-dot"></span>SEM and CFA (lavaan)</a></li><li data-subkey="sec3sub5"><a href="/Linear-Discriminant-Analysis-in-R.html"><span class="progress-dot"></span>LDA (Linear Discriminant Analysis)</a></li><li data-subkey="sec3sub5"><a href="/Cluster-Analysis-in-R.html"><span class="progress-dot"></span>Clustering (k-Means / HC / DBSCAN)</a></li><li data-subkey="sec3sub5"><a href="/Correspondence-Analysis-in-R.html"><span class="progress-dot"></span>Correspondence Analysis</a></li><li data-subkey="sec3sub5"><a href="/t-SNE-and-UMAP-in-R.html"><span class="progress-dot"></span>t-SNE and UMAP</a></li><li data-subkey="sec3sub5"><a href="/Simple-Linear-Regression-in-R.html"><span class="progress-dot"></span>Simple Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Multiple-Regression-in-R.html"><span class="progress-dot"></span>Multiple Regression</a></li><li 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class="progress-dot"></span>Kruskal-Wallis Test</a></li><li data-subkey="sec3sub5"><a href="/Friedman-Test-in-R.html"><span class="progress-dot"></span>Friedman Test</a></li><li data-subkey="sec3sub5"><a href="/Spearman-and-Kendall-Correlation-in-R.html"><span class="progress-dot"></span>Spearman & Kendall Correlation</a></li><li data-subkey="sec3sub5"><a href="/Bootstrap-in-R.html"><span class="progress-dot"></span>Bootstrap (boot package)</a></li><li data-subkey="sec3sub5"><a href="/Quantile-Regression-in-R-2.html"><span class="progress-dot"></span>Quantile Regression</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Operations-in-R.html"><span class="progress-dot"></span>Matrix Operations in R</a></li><li data-subkey="sec3sub5"><a href="/Solving-Linear-Systems-in-R.html"><span class="progress-dot"></span>Solving Linear Systems in R</a></li><li data-subkey="sec3sub5"><a href="/Eigenvalues-and-Eigenvectors-in-R.html"><span class="progress-dot"></span>Eigenvalues & Eigenvectors in R</a></li><li data-subkey="sec3sub5"><a href="/Singular-Value-Decomposition-in-R.html"><span class="progress-dot"></span>Singular Value Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Projections-and-the-Hat-Matrix-in-R.html"><span class="progress-dot"></span>Projections & the Hat Matrix</a></li><li data-subkey="sec3sub5"><a href="/QR-Decomposition-in-R.html"><span class="progress-dot"></span>QR Decomposition in R</a></li><li data-subkey="sec3sub5"><a href="/Quadratic-Forms-in-R.html"><span class="progress-dot"></span>Quadratic Forms</a></li><li data-subkey="sec3sub5"><a href="/Matrix-Derivatives-and-the-Hessian-in-R.html"><span class="progress-dot"></span>Matrix Derivatives & Hessian</a></li><li data-subkey="sec3sub5"><a href="/Exponential-Family-Distributions-in-R.html"><span class="progress-dot"></span>Exponential Family Distributions</a></li><li data-subkey="sec3sub5"><a href="/Sufficient-Statistics-in-R.html"><span class="progress-dot"></span>Sufficient Statistics</a></li><li data-subkey="sec3sub5"><a href="/Complete-and-Ancillary-Statistics-in-R.html"><span class="progress-dot"></span>Complete & Ancillary Statistics</a></li><li data-subkey="sec3sub5"><a href="/UMVUE-in-R-2.html"><span class="progress-dot"></span>UMVUE (Rao-Blackwell & Lehmann-Scheffé)</a></li><li data-subkey="sec3sub5"><a href="/Cramer-Rao-Lower-Bound-in-R-2.html"><span class="progress-dot"></span>Cramér-Rao Lower Bound</a></li><li data-subkey="sec3sub5"><a href="/Asymptotic-Theory-in-R-2.html"><span class="progress-dot"></span>Asymptotic Theory</a></li><li data-subkey="sec3sub5"><a href="/Neyman-Pearson-Lemma-in-R-2.html"><span class="progress-dot"></span>Neyman-Pearson Lemma</a></li><li data-subkey="sec3sub5"><a href="/Likelihood-Ratio-Tests-and-Pivotal-Methods.html"><span class="progress-dot"></span>Likelihood Ratio & Pivotal Methods</a></li><li data-subkey="sec3sub5"><a href="/Decision-Theory-in-R.html"><span class="progress-dot"></span>Decision 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class="progress-dot"></span>Hamiltonian Monte Carlo</a></li><li data-subkey="sec3sub5"><a href="/Stan-in-R.html"><span class="progress-dot"></span>Stan</a></li><li data-subkey="sec3sub5"><a href="/brms-in-R.html"><span class="progress-dot"></span>brms</a></li><li data-subkey="sec3sub5"><a href="/Choosing-Priors-in-R.html"><span class="progress-dot"></span>Choosing Priors</a></li><li data-subkey="sec3sub5"><a href="/Prior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Prior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Compare-Bayesian-Models-in-R.html"><span class="progress-dot"></span>Compare Bayesian Models</a></li><li data-subkey="sec3sub5"><a href="/Posterior-Predictive-Checks-in-R.html"><span class="progress-dot"></span>Posterior Predictive Checks</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Linear-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Linear Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Logistic-Regression-in-R.html"><span class="progress-dot"></span>Bayesian Logistic Regression</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-Hierarchical-Models-in-R.html"><span class="progress-dot"></span>Bayesian Hierarchical Models</a></li><li data-subkey="sec3sub5"><a href="/Multilevel-Models-in-R.html"><span class="progress-dot"></span>Multilevel Models</a></li><li data-subkey="sec3sub5"><a href="/Bayesian-ANOVA-in-R.html"><span class="progress-dot"></span>Bayesian ANOVA</a></li><li class="sidebar-divider sidebar-subsection-toggle" data-subkey="sec3sub6" data-collapsed="false"><span class="subsec-chevron">▼</span> Machine Learning</li><li data-subkey="sec3sub6"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Time Series<span class="section-meta" data-section-meta></span></div><ul 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data-subkey="sec5sub1"><a href="/Functional-Programming-in-R.html"><span class="progress-dot"></span>Functional Programming</a></li><li data-subkey="sec5sub1"><a href="/R-Functional-Programming-Exercises-quiz.html"><span class="progress-dot"></span>Functional Programming Quiz</a></li><li data-subkey="sec5sub1"><a href="/purrr-map-Variants.html"><span class="progress-dot"></span>purrr map() Variants</a></li><li data-subkey="sec5sub1"><a href="/R-Anonymous-Functions.html"><span class="progress-dot"></span>R Anonymous Functions</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Factories.html"><span class="progress-dot"></span>R Function Factories</a></li><li data-subkey="sec5sub1"><a href="/R-Function-Operators.html"><span class="progress-dot"></span>R Function Operators</a></li><li data-subkey="sec5sub1"><a href="/Reduce-Filter-Map-in-R.html"><span class="progress-dot"></span>Reduce, Filter, Map</a></li><li data-subkey="sec5sub1"><a href="/Memoization-in-R.html"><span class="progress-dot"></span>Memoization in R</a></li><li data-subkey="sec5sub1"><a href="/Writing-Composable-R-Code.html"><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"><span class="progress-dot"></span>OOP in R: S3/S4/R6</a></li><li data-subkey="sec5sub2"><a href="/S3-Classes-in-R.html"><span class="progress-dot"></span>S3 Classes</a></li><li data-subkey="sec5sub2"><a href="/S3-Method-Dispatch-in-R.html"><span class="progress-dot"></span>S3 Method Dispatch</a></li><li data-subkey="sec5sub2"><a href="/S4-Classes-in-R.html"><span class="progress-dot"></span>S4 Classes</a></li><li data-subkey="sec5sub2"><a href="/S4-Methods-in-R.html"><span class="progress-dot"></span>S4 Methods & Dispatch</a></li><li data-subkey="sec5sub2"><a href="/R6-Classes-in-R.html"><span class="progress-dot"></span>R6 Classes</a></li><li data-subkey="sec5sub2"><a href="/R6-Advanced.html"><span class="progress-dot"></span>R6 Advanced</a></li><li data-subkey="sec5sub2"><a href="/Operator-Overloading-in-R.html"><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"><span class="progress-dot"></span>R Names & Values</a></li><li data-subkey="sec5sub3"><a href="/R-Assignment-Deep-Dive.html"><span class="progress-dot"></span>R Assignment Deep Dive</a></li><li data-subkey="sec5sub3"><a href="/R-Memory-lobstr.html"><span class="progress-dot"></span>R Memory & lobstr</a></li><li data-subkey="sec5sub3"><a href="/R-Environments.html"><span class="progress-dot"></span>R Environments</a></li><li data-subkey="sec5sub3"><a href="/R-Lexical-Scoping.html"><span class="progress-dot"></span>Lexical Scoping</a></li><li data-subkey="sec5sub3"><a href="/R-Closures.html"><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"><span class="progress-dot"></span>Conditions System</a></li><li data-subkey="sec5sub4"><a href="/R-Debugging.html"><span class="progress-dot"></span>Debugging R Code</a></li><li data-subkey="sec5sub4"><a href="/R-Common-Errors.html"><span class="progress-dot"></span>50 Common R Errors</a></li><li data-subkey="sec5sub4"><a href="/Parallel-Computing-With-R.html"><span class="progress-dot"></span>Parallel Computing</a></li><li data-subkey="sec5sub4"><a href="/Strategies-To-Improve-And-Speedup-R-Code.html"><span class="progress-dot"></span>Speedup R Code</a></li><li data-subkey="sec5sub4"><a href="/Shiny-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Shiny Quiz</a></li></ul></li><li class="sidebar-section"><div class="sidebar-section-header"><span class="sidebar-chevron">▸</span> Classic Tutorials<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"><span class="progress-dot"></span>R Tutorial (Classic)</a></li><li data-subkey="sec6sub0"><a href="/ggplot2-Tutorial-With-R.html"><span class="progress-dot"></span>ggplot2 Short Tutorial</a></li><li data-subkey="sec6sub0"><a href="/Complete-Ggplot2-Tutorial-Part1-With-R-Code.html"><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"><span class="progress-dot"></span>ggplot2 Tutorial 2 - Theme</a></li><li data-subkey="sec6sub0"><a href="/Top50-Ggplot2-Visualizations-MasterList-R-Code.html"><span class="progress-dot"></span>ggplot2 Tutorial 3 - Masterlist</a></li><li data-subkey="sec6sub0"><a href="/Association-Mining-With-R.html"><span class="progress-dot"></span>Association Mining</a></li><li data-subkey="sec6sub0"><a href="/Multi-Dimensional-Scaling-With-R.html"><span class="progress-dot"></span>Multi Dimensional Scaling</a></li><li data-subkey="sec6sub0"><a href="/Optimization-With-R.html"><span class="progress-dot"></span>Optimization</a></li><li data-subkey="sec6sub0"><a href="/Information-Value-With-R.html"><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> Practice Exercises<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> Mastery Quizzes (Certificate)</li><li data-subkey="sec7sub1"><a href="/R-Beginner-Exercises-quiz.html"><span class="progress-dot"></span>R Fundamentals Quiz</a></li><li data-subkey="sec7sub1"><a href="/dplyr-Exercises-in-R-quiz.html"><span class="progress-dot"></span>dplyr Quiz</a></li><li data-subkey="sec7sub1"><a href="/ggplot2-Exercises-in-R-quiz.html"><span class="progress-dot"></span>ggplot2 Quiz</a></li><li data-subkey="sec7sub1"><a href="/Hypothesis-Testing-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Hypothesis Testing Quiz</a></li><li data-subkey="sec7sub1"><a href="/Linear-Regression-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Linear Regression Quiz</a></li><li data-subkey="sec7sub1"><a href="/Machine-Learning-Exercises-in-R-quiz.html"><span class="progress-dot"></span>Machine Learning Quiz</a></li><li data-subkey="sec7sub1"><a href="/tidyr-Exercises-in-R-quiz.html"><span 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<h1>A/B Testing in R: Plan Your Sample Size, Analyse Correctly, and Know When to Stop</h1>
<p class="lead">A/B testing in R is a design discipline with three moves: pick a sample size with <code>pwr</code> <em>before</em> the test runs, analyse the result exactly once with <code>prop.test()</code> or <code>t.test()</code>, and stop on a pre-specified rule instead of whenever the p-value looks favourable. Do those three things and the rest is bookkeeping.</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 May 13, 2026 · Last updated May 13, 2026</div>
<div class="engagement-header" data-difficulty="Intermediate" data-time="35" data-exercises="8" data-xp="120"></div>
<h2>What does an A/B test in R actually look like?</h2>
<p>Before the jargon, let's see the whole pattern in one example. Suppose we ran a test where a new checkout converted 130 out of 2000 visitors and the old one 105 out of 2000. We want a single, honest answer: did the new checkout actually win? The code below does the whole job in one call, it computes the p-value, the confidence interval for the lift, and a sanity check on the effect size.</p>
<div class="webr-container" data-block-title="Run a two-proportion A/B test on checkout data">
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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">Run a two-proportion A/B test on checkout data</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="c1"># Successes and trials for (new, old)</span></span>
<span class="cl">success <span class="o"><-</span> <span class="nf">c</span>(<span class="m">130</span>, <span class="m">105</span>)</span>
<span class="cl">trials <span class="o"><-</span> <span class="nf">c</span>(<span class="m">2000</span>, <span class="m">2000</span>)</span>
<span class="cl"></span>
<span class="cl">test_result <span class="o"><-</span> <span class="nf">prop.test</span>(success, trials, correct <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl">test_result</span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> 2-sample test for equality of proportions without continuity correction</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> data: success out of trials</span></span>
<span class="cl"><span class="c1">#> X-squared = 3.0617, df = 1, p-value = 0.08015</span></span>
<span class="cl"><span class="c1">#> alternative hypothesis: two.sided</span></span>
<span class="cl"><span class="c1">#> 95 percent confidence interval:</span></span>
<span class="cl"><span class="c1">#> -0.001511495 0.026511495</span></span>
<span class="cl"><span class="c1">#> sample estimates:</span></span>
<span class="cl"><span class="c1">#> prop 1 prop 2</span></span>
<span class="cl"><span class="c1">#> 0.065 0.0525</span></span></div>
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<p>Three numbers tell the whole story. The conversion rates were 6.5% vs 5.25%, a lift of 1.25 percentage points. The p-value is 0.08, which is above the usual 0.05 threshold, so we cannot reject "the two checkouts perform equally". The <a class="auto-link" href="Confidence-Intervals-in-R.html" title="Confidence Intervals in R: The Definition Most Textbooks State Incorrectly">95% confidence interval</a> for the lift is roughly -0.2 to 2.7 percentage points, meaning the data are consistent with anything from a small loss to a meaningful win. Calling this a winner would be premature.</p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>Every A/B test has three pillars: plan, analyse, stop.</strong> Plan the sample size <em>before</em> you collect data, analyse the result with the test that matches your metric, and stop on the pre-specified rule. Skip any one and the p-value you report is a number without a meaning.</div></div>
<p><img src="screenshots/AB-Testing-in-R-three-pillars.webp" alt="The three pillars of an A/B test" class="img-responsive img-zoomable" loading="lazy" width="2128" height="852" /></p>
<p><em>Figure 1: The three pillars of an A/B test: plan sample size, run to the planned N, analyse once at the end.</em></p>
<section class="tryit-block">
<p><strong>Try it:</strong> Using the same <code>prop.test()</code> pattern, run the test with only 500 visitors per arm (so 33 conversions in the new group and 26 in the old). The point estimate of the lift barely moves but the confidence interval should widen noticeably. Confirm that.</p>
<div class="webr-container" data-block-title="Your turn: shrink the sample and watch the CI">
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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: shrink the sample and watch the CI</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_success <span class="o"><-</span> <span class="nf">c</span>(<span class="m">33</span>, <span class="m">26</span>)</span>
<span class="cl">ex_trials <span class="o"><-</span> <span class="nf">c</span>(<span class="m">500</span>, <span class="m">500</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code here: call prop.test() and print it</span></span>
<span class="cl">ex_result <span class="o"><-</span> <span class="kc">NULL</span></span>
<span class="cl"></span>
<span class="cl">ex_result</span>
<span class="cl"><span class="c1">#> Expected: p-value around 0.35, CI roughly (-0.016, 0.044)</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Smaller sample 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">Smaller sample 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_success <span class="o"><-</span> <span class="nf">c</span>(<span class="m">33</span>, <span class="m">26</span>)</span>
<span class="cl">ex_trials <span class="o"><-</span> <span class="nf">c</span>(<span class="m">500</span>, <span class="m">500</span>)</span>
<span class="cl"></span>
<span class="cl">ex_result <span class="o"><-</span> <span class="nf">prop.test</span>(ex_success, ex_trials, correct <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl">ex_result</span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> 2-sample test for equality of proportions without continuity correction</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> data: ex_success out of ex_trials</span></span>
<span class="cl"><span class="c1">#> X-squared = 0.87845, df = 1, p-value = 0.3486</span></span>
<span class="cl"><span class="c1">#> alternative hypothesis: two.sided</span></span>
<span class="cl"><span class="c1">#> 95 percent confidence interval:</span></span>
<span class="cl"><span class="c1">#> -0.01578115 0.04378115</span></span>
<span class="cl"><span class="c1">#> sample estimates:</span></span>
<span class="cl"><span class="c1">#> prop 1 prop 2</span></span>
<span class="cl"><span class="c1">#> 0.066 0.052</span></span></div>
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<p><strong>Explanation:</strong> With 4x fewer visitors the point estimates barely change, but the CI widens by roughly 2x. Sample size controls <em>precision</em>, not <em>effect</em>.</p>
</details>
</section>
<h2>How do you pick a sample size before the test?</h2>
<p>The sample size question has four knobs: the baseline rate $p_1$, the minimum effect you care about ($p_2 - p_1$), the <a class="auto-link" href="Hypothesis-Testing-in-R.html" title="Hypothesis Testing in R: Understand the Framework, Not Just the p-Value">significance level</a> $\alpha$ (usually 0.05), and the power $1 - \beta$ (usually 0.80). Fix any three and R will solve for the fourth. For sample size, you fix $p_1$, $p_2$, $\alpha$, and $1 - \beta$, and leave $n$ blank.</p>
<p>R has two common ways to do this, and they give slightly different answers because they use different approximations. Let's run both on the same problem: baseline 5% conversion, we want to detect a lift to 6%, at 80% power.</p>
<div class="webr-container" data-block-title="Base R power.prop.test for sample size">
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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">Base R power.prop.test for sample size</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="c1"># Fix everything except n</span></span>
<span class="cl">size_base <span class="o"><-</span> <span class="nf">power.prop.test</span>(</span>
<span class="cl"> p1 <span class="o">=</span> <span class="m">0.05</span>,</span>
<span class="cl"> p2 <span class="o">=</span> <span class="m">0.06</span>,</span>
<span class="cl"> sig.level <span class="o">=</span> <span class="m">0.05</span>,</span>
<span class="cl"> power <span class="o">=</span> <span class="m">0.80</span>,</span>
<span class="cl"> alternative <span class="o">=</span> <span class="s">"two.sided"</span></span>
<span class="cl">)</span>
<span class="cl"></span>
<span class="cl">n_per_group_base <span class="o"><-</span> <span class="nf">ceiling</span>(size_base<span class="o">$</span>n)</span>
<span class="cl">n_per_group_base</span>
<span class="cl"><span class="c1">#> [1] 8836</span></span></div>
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<p>So base R tells us we need roughly 8,836 visitors per arm, about 17,700 total. Now compare with the <code>pwr</code> package, which uses Cohen's arcsine-transformed effect size $h$ instead of the raw difference. The arcsine transform stretches the scale near 0 and 1, which matters when proportions are small.</p>
<p>$$h = 2 \cdot \arcsin(\sqrt{p_1}) - 2 \cdot \arcsin(\sqrt{p_2})$$</p>
<p>Where:</p>
<ul>
<li>$p_1, p_2$ = the two conversion rates</li>
<li>$h$ = a standardised effect size with known power tables</li>
</ul>
<p><em>If you're not interested in the math, the takeaway is that <code>ES.h()</code> computes $h$ for you.</em></p>
<div class="webr-container" data-block-title="pwr.2p.test with Cohen's h effect size">
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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">pwr.2p.test with Cohen's h effect size</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">library</span>(pwr)</span>
<span class="cl"></span>
<span class="cl">h_effect <span class="o"><-</span> <span class="nf">ES.h</span>(p1 <span class="o">=</span> <span class="m">0.06</span>, p2 <span class="o">=</span> <span class="m">0.05</span>)</span>
<span class="cl"><span class="nf">round</span>(h_effect, <span class="m">4</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.0438</span></span>
<span class="cl"></span>
<span class="cl">pwr_result <span class="o"><-</span> <span class="nf">pwr.2p.test</span>(</span>
<span class="cl"> h <span class="o">=</span> h_effect,</span>
<span class="cl"> sig.level <span class="o">=</span> <span class="m">0.05</span>,</span>
<span class="cl"> power <span class="o">=</span> <span class="m">0.80</span>,</span>
<span class="cl"> alternative <span class="o">=</span> <span class="s">"two.sided"</span></span>
<span class="cl">)</span>
<span class="cl">pwr_result</span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Difference of proportion power calculation for binomial distribution (arcsine transformation)</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> h = 0.04378621</span></span>
<span class="cl"><span class="c1">#> n = 8193.114</span></span>
<span class="cl"><span class="c1">#> sig.level = 0.05</span></span>
<span class="cl"><span class="c1">#> power = 0.8</span></span>
<span class="cl"><span class="c1">#> alternative = two.sided</span></span></div>
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<p>The two answers disagree by about 8%: 8,193 per arm with <code>pwr.2p.test()</code> vs 8,836 per arm with <code>power.prop.test()</code>. The difference comes from the arcsine transformation and from whether a continuity correction is built in. For planning, either is fine. Pick one and use it consistently. The important number is the order of magnitude: you need roughly 8,000 to 9,000 visitors per arm, not 2,000.</p>
<div class="callout callout-tip"><div class="callout-label">Tip</div><div class="callout-body"><strong>Always sanity-check the minimum detectable effect against business reality.</strong> If the sample size that's feasible detects only a 3 percentage point lift, and your product team's intuition says the true lift is ~0.5 percentage points, the test is underpowered before it starts. Kill it or redesign.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> A signup page converts at 4% and you want to detect a lift to 4.5% with 80% power at $\alpha = 0.05$. Compute the per-arm sample size using <code>pwr.2p.test()</code>. You should land around 13,000.</p>
<div class="webr-container" data-block-title="Your turn: sample size for baseline 4%, lift 0.5pp">
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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: sample size for baseline 4%, lift 0.5pp</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="c1"># your code here</span></span>
<span class="cl">ex_h <span class="o"><-</span> <span class="kc">NULL</span></span>
<span class="cl">ex_n <span class="o"><-</span> <span class="kc">NULL</span></span>
<span class="cl"></span>
<span class="cl">ex_n</span>
<span class="cl"><span class="c1">#> Expected: roughly 13,000 per arm</span></span></div>
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<div class="webr-container" data-block-title="Baseline 4% to 4.5% 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">Baseline 4% to 4.5% 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_h <span class="o"><-</span> <span class="nf">ES.h</span>(p1 <span class="o">=</span> <span class="m">0.045</span>, p2 <span class="o">=</span> <span class="m">0.04</span>)</span>
<span class="cl">ex_n <span class="o"><-</span> <span class="nf">pwr.2p.test</span>(h <span class="o">=</span> ex_h, sig.level <span class="o">=</span> <span class="m">0.05</span>, power <span class="o">=</span> <span class="m">0.80</span>)</span>
<span class="cl"><span class="nf">ceiling</span>(ex_n<span class="o">$</span>n)</span>
<span class="cl"><span class="c1">#> [1] 13019</span></span></div>
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<p><strong>Explanation:</strong> Small absolute differences at small baseline rates require enormous samples because the standard error shrinks slower than the effect.</p>
</details>
</section>
<h2>How do you analyse a proportion A/B test correctly?</h2>
<p>Once the test is over and the data is in, there is one question: given the final counts, is the lift real? For proportions (conversion, click-through, signup, churn flag), the right tool is <code>prop.test()</code>. Under the hood it is a Pearson <a class="auto-link" href="Chi-Square-Tests-in-R.html" title="Chi-Square Tests in R: Independence, Goodness-of-Fit, With Effect Sizes">chi-square test</a> on a <a class="auto-link" href="Fishers-Exact-Test-in-R.html" title="Fisher's Exact Test in R: 2×2 Tables, Odds Ratios & Small Samples">2x2 contingency table</a>, but it reports something more useful: the confidence interval for the <em>difference</em> in proportions, which is the lift itself.</p>
<p>Let's reuse the <code>success</code> and <code>trials</code> vectors from block 1 and pull out the pieces analysts actually cite in a report.</p>
<div class="webr-container" data-block-title="Pull lift, CI, and p-value from prop.test">
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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">Pull lift, CI, and p-value from prop.test</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">conv_test <span class="o"><-</span> <span class="nf">prop.test</span>(success, trials, correct <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl"></span>
<span class="cl">lift <span class="o"><-</span> <span class="nf">diff</span>(<span class="nf">rev</span>(conv_test<span class="o">$</span>estimate))</span>
<span class="cl">ci_lower <span class="o"><-</span> conv_test<span class="o">$</span>conf.int[<span class="m">1</span>]</span>
<span class="cl">ci_upper <span class="o"><-</span> conv_test<span class="o">$</span>conf.int[<span class="m">2</span>]</span>
<span class="cl">p_value <span class="o"><-</span> conv_test<span class="o">$</span>p.value</span>
<span class="cl"></span>
<span class="cl"><span class="nf">cat</span>(<span class="nf">sprintf</span>(<span class="s">"Lift: %.2fpp\n"</span>, lift <span class="o">*</span> <span class="m">100</span>))</span>
<span class="cl"><span class="nf">cat</span>(<span class="nf">sprintf</span>(<span class="s">"95%% CI: (%.2fpp, %.2fpp)\n"</span>, ci_lower <span class="o">*</span> <span class="m">100</span>, ci_upper <span class="o">*</span> <span class="m">100</span>))</span>
<span class="cl"><span class="nf">cat</span>(<span class="nf">sprintf</span>(<span class="s">"p-value: %.4f\n"</span>, p_value))</span>
<span class="cl"><span class="c1">#> Lift: 1.25pp</span></span>
<span class="cl"><span class="c1">#> 95% CI: (-0.15pp, 2.65pp)</span></span>
<span class="cl"><span class="c1">#> p-value: 0.0801</span></span></div>
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<p>The three numbers, reported together, tell a complete story. The lift point estimate is 1.25 percentage points. The 95% CI is (-0.15pp, 2.65pp), which just barely crosses zero, and that is exactly why the p-value (0.08) does not clear 0.05. Reporting the lift and its CI is more informative than reporting the p-value alone, because the CI tells the reader the <em>range of plausible lifts</em>, not just whether we rejected the null.</p>
<div class="callout callout-note"><div class="callout-label">Note</div><div class="callout-body"><strong><code>chisq.test()</code> and <code>prop.test()</code> give the same p-value on a 2x2 table.</strong> Use <code>prop.test()</code> for A/B tests because it reports the lift CI. Use <code>chisq.test()</code> when the table is larger than 2x2 (for example, a three-arm A/B/C test compared to control). Set <code>correct = FALSE</code> to match the formula in most textbooks; the default <code>correct = TRUE</code> applies Yates's continuity correction and is slightly more conservative.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Suppose the final counts are 48/1000 in control and 60/1000 in treatment. Run <code>prop.test()</code> and decide: does the 95% CI cross zero? Is the p-value below 0.05?</p>
<div class="webr-container" data-block-title="Your turn: interpret a close-call result">
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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: interpret a close-call result</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_succ <span class="o"><-</span> <span class="nf">c</span>(<span class="m">60</span>, <span class="m">48</span>)</span>
<span class="cl">ex_trial <span class="o"><-</span> <span class="nf">c</span>(<span class="m">1000</span>, <span class="m">1000</span>)</span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code here</span></span>
<span class="cl">ex_test <span class="o"><-</span> <span class="kc">NULL</span></span>
<span class="cl"></span>
<span class="cl">ex_test</span>
<span class="cl"><span class="c1">#> Expected: lift ~1.2pp, CI crosses zero, p around 0.22</span></span></div>
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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">Close-call analysis 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_succ <span class="o"><-</span> <span class="nf">c</span>(<span class="m">60</span>, <span class="m">48</span>)</span>
<span class="cl">ex_trial <span class="o"><-</span> <span class="nf">c</span>(<span class="m">1000</span>, <span class="m">1000</span>)</span>
<span class="cl"></span>
<span class="cl">ex_test <span class="o"><-</span> <span class="nf">prop.test</span>(ex_succ, ex_trial, correct <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl">ex_test<span class="o">$</span>p.value</span>
<span class="cl"><span class="c1">#> [1] 0.2163797</span></span>
<span class="cl">ex_test<span class="o">$</span>conf.int</span>
<span class="cl"><span class="c1">#> [1] -0.007130596 0.031130596</span></span>
<span class="cl"><span class="c1">#> attr(,"conf.level")</span></span>
<span class="cl"><span class="c1">#> [1] 0.95</span></span></div>
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<p><strong>Explanation:</strong> The CI spans -0.7pp to +3.1pp, so zero is inside the range of plausible lifts. Reporting "lift = 1.2pp, p = 0.22" captures this honestly; reporting "lift = 1.2pp, p < 0.05" would be false.</p>
</details>
</section>
<h2>How do you analyse a continuous-metric A/B test?</h2>
<p>Not every A/B metric is a proportion. Revenue per user, time on site, pages viewed, and API latency are all continuous, and proportions tests do not apply. The default tool is Welch's <a class="auto-link" href="t-Tests-in-R.html" title="t-Tests in R: Every Variant With the Decision Rule for Choosing Between Them">two-sample t-test</a>, which does not assume equal variances between arms, a property you want because revenue distributions are always more variable in one arm than the other.</p>
<p>Let's simulate two revenue streams and compare them. We draw from log-normal distributions because real revenue is heavy-tailed, not normal.</p>
<div class="webr-container" data-block-title="Welch t-test on simulated revenue per user">
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<div class="webr-editor" data-language="r"><span class="cl"><span class="nf">set.seed</span>(<span class="m">4248</span>)</span>
<span class="cl"></span>
<span class="cl">rev_A <span class="o"><-</span> <span class="nf">rlnorm</span>(<span class="m">1500</span>, meanlog <span class="o">=</span> <span class="m">2.8</span>, sdlog <span class="o">=</span> <span class="m">0.6</span>) <span class="c1"># old page</span></span>
<span class="cl">rev_B <span class="o"><-</span> <span class="nf">rlnorm</span>(<span class="m">1500</span>, meanlog <span class="o">=</span> <span class="m">2.9</span>, sdlog <span class="o">=</span> <span class="m">0.6</span>) <span class="c1"># new page</span></span>
<span class="cl"></span>
<span class="cl">t_result <span class="o"><-</span> <span class="nf">t.test</span>(rev_B, rev_A, var.equal <span class="o">=</span> <span class="kc">FALSE</span>)</span>
<span class="cl">t_result</span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> Welch Two Sample t-test</span></span>
<span class="cl"><span class="c1">#></span></span>
<span class="cl"><span class="c1">#> data: rev_B and rev_A</span></span>
<span class="cl"><span class="c1">#> t = 4.5876, df = 2994.1, p-value = 4.688e-06</span></span>
<span class="cl"><span class="c1">#> alternative hypothesis: true difference in means is not equal to 0</span></span>
<span class="cl"><span class="c1">#> 95 percent confidence interval:</span></span>
<span class="cl"><span class="c1">#> 1.276207 3.198389</span></span>
<span class="cl"><span class="c1">#> sample estimates:</span></span>
<span class="cl"><span class="c1">#> mean of x mean of y</span></span>
<span class="cl"><span class="c1">#> 21.76938 19.53208</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># Cohen's d (standardised effect size)</span></span>
<span class="cl">cohen_d <span class="o"><-</span> (<span class="nf">mean</span>(rev_B) <span class="o">-</span> <span class="nf">mean</span>(rev_A)) <span class="o">/</span></span>
<span class="cl"> <span class="nf">sqrt</span>(((<span class="nf">length</span>(rev_A) <span class="o">-</span> <span class="m">1</span>) <span class="o">*</span> <span class="nf">var</span>(rev_A) <span class="o">+</span></span>
<span class="cl"> (<span class="nf">length</span>(rev_B) <span class="o">-</span> <span class="m">1</span>) <span class="o">*</span> <span class="nf">var</span>(rev_B)) <span class="o">/</span></span>
<span class="cl"> (<span class="nf">length</span>(rev_A) <span class="o">+</span> <span class="nf">length</span>(rev_B) <span class="o">-</span> <span class="m">2</span>))</span>
<span class="cl"><span class="nf">round</span>(cohen_d, <span class="m">3</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.167</span></span></div>
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<p>The new page raised revenue per user from $19.53 to $21.77, a lift of $2.24 per user with a 95% CI of ($1.28, $3.20). The p-value is tiny and <a class="auto-link" href="Statistical-Power-Analysis-in-R.html" title="Power Analysis in R: Calculate the Sample Size You Need Before You Collect Any Data">Cohen's d</a> is 0.17, which in Cohen's conventions is a "small" effect, but <em>small does not mean unimportant</em>. A 0.17 standardised effect on a large revenue stream is often the difference between a profitable quarter and a flat one.</p>
<div class="callout callout-warning"><div class="callout-label">Warning</div><div class="callout-body"><strong>Revenue distributions have extreme outliers that can mislead a t-test.</strong> A single whale customer can swing the mean by 20% and turn a real effect into noise (or noise into a "win"). Three defences: (1) cap extreme values at the 99th percentile before the test, (2) run <code>wilcox.test()</code> as a non-parametric sanity check, or (3) model on <code>log(revenue + 1)</code> and compare geometric means.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Two arms have means 45.0 and 47.5, standard deviations 12.0 and 12.5, and 800 users each. Compute Cohen's d for this effect.</p>
<div class="webr-container" data-block-title="Your turn: compute Cohen's d">
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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: compute Cohen's d</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_m1 <span class="o"><-</span> <span class="m">45.0</span>; ex_s1 <span class="o"><-</span> <span class="m">12.0</span>; ex_n1 <span class="o"><-</span> <span class="m">800</span></span>
<span class="cl">ex_m2 <span class="o"><-</span> <span class="m">47.5</span>; ex_s2 <span class="o"><-</span> <span class="m">12.5</span>; ex_n2 <span class="o"><-</span> <span class="m">800</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code here</span></span>
<span class="cl">ex_d <span class="o"><-</span> <span class="kc">NULL</span></span>
<span class="cl"></span>
<span class="cl">ex_d</span>
<span class="cl"><span class="c1">#> Expected: about 0.204</span></span></div>
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<div class="webr-container" data-block-title="Cohen's d 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">Cohen's d 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_d <span class="o"><-</span> (ex_m2 <span class="o">-</span> ex_m1) <span class="o">/</span></span>
<span class="cl"> <span class="nf">sqrt</span>(((ex_n1 <span class="o">-</span> <span class="m">1</span>) <span class="o">*</span> ex_s1<span class="o">^</span><span class="m">2</span> <span class="o">+</span> (ex_n2 <span class="o">-</span> <span class="m">1</span>) <span class="o">*</span> ex_s2<span class="o">^</span><span class="m">2</span>) <span class="o">/</span></span>
<span class="cl"> (ex_n1 <span class="o">+</span> ex_n2 <span class="o">-</span> <span class="m">2</span>))</span>
<span class="cl"><span class="nf">round</span>(ex_d, <span class="m">3</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.204</span></span></div>
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<p><strong>Explanation:</strong> Cohen's d is the raw mean difference divided by the pooled standard deviation. It is unit-free, which is why you compare it across experiments regardless of metric.</p>
</details>
</section>
<h2>What happens if you peek at the p-value every day?</h2>
<p>Here is the mistake that ruins more A/B tests than any other: running the test, checking the p-value every day, and stopping the moment it drops below 0.05. This sounds harmless. It is not. Each extra look is another chance for random noise to cross the threshold, and the effective false-positive rate climbs fast.</p>
<p>The cleanest way to see this is a simulation. We will assume the null is true (both arms converge to the same rate), run the same experiment 5,000 times, peek at it 10 times per run, and count how often <em>any</em> peek crossed $\alpha = 0.05$.</p>
<div class="webr-container" data-block-title="Simulate peeking inflation of Type I error">
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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">Simulate peeking inflation of Type I error</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">set.seed</span>(<span class="m">10820</span>)</span>
<span class="cl"></span>
<span class="cl">n_sims <span class="o"><-</span> <span class="m">5000</span></span>
<span class="cl">look_points <span class="o"><-</span> <span class="nf">seq</span>(<span class="m">200</span>, <span class="m">2000</span>, by <span class="o">=</span> <span class="m">200</span>) <span class="c1"># 10 peeks</span></span>
<span class="cl">p_rate <span class="o"><-</span> <span class="m">0.05</span> <span class="c1"># null is true: same for both arms</span></span>
<span class="cl"></span>
<span class="cl">peek_hits <span class="o"><-</span> <span class="m">0</span></span>
<span class="cl"><span class="kr">for</span> (i <span class="kr">in</span> <span class="nf">seq_len</span>(n_sims)) {</span>
<span class="cl"> <span class="c1"># Simulate the full run in one pass, then take cumulative counts</span></span>
<span class="cl"> xA <span class="o"><-</span> <span class="nf">cumsum</span>(<span class="nf">rbinom</span>(<span class="nf">max</span>(look_points), <span class="m">1</span>, p_rate))</span>
<span class="cl"> xB <span class="o"><-</span> <span class="nf">cumsum</span>(<span class="nf">rbinom</span>(<span class="nf">max</span>(look_points), <span class="m">1</span>, p_rate))</span>
<span class="cl"></span>
<span class="cl"> crossed <span class="o"><-</span> <span class="kc">FALSE</span></span>
<span class="cl"> <span class="kr">for</span> (n <span class="kr">in</span> look_points) {</span>
<span class="cl"> pv <span class="o"><-</span> <span class="nf">prop.test</span>(<span class="nf">c</span>(xA[n], xB[n]), <span class="nf">c</span>(n, n), correct <span class="o">=</span> <span class="kc">FALSE</span>)<span class="o">$</span>p.value</span>
<span class="cl"> <span class="kr">if</span> (<span class="o">!</span><span class="nf">is.nan</span>(pv) <span class="o">&&</span> pv <span class="o"><</span> <span class="m">0.05</span>) { crossed <span class="o"><-</span> <span class="kc">TRUE</span>; <span class="kr">break</span> }</span>
<span class="cl"> }</span>
<span class="cl"> <span class="kr">if</span> (crossed) peek_hits <span class="o"><-</span> peek_hits <span class="o">+</span> <span class="m">1</span></span>
<span class="cl">}</span>
<span class="cl"></span>
<span class="cl">peek_alpha <span class="o"><-</span> peek_hits <span class="o">/</span> n_sims</span>
<span class="cl"><span class="nf">round</span>(peek_alpha, <span class="m">3</span>)</span>
<span class="cl"><span class="c1">#> [1] 0.221</span></span></div>
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<p>Under the null, 10 peeks turn a nominal 5% false-positive rate into roughly 22%. One in five "wins" declared by a peeking analyst is pure noise. This is the peeking problem, and it does not go away by adding a "just checking" disclaimer. The test is broken.</p>
<p><img src="screenshots/AB-Testing-in-R-peeking-inflation.webp" alt="Peeking inflates Type I error" class="img-responsive img-zoomable" loading="lazy" width="1246" height="1122" /></p>
<p><em>Figure 2: Each extra peek at the data adds a chance for a <a class="auto-link" href="Type-I-and-Type-II-Errors-in-R.html" title="Type I and Type II Errors in R: Visualise the Trade-Off Between α and Power">false positive</a>; cumulative alpha climbs far above 0.05.</em></p>
<div class="callout callout-insight"><div class="callout-label">Key Insight</div><div class="callout-body"><strong>A single look at alpha = 0.05 is 5%. Ten looks is not 50%, but it is nowhere near 5%.</strong> The exact inflated rate depends on look spacing and correlation structure, but every one of those designs still breaks the guarantee you meant to give. The fix is to decide in advance what "look at the data" means, and pay the price (wider rejection thresholds) once, in planning.</div></div>
<section class="tryit-block">
<p><strong>Try it:</strong> Predict what happens with 5 peeks instead of 10. Re-run the simulation with <code>look_points <- seq(400, 2000, by = 400)</code> and see whether <code>peek_alpha</code> falls closer to 0.05 or stays well above.</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">Your turn: 5 peeks inflation</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">set.seed</span>(<span class="m">10820</span>)</span>
<span class="cl"></span>
<span class="cl">ex_look_points <span class="o"><-</span> <span class="nf">seq</span>(<span class="m">400</span>, <span class="m">2000</span>, by <span class="o">=</span> <span class="m">400</span>)</span>
<span class="cl">ex_peek_hits <span class="o"><-</span> <span class="m">0</span></span>
<span class="cl"></span>
<span class="cl"><span class="c1"># your code: adapt the loop above to run for 2000 sims with 5 peeks</span></span>
<span class="cl"><span class="c1"># (use 2000 sims to keep it fast)</span></span>
<span class="cl"></span>
<span class="cl">ex_peek_alpha <span class="o"><-</span> ex_peek_hits <span class="o">/</span> <span class="m">2000</span></span>
<span class="cl">ex_peek_alpha</span>
<span class="cl"><span class="c1">#> Expected: around 0.14 to 0.17 (still well above 0.05)</span></span></div>
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<details>
<summary>Click to reveal solution</summary>
<div class="webr-container" data-block-title="Five-peek simulation 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">Five-peek simulation 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"><span class="nf">set.seed</span>(<span class="m">10820</span>)</span>
<span class="cl"></span>
<span class="cl">ex_look_points <span class="o"><-</span> <span class="nf">seq</span>(<span class="m">400</span>, <span class="m">2000</span>, by <span class="o">=</span> <span class="m">400</span>)</span>
<span class="cl">ex_n_sims <span class="o"><-</span> <span class="m">2000</span></span>
<span class="cl">ex_peek_hits <span class="o"><-</span> <span class="m">0</span></span>
<span class="cl"></span>
<span class="cl"><span class="kr">for</span> (i <span class="kr">in</span> <span class="nf">seq_len</span>(ex_n_sims)) {</span>
<span class="cl"> xA <span class="o"><-</span> <span class="nf">cumsum</span>(<span class="nf">rbinom</span>(<span class="nf">max</span>(ex_look_points), <span class="m">1</span>, <span class="m">0.05</span>))</span>
<span class="cl"> xB <span class="o"><-</span> <span class="nf">cumsum</span>(<span class="nf">rbinom</span>(<span class="nf">max</span>(ex_look_points), <span class="m">1</span>, <span class="m">0.05</span>))</span>
<span class="cl"> crossed <span class="o"><-</span> <span class="kc">FALSE</span></span>
<span class="cl"> <span class="kr">for</span> (n <span class="kr">in</span> ex_look_points) {</span>
<span class="cl"> pv <span class="o"><-</span> <span class="nf">prop.test</span>(<span class="nf">c</span>(xA[n], xB[n]), <span class="nf">c</span>(n, n), correct <span class="o">=</span> <span class="kc">FALSE</span>)<span class="o">$</span>p.value</span>
<span class="cl"> <span class="kr">if</span> (<span class="o">!</span><span class="nf">is.nan</span>(pv) <span class="o">&&</span> pv <span class="o"><</span> <span class="m">0.05</span>) { crossed <span class="o"><-</span> <span class="kc">TRUE</span>; <span class="kr">break</span> }</span>
<span class="cl"> }</span>
<span class="cl"> <span class="kr">if</span> (crossed) ex_peek_hits <span class="o"><-</span> ex_peek_hits <span class="o">+</span> <span class="m">1</span></span>