General principles
- The calculators show numerical results and, where relevant, formulas and an explanation of the interpretation.
- Sample sizes are rounded up to guarantee the minimum required under the assumptions entered.
- Intervals and tests depend on assumptions such as independence, randomization, approximate normality or sufficiently large expected counts.
- Results should be reviewed with methodological judgment, especially in clinical, regulatory, financial or high-impact studies.
Distributions and tables
The probability distribution pages compute density or mass, cumulative probability, tails and inverses whenever the distribution allows it. The statistical tables provide critical values and quantiles for common inference uses.
To avoid misinterpretation, always check the parameterization: for example, mean and standard deviation in the normal distribution, degrees of freedom in t, chi-square and F, or rate λ in Poisson and exponential.
Intervals, tests and sample sizes
Confidence intervals combine a point estimate, standard error and critical value. Hypothesis tests compare an observed statistic against a null distribution to obtain a p-value or critical region. Sample size tools estimate the minimum number needed based on confidence, margin of error, variability, power and minimum detectable effect.
If you change the confidence level, power or minimum detectable effect, the required sample size can vary substantially. It is therefore advisable to run a sensitivity analysis before finalizing a design.
Numerical precision and rounding
Calculations run in the browser using JavaScript and statistical libraries where needed. Results are formatted for human reading, so small differences may exist compared to specialized software due to rounding, floating-point precision or alternative parameterizations.
When a result will be published or used for an important decision, cross-check it with a second tool or statistical software and document the inputs used.
Limitations
- Not all calculators cover complex designs, stratification, clustering, weighting or corrections for multiple comparisons.
- A statistically significant result does not imply practical relevance or causality unless the design justifies it.
- In A/B testing, stopping an experiment as soon as significance is observed can inflate false positives.
- In small samples or extreme data, it is advisable to consider exact methods or simulations.
Documentation and external references
These sources were used as a reference for the implemented methods and may be useful for further study or to cross-check results:
- NIST/SEMATECH e-Handbook of Statistical Methods — official technical reference from the U.S. National Institute of Standards and Technology on applied statistical methods.
- SciPy stats (Python) — documentation of the reference statistical library in Python, with implementations of distributions, tests and density functions.
- CRAN — The Comprehensive R Archive Network — official repository of packages and documentation for the R language, the most widely used in academic and applied statistics.
- Statistics — Wikipedia — an encyclopedic starting point for general concepts and bibliographic references for each method.