Errors and Power

Explore probabilities of Type I and Type II errors and connections to sample size, significance level and true parameter value.

Sampling Distributions and the Central Limit Theorem

Normal Distribution 

Explore how mean and standard deviation change the shape and find percentiles (critical values) or probabilities.

Binomial Distribution

Find the probability for the number of successes in n Bernoulli trials. Explore how the distribution depends on n and p.

Time Series

Plot a simple time series and add a smooth or linear trend.

Random Numbers 

Generate random numbers or flips of a (biased) coin. Keep track of generated numbers with a bar chart.

​​Exploratory Analysis (One and Two Samples), Random Numbers

Multiple Linear Regression

In the meantime, use the Multivariate Relationships app, which has some capablities to fit a multiple linear regression model with two explanatory variables.

Visualize and run a permutation test for testing independence in a contingency table using Pearson's Chi-squared statistic.

Chi-Squared Distribution 

Explore how the shape depends on the degrees of freedom. Find and visualize percentiles and probabilities.

t Distribution 

Explore how the degrees of freedom effect the shape and find percentiles, probabilities and P-values

Distributions: Explore the Shape, Find Probabilities and Percentiles

Scatterplots & Correlation

Construct interactive scatterplots, hover over points, move them around or overlay a smooth trend line. Find the correlation coefficient r. Built the sampling distribution of r via bootstrapping or permutation, one resample at a time.

Exponential Regression

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

Fit a logistic regression model with a single quantitative predictor. Obtain parameter estimates, a graph of the fitted probabilities and construct confidence intervals.

Bootstrap Confidence Intervals & Permutation Tests

Sampling Distribution for the 

Multivariate Relationships

Construct interactive scatterplots to explore the relationship between two quantitative variables while accounting for a third (categorical or quantitative) grouping variable.

Fisher's Exact Test

Visualize and run Fisher's exact test for

2 x 2 contingency tables.

For continuous variables. Choose from many different population distributions (or built your own) and explore the sampling distribution.

Association, Linear Regression and Logistic Regression

Interactive Web Apps

Inference for a Mean 

Find confidence intervals and test hypo-theses about a population mean. Visualize the interval or P-value.

Permutation Test for 

Sample Mean (Continuous Population)

Inference for a Proportion

Find confidence intervals and test hypo-theses about a population proportion. Visualize the interval or P-value.

Linear Regression 

Construct interactive scatterplots and super-impose a regression line. Obtain the regression equation, rr-squared and obtain predictions. Display & analyze residuals.

Bootstrap for One Sample

Create the bootstrap distribution of the mean, median or standard deviation and find the percentile confidence interval.

Compare Two Proportions

Confidence interval and significance test for the difference of two proportions. Independent & dependent samples.

Categorical Variables

Construct 2x2 contingency tables, obtain conditional proportions and get a bar graph. Find the difference or ratio of proportions. Built the sampling distribution via resampling.

Guess the Correlation

Randomly generate scatterplots to guess the correlation coefficient r. Optionally, display the regression line. How do your guesses correlate with the actual values?

Coming soon ...

Sampling Distribution for the

Independence (Categ. Data)

Poisson Distribution 

Explore how the shape of the Poisson Distribution depends on λ and find probabilities of various kinds

ANOVA (One-Way)

Analysis of Variance for one factor, including multiple comparisons of means (Tukey, Dunnett). (Under Construction)

Permutation Test

Visualize and run a permutation test comparing two samples with a quantitative response.

See how the sampling distribution builds up with repeated sampling and explore how its shape depends on n and p.

F Distribution 

Explore how the shape depends on the two sets of degrees of freedom. Find and visualize percentiles and probabilities.

One Sample Inference: Confidence Intervals & Significance Tests

Two Sample Inference: Confidence Intervals & Significance Tests for Comparing Groups

Explore Categorical Data

Construct frequency and contingency tables and bar graphs to explore distributions of categorical variables. For one or two variables. 

Compare Two Means

Confidence interval and significance test for the difference of two means. Independent & dependent samples.

Explore Quantitative Data

Find summary statistics and construct inter-active histograms, boxplots, dotplots or stem & leaf plots. For one or several samples. 

Explore Linear Regression 

Create scatterplots from scratch by clicking in an empty plot and creating points. Investigate the effect of outliers on the regression line. Simulate linear or non-linear relationships.

For discrete variables. Define your own discrete distribution (such as uniform or skewed) and explore the sampling distribution. 

Bootstrap for Two Samples

Create the bootstrap distribution for the difference of means or medians, and find percentile confidence intervals.

Association Between Two 

Coming soon ...

Explore statistical concepts in an interactive way

The following apps have graphs that update with clicks on buttons or sliders. Each explores a different statistical topic and allows results to be saved. ​Click on a picture to start the corresponding app.

Chi-squared Test

Test for independence, homogeneity or goodness of fit in contingency tables. Analyze observed & expected counts and residuals.

Explore Coverage 

What does 95% confidence mean? What affects the width of an interval? Visualize with intervals for proportions or means.

Sample Proportion

Sample Mean (Discrete Population)

Nonparametric Tests

In the meantime, the Boostrap for One Samples can do the Wilcoxon Test.

Sampling Distribution for the

Mean vs. Median

Explore the relationship between the mean and median for data coming from a variety of distributions, or enter your own data.