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Art of Stat
Web Apps
Explore many statistical concepts in an interactive way. Use the apps to construct graphs, obtain summary statistics, find probabilities under the normal distribution, get confidence intervals or fit linear regression models. Take screenshots of your analyses or download graphs as .png files.
Exploratory Data Analysis
Explore Categorical Data
Explore Quantitative Data
Explore Quantitative Data
Construct frequency and contingency tables and bar graphs to explore distributions of categorical variables.
For one or two variables.
Explore Quantitative Data
Explore Quantitative Data
Explore Quantitative Data
Find summary statistics and construct interactive histograms, boxplots, dotplots or stem & leaf plots.
For one, two, or more groups.
Time Series Plots
Explore Quantitative Data
Time Series Plots
Plot a simple time series and add a smooth or linear trend. Use preloaded data or provide your own.
Mean vs. Median
Mean vs. Median
Time Series Plots
Explore the relationship between the mean and median for data coming from a variety of distributions, or enter your own data
Random Numbers
Mean vs. Median
Random Numbers
Generate random numbers or flips of a (biased) coin. Keep track of generated numbers with a bar chart.
Association, Correlation & Regression
Association Between Two Categorical Variables
Association Between Two Categorical Variables
Association Between Two Categorical Variables
Construct 2 x 2 contingency tables, obtain conditional proportions and get a bar graph. Find the difference or ratio of proportions to describe the strength of the association. Built the sampling distribution of the difference or ratio via resampling.
Scatterplots & Correlation
Association Between Two Categorical Variables
Association Between Two Categorical Variables
Construct interactive scatterplots, hover over points, move them around (or remove them) and overlay a smooth trend line. Find the correlation coefficient r and see if it is robust to outliers. Built the sampling distribution of r via resampling.
Guess the Correlation
Association Between Two Categorical Variables
Explore Linear Regression
Randomly generate scatterplots to guess the correlation coefficient r. Optionally, display the regression line. How do your guesses correlate with the actual values?
Explore Linear Regression
Explore Linear Regression
Explore Linear Regression
Create scatterplots from scratch by clicking in an empty plot to add or remove points. Investigate the effect of outliers on the correlation coefficient or regression line. Simulate linear or non-linear relationships.
Linear Regression
Explore Linear Regression
Multivariate Relationships
Fit a simple linear regression model and obtain the regression equation and related statistics such as r-squared. Make predictions and construct confidence intervals. Display and analyze residuals.
Multivariate Relationships
Explore Linear Regression
Multivariate Relationships
Construct interactive scatterplots to explore the relationship between two quantitative variables, while accounting for a third (categorical or quantitative) grouping variable. Multiple Linear Regression.
Multiple Linear Regression
Multiple Linear Regression
Multiple Linear Regression
Under Construction. In the meantime, use the Multivariate Relationships app, which has some capablities to fit a multiple linear regression model with two explanatory variables.
Exponential Regression
Multiple Linear Regression
Multiple Linear Regression
Under Construction. Fit and visualize a simple exponential regression model to data such as the number of COVID-19 infections in New York City in March 2020 (Example 16, Chapter 13).
Logistic Regression
Multiple Linear Regression
Logistic Regression
Under Construction. Fit a logistic regression model with a single quantitative predictor. Obtain parameter estimates, a graph of the fitted probabilities and construct confidence intervals.
Distributions: Explore Shapes & Find Probabilities
Binomial Distribution
Binomial Distribution
Binomial Distribution
Explore how the shape of the Binomial distribution depends on the parameter n (the sample size) and p (the probability of success in a Bernoulli trial). Find and visualize probabilities of various kinds.
Normal Distribution
Binomial Distribution
Binomial Distribution
See how the shape of the normal distribution depends on the mean and standard deviation. Find and visualize one- and two-tailed probabilities and percentiles (critical values).
t Distribution
Binomial Distribution
Chi-Squared Distribution
See how the shape of the t distribution depends on the degrees of freedom. Find and visualize one-and two-tailed probabilities and percentiles (critical values).
Chi-Squared Distribution
Chi-Squared Distribution
Chi-Squared Distribution
See how the shape of the Chi-Squared distribution depends on the degrees of freedom. Find and visualize probabilities and percentiles (critical values).
F Distribution
Chi-Squared Distribution
Poisson Distribution
See how the shape of the F distribution depends on the degrees of freedom. Find and visualize probabilities and percentiles (critical values).
Poisson Distribution
Chi-Squared Distribution
Poisson Distribution
Explore how the shape of the Poisson distribution depends on the parameter λ (the mean). Find and visualize probabilities of various kinds.
Sampling Distributions and the Central Limit Theorem
Sampling Distribution of the Sample Proportion
Sampling Distribution of the Sample Mean (Continuous Population)
Sampling Distribution of the Sample Mean (Continuous Population)
Experience how the sampling distribution of the sample proportion builds up one sample at a time. Use sliders to explore the shape of the sampling distribution as the sample size n increases, or as the population proportion p changes. Overlay a normal distribution to explore the Central Limit Theorem.
Sampling Distribution of the Sample Mean (Continuous Population)
Sampling Distribution of the Sample Mean (Continuous Population)
Sampling Distribution of the Sample Mean (Continuous Population)
Experience how the sampling distribution of the sample mean builds up one sample at a time. Use a variety of real or theoretical continuous population distributions (or create your own) to draw samples from. Move sliders to explore when the Central Limit Theorem kicks in.
Sampling Distribution of the Sample Mean (Discrete Population)
Sampling Distribution of the Sample Mean (Continuous Population)
Sampling Distribution of the Sample Mean (Discrete Population)
Experience how the sampling distribution of the sample mean builds up one sample at a time. Use a variety of real or theoretical discrete population distributions (or create your own) to draw samples from. Move sliders to explore when the Central Limit Theorem kicks in.
Confidence Intervals & Significance Tests (One Sample)
Inference for a Proportion
Inference for a Proportion
Inference for a Proportion
Find confidence intervals or test hypotheses about a population proportion. Obtain the margin of error or the z-test statistic and visualize the interval or the P-value on a graph.
Inference for a Mean
Inference for a Proportion
Inference for a Proportion
Find confidence intervals or test hypotheses about a population mean. Enter your own data or summary statistics. Use plots to check assumptions and visualize the interval or the P-value on a graph.
Bootstrap for One Sample
Inference for a Proportion
Bootstrap for One Sample
Create and visualize, step-by-step, the bootstrap distribution of the mean, median or standard deviation. Obtain summary statistics and find a percentile confidence interval.
Explore Coverage
Explore Coverage
Bootstrap for One Sample
What does "95% confidence" mean? What affects the width of an interval? Explore these concepts for confidence intervals of proportions or means, using sliders to change parameters or the sample size.
Errors and Power
Explore Coverage
Errors and Power
Visualize and explore relationships between Type I and Type II errors and the power of a test for proportions or means. See how they depend on sample size and the true values of population parameters.
Confidence Intervals & Tests Comparing Two Groups
Compare Two Proportions
Bootstrap for Two Samples
Compare Two Proportions
Confidence intervals or hypotheses tests about the difference of two population proportions. Obtain the margin of error or the z-test statistic and visualize the interval or the P-value on a graph.
For two independent or dependent samples.
Compare Two Means
Bootstrap for Two Samples
Compare Two Proportions
Confidence intervals or hypotheses tests about the difference of two population means. Enter your own data or summary statistics. Visualize the interval or the P-value on a graph.
For two independent or dependent samples.
Bootstrap for Two Samples
Bootstrap for Two Samples
Bootstrap for Two Samples
Create and visualize, step-by-step, the bootstrap distribution for the difference of means or medians. Obtain summary statistics (such as percentiles) of the bootstrap distribution and find a percentile confidence interval.
Permutation Test
Difference & Ratio of Proportions
Bootstrap for Two Samples
Create and visualize, step-by-step, the permutation distribution for statistics such as the difference of means or medians or the t-statistic. Obtain summary statistics (such as percentiles) and find the permutation P-value.
Difference & Ratio of Proportions
Difference & Ratio of Proportions
Difference & Ratio of Proportions
Find the bootstrap or permutation distribution for the difference or ratio of two proportions, or for the odds-ratio. Use these to obtain percentile confidence intervals or permutation P-values for testing whether there is no association.
Fisher's Exact Test
Difference & Ratio of Proportions
Difference & Ratio of Proportions
Visualize and run Fisher's exact test for
2 x 2 contingency tables. Obtain the exact P-value for one- or two-sided tests.
Inference for Comparing Several Groups
Chi-Squared Test
Kruskal-Wallis Test
Chi-Squared Test
Test for independence, homogeneity or goodness of fit in contingency tables. Enter your own data as raw observations or as a contingency table. Obtain observed and expected counts and find residuals.
ANOVA (One-Way)
Kruskal-Wallis Test
Chi-Squared Test
Obtain the ANOVA table, F-statistic and side-by-side boxplots to check assumptions. Carry out pairwise comparisons, including simultaneous confidence intervals for pairwise differences of means.
Kruskal-Wallis Test
Kruskal-Wallis Test
Permutation Test for Independence
Under Construction
Permutation Test for Independence
Permutation Test for Independence
Permutation Test for Independence
Visualize and run a permutation test for testing independence in a contingency table. Obtain the permutation P-value of the Pearson Chi-squared statistic without relying on asymptotical results.
Permutation Test for Independence