Statistical results are only useful if they're correct. This page documents exactly how Stat-Tools computes its numbers, how we check them, and where our methods come from — so you can trust (and verify) what you get.
How the calculations are performed
Stat-Tools uses two computation paths, both built on well-established algorithms:
- Inferential statistics — t-tests, ANOVA, chi-square, correlation, regression, confidence intervals, effect sizes, sample size, Mann-Whitney and the two-proportions z-test — run on a Python 3 backend using SciPy and NumPy, the same scientific-computing libraries used by researchers and data scientists worldwide. We rely on SciPy's peer-reviewed implementations rather than re-deriving formulas by hand.
- Probability, combinatorics and distribution tools — the probability, permutations/combinations, binomial, Poisson, and distribution-explorer tools — compute directly in your browser using exact closed-form formulas. For factorials and binomial coefficients we use exact big-integer arithmetic (no floating-point rounding), and for continuous densities we use numerically stable techniques such as the log-gamma function.
How results are validated
Every calculator's output is cross-checked against a reference implementation before it ships:
- Backend results are compared directly to SciPy, whose algorithms match those in R, SPSS, Stata, and Excel's Analysis ToolPak — so a t-statistic or p-value here should equal what those packages produce for the same input.
- In-browser tools are checked against the same SciPy reference across a range of inputs, including edge cases (tiny samples, extreme probabilities, boundary values) where naïve formulas tend to break.
If you ever find a result that disagrees with R, SPSS, or a textbook, please tell us at [email protected] — we treat every discrepancy report as a bug and investigate it.
Precision and rounding
Calculations are carried out at full machine (or exact integer) precision internally. Only the displayed values are rounded — typically to 4–6 significant figures — so what you see on screen may be rounded, but the computation behind it is not. Very large or very small numbers are shown in scientific notation to avoid misleading zeros.
How our guides are written
Our plain-English guides are written to be correct first and readable second. Every definition, formula, and rule of thumb is checked against standard references before publication, and we deliberately flag common misconceptions (for example, that a p-value is not the probability the null hypothesis is true). We date each guide and revise it when we find an error or a clearer explanation.
Sources & further reading
Our methods and explanations are grounded in widely used, authoritative references:
- NIST/SEMATECH e-Handbook of Statistical Methods — the U.S. National Institute of Standards and Technology's reference handbook.
- OpenStax — Introductory Statistics — a peer-reviewed, openly licensed textbook.
- SciPy
statsdocumentation — the reference implementation our backend uses.
Limitations & responsible use
Stat-Tools is a free educational resource, not a substitute for professional statistical consulting. A calculator can return the right number for the wrong test — choosing an appropriate method, checking its assumptions, and interpreting results in context still require judgement. For high-stakes research, clinical, or business decisions, please have your analysis reviewed by a qualified statistician. Use our test chooser and guides as a starting point, not the final word.
Corrections
Accuracy is a moving target and we'd rather know when we're wrong. Report any error — a calculation, a formula, a definition, a typo — to [email protected] and we'll fix it and credit the fix.
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