Statistics calculators and research tools that show their working

Yes. The distribution functions behind every p-value on this site are computed with series and continued-fraction expansions of the incomplete gamma and incomplete beta functions, which are accurate to roughly fourteen decimal places across the range students work in. Results agree with R, SPSS, and published statistical tables. What the tools cannot do is check whether the test you chose was the right one for your design, so read the assumptions section on each page before you report a number.

No, and this is the single most common misreading in student work. A p-value is the probability of getting a result at least as extreme as the one you observed, calculated on the assumption that the null hypothesis is true. It says nothing directly about the probability that the null hypothesis is true, and nothing about how large or important your effect is. That is why every calculator here also reports a confidence interval or an effect size.

No. Every page is a single self-contained HTML file with the calculation running in your own browser. Nothing you type is uploaded anywhere, there is no account, and no data leaves your device. You can save a page and it will keep working with no internet connection, which is useful in an exam hall or a lab with restricted network access.

It means your data did not provide enough evidence to reject the null hypothesis at the significance level you chose. A real effect can easily go undetected when the sample is small, the measurement is noisy, or the effect is genuinely modest. Report the effect size and the confidence interval alongside the p-value so a reader can see how much uncertainty is left, and use the sample size calculator before your next study rather than after.

It depends on how far from normal and how large your sample is. Because of the central limit theorem, tests on means become robust to non-normality as the sample grows, so a moderate departure with sixty or more observations per group is usually tolerable. With small samples, strong skew, or ordinal outcomes, a rank-based alternative such as the Mann-Whitney U test, the Wilcoxon signed-rank test, or Spearman's rho is the safer choice. The decision tree page walks you through this fork.

Report the test statistic with its degrees of freedom, the p-value, and an effect size, in that order: for example, t(24) = 2.31, p = .028, d = 0.47. Statistic symbols are italicised, p-values are written without a leading zero because they cannot exceed one, and exact p-values are preferred over threshold statements unless the value is very small, in which case p < .001 is standard. Confidence intervals are given in square brackets with the level stated.