Bell Statistics

Statistics calculators

Sixteen calculators for designing an experiment and for analysing one after it has run. Every page does both, shows the formula it used, and names the method — so you can tell somebody where the number came from.

Most online calculators give you a number and no way to check it. These give you the number, the formula behind it, a worked example, and the name of the specific method — Welch rather than Student, Wilson rather than Wald, Schoenfeld rather than Freedman — because when a colleague's tool disagrees in the third decimal, the method is the only thing that settles it.

Each calculator has two tabs. The sample-size tab answers the question you should be asking before the experiment: how many observations do I need to detect a difference worth acting on? The analysis tab answers the one you ask afterwards: given what I actually saw, what can I conclude? They share the same distribution code, so the design and the analysis cannot quietly disagree.

If you are not sure which test you need, start with what your outcome looks like. A rate or a proportion — conversion, click-through, retention — points at the A/B test sample size calculator or the chi-square test. A measured quantity like revenue or duration points at the two-sample t-test. If your question is "are these two the same?" rather than "do they differ?", no ordinary significance test answers it and you want an equivalence test.

We build measurement systems for a living — see our A/B testing work or the case studies. These tools are the parts of that work that fit on one page.

Rates and proportions

Conversion, click-through, retention — anything counted rather than measured. Start here for most product experiments.

  • A/B test sample size

    Size a two-proportion experiment before you launch, then read the lift and its interval once it lands.

  • One-proportion z-test

    Test one observed rate against a fixed target — an SLA, a benchmark, a contractual floor.

  • Chi-square test

    Test a contingency table of counts for association — any number of rows and columns.

  • Fisher's exact test

    The right test for a 2×2 table of small counts — exact p-values, no normal approximation.

  • McNemar's test

    Paired yes/no data — only the pairs that disagree carry any information, and this sizes on them.

  • Proportion confidence interval

    A defensible interval around one rate — Wilson, Agresti-Coull, Jeffreys and Clopper-Pearson, side by side.

Means and distributions

Revenue per user, session length, order value. Continuous outcomes, and the rank-based alternative for when the distribution misbehaves.

  • Two-sample t-test

    Compare the average of two independent groups — plan the sample size, then test the result.

  • Paired t-test

    Before-and-after or matched pairs — size the study on the difference SD, then test it.

  • One-way ANOVA

    Three or more independent groups on one continuous outcome — size it, then run the F test.

  • Wilcoxon test

    Rank-sum (Mann-Whitney) and signed-rank tests, with the exact distribution for small samples.

Equivalence and non-inferiority

For proving two options are close enough, or that a change is not meaningfully worse. A non-significant test does not show this — these do.

  • Equivalence: two means

    Show two averages are close enough to be interchangeable — set the margin, size the study, run both one-sided tests.

  • Equivalence: two proportions

    Prove two rates are close enough to swap — margin in percentage points, sample size, then both one-sided tests.

  • Non-inferiority: two means

    Show an average is not meaningfully worse — one margin, one direction, one-sided p-value and confidence bound.

  • Non-inferiority: two proportions

    Show a rate has not dropped by more than you can afford — margin, direction, sample size and confidence bound.

Relationships and time to event

Association between two variables, and survival or retention curves where what matters is how long something lasts.

  • Correlation test

    Pearson r or Spearman rho, with the Fisher-z interval that says how little a small sample knows.

  • Log-rank test

    Compare two survival, retention or churn curves — events needed to plan, chi-square and p-value to analyse.

Frequently asked questions

Are these calculators free?
Yes, all of them, with no sign-up and no limits. They run entirely in your browser — nothing you type is sent to us or to anyone else, which also means you can use them on data you would not paste into a third-party website.
How do I know which test to use?
Work from three things: what the outcome looks like, how many groups you are comparing, and whether the observations are paired. A counted outcome across two independent groups is a proportions test; a measured outcome across two independent groups is a two-sample t-test; the same subjects measured twice is a paired test. Each calculator's page opens with a section on when to use it and when to use something else instead.
Why does every calculator have two tabs?
Because sample size and analysis are the same statistics asked in opposite directions, and separating them across different tools is how people end up analysing an experiment with a method that does not match the one they designed it with. Sharing the code makes that mismatch impossible.
How do I know the numbers are right?
Every calculation is checked against published reference values from R in an automated test suite that runs before anything is deployed, and each page names the exact method used so you can reproduce it. Where authorities genuinely disagree — McNemar sample size, or the two-sided convention for Fisher's exact test — the page says which convention it follows and who disagrees.
Can I share a result with someone?
Yes. The inputs are kept in the page's address, so copying the URL from the address bar sends someone the same calculation with the same numbers filled in. Pasted raw data is deliberately excluded from the link, so sharing a result never shares the underlying observations.

A calculator answers one question. We answer the rest.

Sample size is the easy part. Choosing the metric, handling the users who appear in both arms, deciding what to do when the result lands in the awkward middle — that is the work. If you would rather have that designed properly the first time, talk to us.