StatLens For instructors →
Visualize and summarize your data. Start by asking: what kind of variable do I have? Part I, Chapters 1–5
One Quantitative Variable
Histogram, boxplot, dotplot, and summary statistics.
Quantitative by Group
Side-by-side boxplots, dotplots, or histograms to compare groups.
One Categorical Variable
Bar chart and frequency table.
Two Categorical Variables
Contingency table with proportion toggles and grouped bar charts.
Two Quantitative Variables
Scatterplot with least-squares regression line, r, and R².
Data Explorer
Load a dataset, select variables, choose your own visualization.
Browse All Datasets
Search and explore 100+ bundled datasets by type and variable.
StatLens Labs
Dotplot Editor
Click to add and remove points. Watch how mean, median, and spread respond to outliers.
Guess the Correlation
See a scatterplot, guess r. Build your visual intuition for correlation.
Regression by Eye
Drag a line to fit the data. Toggle between squared and absolute errors, then compare to the optimal fit.
Scatterplot Editor
Drag, add, and remove data points and watch the least-squares line, r, and R² respond — feel leverage vs. influence.
Are You Psychic?
Predict coin flips to test your "psychic" powers, then see if your result could be due to chance.
Stump the Chump
Try to fake a random coin-flip sequence, then compare it to a real one. Can you fool the computer?
Build confidence intervals by bootstrapping and test hypotheses by randomization. Part II, Chapters 6–9
| Data Type | Bootstrap CI | Randomization Test |
|---|---|---|
| One Quantitativeone numeric variable | CI for a Mean | Test for a Mean |
| One Categoricalone yes/no variable | CI for a Proportion | Test for a Proportion |
| Compare Two Groupsnumeric outcome by group | CI: Diff in Means CI: Diff in Props | Test: Diff in Means Test: Diff in Props |
| Paired Samplesbefore/after on same subjects | CI: Paired Diff | Test: Paired Diff |
StatLens Labs
Sampling Distributions
Draw samples, watch the sampling distribution build. See the Central Limit Theorem in action.
Why the Percentile CI Works
You read a confidence interval off the middle 95% of a bootstrap distribution — but why does that capture the truth? Build both distributions on one axis and check.
CIs by Repeated Sampling
Sample again and again from a population you choose. The middle 95% of the sample means is where the margin of error comes from — the picture the bootstrap is imitating.
Why ± 2 SE Works
The same question for the ± 2 SE interval, with no numbers in the way: an axis marked in standard errors, a statistic you drag, and two verdicts that are never allowed to disagree.
What Estimating the SE Costs
And then: where does the multiplier come from? Lay the margins of error against one ruler of true standard errors and watch the one you can actually compute come up short — for a mean, and for a proportion.
What Is a Randomization Test?
Shuffle cards, build a null distribution step by step, and see what a p-value really means.
Are You Psychic?
Predict coin flips, then use simulation to test whether your result could be due to chance. A gentle intro to hypothesis testing.
Decision Errors
Run many studies with or without a real effect. Watch detections, missed effects (Type II), and false alarms (Type I).
Theory-based tests and confidence intervals using t and z distributions. Part III, Chapters 10–13
| Data Type | Confidence Interval | Hypothesis Test |
|---|---|---|
| One Quantitativeone numeric variable | CI for a Mean | One-Sample t-Test |
| One Categoricalone yes/no variable | CI for a Proportion | One-Proportion z-Test |
| Compare Two Groupsnumeric outcome by group | CI: Diff in Means CI: Diff in Props | Two-Sample t-Test Two-Proportion z-Test |
| Paired Samplesbefore/after on same subjects | CI: Paired Diff | Paired t-Test |
See all Traditional Inference tools — CIs and tests →
Distribution Calculators
Normal
Probabilities and critical values for the normal distribution.
t
Student’s t-distribution with adjustable df.
StatLens Labs
Galton Board
Balls bounce left or right down a field of pegs and pile up into a binomial distribution — with the exact probabilities, and the normal approximation, drawn over the pile.
Law of Large Numbers
Toss a coin or roll a die over and over: the running proportion settles toward what the model predicts — while the running total wanders further from zero. A coin is never “due”.
Confidence Interval Coverage
Draw 100 confidence intervals and see what “95% confidence” really means.
Power & Error Visualizer
Visualize how α, sample size, and effect size determine power.
Conclusion Practice
Practice writing formal and practical conclusions for hypothesis tests.
Apply inference methods to chi-square, ANOVA, correlation, and regression — richer data structures, same simulation and theory-based approaches. Part IV, Chapters 14–20
| Topic | Explore | Simulation | Traditional |
|---|---|---|---|
| Chi-Squaretwo categorical variables | Two-Way Table | χ² Randomization | Chi-Square Test |
| Goodness-of-Fitone categorical variable vs. a distribution | One Category | χ² Goodness-of-Fit | χ² GOF Test |
| ANOVA3+ groups, numeric outcome | Compare Groups | ANOVA Randomization | ANOVA F-Test Multiple Comparisons |
| Correlation & Regressiontwo quantitative variables | Scatterplot | CI for Slope Test for Correlation | CI for Slope Regression Slope t-Test Multiple Regression |
See all Traditional Inference tools — CIs and tests →
Distribution Calculators
StatLens Labs
Guess the Correlation
See a scatterplot, guess r. Build your visual intuition for correlation.
Regression by Eye
Drag a line to fit the data. Toggle between squared and absolute errors, then compare to the optimal fit.
Scatterplot Editor
Drag, add, and remove data points and watch the least-squares line, r, and R² respond — feel leverage vs. influence.
Probability distributions, and the ideas behind hypothesis testing errors and power. Most of these chapters involve pencil-and-paper work — the tools below support the computational sections. Parts V–VI, Chapters 21–28
Distribution Calculators
Binomial
Discrete PMF, cumulative probabilities, normal approximation overlay.
Normal
Probabilities and critical values for the normal distribution.
All Distribution Calculators
Normal, t, chi-square, F, binomial, and power visualizer — all in one place.
StatLens Labs
Confidence Interval Coverage
Draw 100 confidence intervals and see what “95% confidence” really means.
Power & Error Visualizer
Visualize Type I error, Type II error, and power. Adjust α, sample size, and effect size to see how they interact.
Power Lab (simulation)
Run thousands of studies; watch the empirical power converge to the theory and the p-values dance around α.
Decision Errors
A plainer, jargon-light view of hits, missed effects, and false alarms across many studies.
Start by asking: what type of variable do I have? Each row shows all the tools for that situation.
| Data Type | Explore | Bootstrap CI | Randomization Test | Traditional CI | Traditional Test |
|---|---|---|---|---|---|
| One Quantitativeone numeric variable (heights, scores, prices…) | Descriptive Stats | CI for a Mean | Test for a Mean | CI for a Mean | One-Sample t-Test |
| One Categoricalone variable with categories (yes/no, color, party…) | Frequency Table | CI for a Proportion | Test for a Proportion | CI for a Proportion | One-Proportion z-Test |
| Compare Two Groupsnumeric outcome split by group (treatment vs. control…) | Compare Groups | CI: Diff in Means CI: Diff in Props | Test: Diff in Means Test: Diff in Props | CI: Diff in Means CI: Diff in Props | Two-Sample t-Test Two-Proportion z-Test |
| Paired Samplestwo measurements on the same subjects (before/after…) | Descriptive Statson the differences | CI: Paired Diff | Test: Paired Diff | CI: Paired Diff | Paired t-Test |
| Two Categoricaltwo category variables (group × outcome…) | Two-Way Table | — | χ² Randomization | — | χ² Test |
| One Categorical (fit)one category variable vs. a claimed distribution | One Category | — | χ² Goodness-of-Fit | — | χ² GOF Test |
| Multiple Groups3+ groups, numeric outcome (ANOVA) | Compare Groups | — | ANOVA Randomization | Pairwise CIsTukey / Bonferroni | ANOVA F-Test |
| Two Quantitativetwo numeric variables (x & y, scatterplot data…) | Scatterplot | CI for Slope | Test for Correlation | CI for Slope | Regression Slope t-Test Multiple Regression |
StatLens Labs
Dotplot Editor
Click to add and remove points. Watch how mean, median, and spread respond.
Guess the Correlation
See a scatterplot, guess r.
Regression by Eye
Drag a line to fit the data, then compare to the optimal fit.
Scatterplot Editor
Drag, add, and remove points; watch the line, r, and R² respond. Leverage vs. influence.
Are You Psychic?
Predict coin flips, then test if your result could be chance.
Stump the Chump
Fake a random sequence, then compare to a real one.
Conceptual Demonstrations see all →
Sampling Distributions
Draw samples, see the CLT in action.
Sampling Bias Lab
Gettysburg words: pick by eye vs. random.
Sampling Designs
Dig for rocks: simple random, stratified, cluster, multistage — what each costs and how precise it is.
CI Coverage
What “95% confidence” really means.
Why the Percentile CI Works
Why the middle 95% of a bootstrap distribution is a confidence interval.
Why ± 2 SE Works
The capture argument on a symbolic axis — no numbers.
What Estimating the SE Costs
Swapping s for σ, or p̂ for p, and what it costs the interval.
CIs by Repeated Sampling
Where the margin of error comes from, by sampling the population itself.
Law of Large Numbers
The proportion settles; the running total does not. A coin is never “due”.
Galton Board
Balls bounce down a peg field and pile up into the binomial distribution.
Decision Errors
Run many studies; see hits, misses, and false alarms.
Power Lab
Empirical power converging to theory; the p-value dance.
Randomization Test
Shuffle cards, build a null distribution.
Distribution Calculators
Normal
z probabilities and critical values.
t
t-distribution with adjustable df.
Chi-Square
χ² distribution calculator.
F
F distribution calculator.
Binomial
PMF, cumulative P, normal overlay.
Power & Error
Visualize α, β, and power trade-offs.
Practice & Games see all →
Start by asking: what parameter am I estimating or testing?
| Parameter | Explore | Bootstrap CI | Randomization Test | Traditional CI | Traditional Test |
|---|---|---|---|---|---|
| One Mean μ | Descriptive Stats | CI for a Mean | Test for a Mean | CI for a Mean | One-Sample t-Test |
| One Proportion p | Frequency Table | CI for a Proportion | Test for a Proportion | CI for a Proportion | One-Proportion z-Test |
| Diff in Means μ₁ − μ₂ | Compare Groups | CI: Diff in Means | Test: Diff in Means | CI: Diff in Means | Two-Sample t-Test |
| Paired Mean Diff μd | Descriptive Statson differences | CI: Paired Diff | Test: Paired Diff | CI: Paired Diff | Paired t-Test |
| Diff in Proportions p₁ − p₂ | Two-Way Table | CI: Diff in Props | Test: Diff in Props | CI: Diff in Props | Two-Proportion z-Test |
| Independence χ² | Two-Way Table | — | χ² Randomization | — | χ² Test |
| Goodness-of-Fit χ² | One Category | — | χ² Goodness-of-Fit | — | χ² GOF Test |
| ANOVA F | Compare Groups | — | ANOVA Randomization | Pairwise CIsTukey / Bonferroni | ANOVA F-Test |
| Slope / Correlation β₁, ρ | Scatterplot | CI for Slope | Test for Correlation | CI for Slope | Regression Slope t-Test Multiple Regression |
StatLens Labs
Dotplot Editor
Click to add and remove points. Watch how mean, median, and spread respond.
Guess the Correlation
See a scatterplot, guess r.
Regression by Eye
Drag a line to fit the data, then compare to the optimal fit.
Scatterplot Editor
Drag, add, and remove points; watch the line, r, and R² respond. Leverage vs. influence.
Are You Psychic?
Predict coin flips, then test if your result could be chance.
Stump the Chump
Fake a random sequence, then compare to a real one.
Conceptual Demonstrations see all →
Sampling Distributions
Draw samples, see the CLT in action.
Sampling Bias Lab
Gettysburg words: pick by eye vs. random.
Sampling Designs
Dig for rocks: simple random, stratified, cluster, multistage — what each costs and how precise it is.
CI Coverage
What “95% confidence” really means.
Why the Percentile CI Works
Why the middle 95% of a bootstrap distribution is a confidence interval.
Why ± 2 SE Works
The capture argument on a symbolic axis — no numbers.
What Estimating the SE Costs
Swapping s for σ, or p̂ for p, and what it costs the interval.
CIs by Repeated Sampling
Where the margin of error comes from, by sampling the population itself.
Law of Large Numbers
The proportion settles; the running total does not. A coin is never “due”.
Galton Board
Balls bounce down a peg field and pile up into the binomial distribution.
Decision Errors
Run many studies; see hits, misses, and false alarms.
Power Lab
Empirical power converging to theory; the p-value dance.
Randomization Test
Shuffle cards, build a null distribution.
Distribution Calculators
Normal
z probabilities and critical values.
t
t-distribution with adjustable df.
Chi-Square
χ² distribution calculator.
F
F distribution calculator.
Binomial
PMF, cumulative P, normal overlay.
Power & Error
Visualize α, β, and power trade-offs.
Practice & Games see all →