Every algorithm and convention described on the methodology page traces to one of the sources below.
Algorithms
- Welford, B. P. (1962). "Note on a method for calculating corrected sums of squares and products." Technometrics, 4(3), 419–420. — the one-pass variance algorithm used throughout this site.
- Knuth, D. E. (1998). The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd ed., section 4.2.2. — the presentation of Welford's method that popularised it, and the analysis of why the naive formula fails.
- Chan, T. F., Golub, G. H., & LeVeque, R. J. (1983). "Algorithms for computing the sample variance: analysis and recommendations." The American Statistician, 37(3), 242–247. — the numerical comparison of variance algorithms.
- Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions, formula 7.1.26. — the error function approximation behind the normal CDF.
- Acklam, P. J. (2003). "An algorithm for computing the inverse normal cumulative distribution function." — the inverse normal approximation.
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed., sections 6.1 and 6.4. — the Lanczos log-gamma and the continued-fraction incomplete beta used for the t distribution.
- Hyndman, R. J., & Fan, Y. (1996). "Sample quantiles in statistical packages." The American Statistician, 50(4), 361–365. — the nine quantile definitions, and the source of the "type 7" naming used on the methodology page.
Statistical conventions
- NIST/SEMATECH (2012). e-Handbook of Statistical Methods. — reference definitions for standard deviation, pooled standard deviation and weighted statistics.
- Higgins, J. P. T., et al. (eds.) (2023). Cochrane Handbook for Systematic Reviews of Interventions, version 6.4, section 6.5.2. — recovering a standard deviation from a reported confidence interval or standard error.
- Wan, X., Wang, W., Liu, J., & Tong, T. (2014). "Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range." BMC Medical Research Methodology, 14, 135. — the range-to-SD and IQR-to-SD estimators implemented in the engine.
- Kish, L. (1965). Survey Sampling. — the effective sample size used for reliability weights.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed. — pooled standard deviation as the denominator of the standardised effect size.
Software behaviour referenced
- Microsoft. STDEV.S, STDEV.P, VAR.S, VAR.P and AGGREGATE function documentation.
- R Core Team. `quantile` documentation — the type 1–9 definitions.
- Texas Instruments. TI-84 Plus CE Graphing Calculator Guidebook — 1-Var Stats output and the Sx / σx labels.
- Casio. fx-991EX / fx-570EX Quick Start Guide, Statistics.
A note on how these are used
Citations here are to the sources actually consulted in building the calculators, not a reading list assembled afterwards. Where a source's convention differs from another's — the quantile definitions being the clearest case — the methodology page states which one this site follows and why, rather than presenting one choice as if it were the only one.