Why this page exists
A statistics calculator that will not tell you what it computed is asking for trust it has not earned. This page lists every algorithm used on the site, its accuracy, and the cases where it is the wrong tool. Where a choice was made that affects results — Student's versus Welch's, corrected versus uncorrected chi-square — the choice and its consequence are stated rather than buried.
Numerical methods
| Quantity | Method | Accuracy |
|---|---|---|
| Normal CDF Φ(z) | Abramowitz & Stegun 7.1.26 rational approximation to erf | ~1.5 × 10⁻⁷ absolute |
| Inverse normal CDF | Acklam's algorithm | ~1.15 × 10⁻⁹ relative |
| t and F distributions | Regularised incomplete beta function | 10⁻¹⁴ relative tolerance |
| Chi-square distribution | Regularised lower incomplete gamma function | 10⁻¹⁴ relative tolerance |
| ln Γ(x) | Lanczos approximation | ~10⁻¹⁵ relative |
The incomplete beta and gamma functions each use either a series expansion or a continued-fraction expansion depending on argument size, following the standard treatment in Numerical Recipes, capped at 500 iterations. All results are displayed to two decimal places, or three for p-values, which is well inside the error of every method above — the approximations are not the limiting factor in anything this site displays.
Choices that affect your results
Two-sample t-test: Student's, not Welch's
The two-sample test assumes equal population variances and pools them. Current methodological guidance generally recommends Welch's test as the default, because it is barely less powerful when variances are equal and substantially more reliable when they are not. If your group standard deviations differ by more than roughly a factor of two, or the groups are both unequal in variance and unequal in size, the p-value here will be wrong — sometimes optimistically.
2×2 chi-square: no continuity correction
The uncorrected Pearson statistic is reported. With expected cell counts below about 5 this is anti-conservative — it will reach significance too easily — and Fisher's exact test is the appropriate alternative.
Power analysis: normal approximation
Sample size and power use the normal-approximation method from Cohen (1988) rather than the exact non-central t and F distributions. Against G*Power, results typically differ by one to two participants per group, with the gap widest at small n and low power. Always round up, and verify in dedicated software before submitting a figure for a grant, ethics application or pre-registration.
Effect size conversion: equal group sizes assumed
The d ↔ r conversion uses r = d / √(d² + 4). The constant 4 is a = (n₁ + n₂)² / (n₁ n₂), which equals 4 only when the groups are the same size. For badly unbalanced designs, compute a from your actual group sizes.
Quartiles: linear interpolation
Quartiles use the linear interpolation method, matching R's type 7 and Excel's QUARTILE.INC. Other packages use different conventions, so quartile and IQR values can differ slightly on small samples.
Histogram binning: Sturges' rule
Bin count is ⌈log₂(n) + 1⌉. This under-bins large (n above roughly 200) and skewed datasets; the Freedman–Diaconis rule is generally better for those.
Box plot outliers: Tukey's 1.5 × IQR
Points beyond 1.5 × IQR from the nearer quartile are flagged. In genuinely normal data roughly 0.7% of points will be flagged by chance, so a flag is a prompt to investigate, not evidence of an error.
Admission reference data
The admission pages mix three kinds of information, and they do not carry equal weight:
- Published conversion rules — the JUPAS points scale, the hensachi formula, the CSAT standard score formula, the 五標 percentile definitions. These are fixed, documented and reproduced exactly.
- Verified per-programme figures — a small number of specific admission statistics taken directly from the institution's own published documents, each with a source link on the row. Where sources conflicted, the entry was removed rather than reconciled.
- Placement bands — approximate ranges synthesised from public sources. These are not official, shift annually, and are labelled as such on every page that shows them.
Reference datasets carry an inline freshness marker recording when the entry was last verified and how often it should be rechecked. A scheduled quarterly job flags entries past their review interval so that stale figures surface rather than quietly ageing.
Deliberate gaps
- No Gaokao converter. Provinces use different papers, totals and cut-off lines, so no cross-province conversion is valid. The Gaokao page explains the tier system and links to provincial score lookups instead.
- No non-parametric tests. Mann–Whitney, Wilcoxon signed-rank and Kruskal–Wallis are not implemented. The test wizard names them and directs you elsewhere when they are what you need.
- No post-hoc tests after ANOVA. A significant ANOVA tells you some pair of groups differs, not which. Tukey's HSD and equivalents are not provided.
- No confidence intervals on effect sizes. Point estimates only.
Corrections
If a formula, figure or citation on this site is wrong, please report it via the contact page. Errors in the mathematics are treated as urgent; errors in reference data are corrected and the verification date updated.
Primary references
- Abramowitz, M. & Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards.
- Acklam, P. J. (2003). An algorithm for computing the inverse normal cumulative distribution function.
- Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (2007). Numerical Recipes (3rd ed.). Cambridge University Press.
- Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum.
- Tukey, J. W. (1977). Exploratory Data Analysis. Addison-Wesley.
- Delacre, M., Lakens, D. & Leys, C. (2017). Why psychologists should by default use Welch's t-test. International Review of Social Psychology, 30(1), 92–101.
- Hyndman, R. J. & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365.