Change the data and watch the statistics respond. Every view compares all data with the included data, so you can see which numbers move a lot when a single point changes and which barely move.
Terms with a dotted underline explain themselves: hover over them, tab to them, or tap them.
Rules are applied to the currently included values. Pressing the same rule again can remove more points, because the mean and SD have changed.
Pearson's r uses the actual values, so one extreme point can pull it a long way. Spearman's ρ uses ranks, so an extreme point counts no more than any other.
Two groups are compared with an independent-samples t-test. Drag the orange treatment curve to change the difference between means, and drag its round handle to change the spread. Power is the chance that a study of this size finds a significant result when the effect is real.
Power figures use the exact noncentral t distribution. The curves in the middle chart are normal approximations, drawn on the scale of the observed effect size so you can see them narrow as n grows.
A significance test works like a diagnostic test for effects. Its sensitivity is the power and its specificity is 1 − α. But how often a significant result is a false alarm also depends on how many of the hypotheses we test are true to begin with: the prior probability, like disease prevalence in screening. Everything below uses the effect, n and α set above.
Sign and exaggeration errors use the normal approximation shown in the middle chart above.
Draw repeated samples from a population whose true mean μ we know. Each sample gives a mean and a confidence interval. The standard error is the spread of those sample means; the confidence level is the long-run proportion of intervals that contain μ.
With a skewed or outlier-prone population and a small n, the t-based interval can contain μ less often than the stated level. Increase n and the coverage recovers (the central limit theorem at work).
Three sample sizes from the same population, at the confidence level set above, all drawn on one shared axis. Draw samples for all three at once and compare how wide their intervals are and how often they contain μ.
An a-priori power analysis is done before collecting data. Pick the design that matches your study, say how big an effect you want to be able to detect, and the calculator finds the sample size. You can also turn it around: find the power of a sample you already have in mind, or the smallest effect that sample could reliably detect. Everything updates as you change it.
A causal diagram (DAG) writes down what we believe causes what. Draw one, choose the exposure and the outcome, then choose what to adjust for. The tool traces every path between them, simulates data from your diagram and fits the regression, so you can see whether the coefficient recovers the true causal effect.
Only the exposure's coefficient is the target. The covariates are there to block paths, so their own coefficients are not their causal effects (the Table 2 fallacy).