Math & Statistics

Statistics & Probability Calculators

Calculators for summarising data, testing hypotheses and working with probability. The Mean, Median and Mode Calculator and the Standard Deviation Calculator summarise a sample; the Percentile Calculator and the Z-Score Calculator place one value within it. For estimates, the Confidence Interval Calculator and the Margin of Error Calculator give the range a population value is likely to fall in, and the Sample Size Calculator works back from the precision you need to the number of observations. The t-test Calculator, the Chi-Square Calculator and the One-Way ANOVA Calculator report a test statistic, its degrees of freedom and a p-value, and the p-value Calculator converts any test statistic into one. The Linear Regression Calculator and the Correlation Coefficient Calculator fit and measure relationships between two variables. The distribution calculators, from the Normal Distribution Calculator to the Binomial Distribution Calculator and the Poisson Distribution Calculator, return probabilities and quantiles, and the Bayes' Theorem Calculator updates a probability on new evidence.

Statistics

Standard Deviation CalculatorSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
Mean, Median and Mode CalculatorAll three centres marked on your own data, every mode rather than just the first, and the mean-median gap read as a direct measure of skew.
Probability CalculatorTwo events, repeated trials and Bayes, with the three usual errors handled — the dropped overlap in P(A or B), n×p instead of the complement, and the base rate that makes a 99% test 17% right.
Z-Score CalculatorA z-score from your data or from a known mean and SD — with the normal-table percentile checked against the share of your data that actually falls below it, and a warning when they disagree.
Percent Error CalculatorPercent error against an accepted value, rounded to your significant figures, for one measurement, repeated trials, or a whole column of pairs.
Variance CalculatorSample and population variance from your data, with a live simulation that shows exactly how much the wrong divisor costs — 20% low at n = 5, closing as the sample grows.
Percentile CalculatorThe value at any percentile and the percentile rank of any value, with all nine standard methods — and the round trip that shows the two directions are not inverses.
Confidence Interval CalculatorIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
Sample Size CalculatorResponses needed for a target margin of error, with the finite-population correction and a table of the whole cost curve — because n scales with 1/margin², so the last point of precision costs more than the first ten.
Cross-Validation CalculatorRun k-fold cross-validation over a penalty path and repeat it across twelve fold assignments, so the variability of the chosen tuning parameter is visible.
Partial Least Squares CalculatorBuild regression components against the outcome rather than from predictor variance, with the unsupervised decomposition computed beside it for comparison.
Principal Component Regression CalculatorRegress on principal components with each component's variance share printed beside its correlation with the outcome, and the error at every component count.
Elastic Net CalculatorFit an elastic net across the mixing dial from pure lasso to pure ridge, with selection stability and cross-validated error computed at each end.
Lasso Regression CalculatorFit a lasso across a penalty path with cross-validation, and see how often each selected predictor survives when the rows are resampled.
Ridge Regression CalculatorFit ridge regression across a penalty path chosen by cross-validation, with least squares beside it and the shrinkage applied to every coefficient shown.
Tobit CalculatorFit a censored regression to an outcome piled at a limit, with both least-squares alternatives beside it and the shortfall in each measured.
Zero-Inflated CalculatorFit a zero-inflated Poisson model, with the zeros a plain Poisson would predict against the zeros actually observed and a Vuong comparison.
Negative Binomial Regression CalculatorFit a negative binomial model to overdispersed counts, with the Poisson fit beside it and the ratio between their standard errors per coefficient.
Multinomial Logistic CalculatorFit a multinomial model to an unordered outcome, with odds ratios per category and a measured demonstration that the baseline rewrites every coefficient.
Ordinal Regression CalculatorFit a proportional-odds model to an ordered outcome, with the separate binary fits at each cutpoint and a test of the assumption they rest on.
Quantile Regression CalculatorFit a slope at the 10th, 25th, 50th, 75th and 90th percentiles with least squares beside it, and see what one outlier does to each.
Cluster Sample Size CalculatorClusters needed for a target power, with the individual-randomisation comparison and the whole trade-off between more clusters and bigger ones.
Design Effect CalculatorThe design effect 1 + (m − 1)ρ with the effective sample size it implies, the ceiling of 1/ρ observations per cluster, and the gain from each extra person.
Mixed Model CalculatorA random-intercept model from variance components, with the intraclass correlation, the design effect it implies, both standard errors and shrunk estimates.
Survival Sample Size CalculatorSchoenfeld's event requirement converted to an enrolment target, with the sensitivity to the hazard ratio and the power achieved at every event count.
Competing Risks CalculatorAalen-Johansen cumulative incidence with the naive 1 − Kaplan-Meier beside it for every cause, and the probability identity printed as a check on both.
Cox Regression CalculatorProportional hazards by partial likelihood with hazard ratios, concordance, and the Schoenfeld residual test computed on every fit rather than offered as an extra.
Odds Ratio to Risk Ratio CalculatorConvert an odds ratio to a risk ratio at any baseline risk, with the interval converted too and the same odds ratio shown across eleven baselines.
Log Loss CalculatorLog loss with the Brier score on the same forecasts, the penalty each applies at increasing confidence, and how much of the total a single case carries.
Brier Score CalculatorThe Brier score split into reliability, resolution and uncertainty, with the term the decomposition drops printed rather than absorbed, plus a skill score.
Hosmer-Lemeshow CalculatorThe Hosmer-Lemeshow calibration test with the group count swept from five to fifteen, per-group promised-against-observed rates, and the Brier reliability alongside.
Poisson Regression CalculatorPoisson regression with rate ratios, the Pearson dispersion check, and quasi-Poisson standard errors printed beside the Poisson ones for every coefficient.
Logistic Regression CalculatorLogistic regression with odds ratios converted to risk ratios at your own event rate, a likelihood ratio test, AUC, and separation reported rather than hidden.
VIF CalculatorVariance inflation factors with Belsley-Kuh-Welsch condition indices and variance-decomposition proportions, so a near-dependency is named rather than only detected.
White Test CalculatorWhite's heteroscedasticity test with squares and cross-products in the auxiliary regression, beside Breusch-Pagan, so the generality and its cost in degrees of freedom are both visible.
Breusch-Pagan CalculatorThe Breusch-Pagan test in both forms — Koenker's studentised statistic and the original 1979 chi-square — with White's test alongside and a residual plot.
Influence Diagnostics CalculatorDFFITS, DFBETAS and the covariance ratio in standard-error units beside Cook's distance, so a row that moves one coefficient rather than the whole fit is visible.
Cook's Distance CalculatorCook's distance for every row against both cutoffs in circulation, with the leverage and studentised residual it is built from and the F percentile it corresponds to.
Leverage CalculatorHat values for every observation with the 2p/n and 3p/n cutoffs, the studentised residuals alongside, and the Σh = p identity printed as a check.
Covariance Matrix CalculatorSample covariance and correlation matrices with eigenvalues, determinant and condition number, so redundancy spread across several variables cannot hide from a pairwise scan.
Distance Correlation CalculatorDistance correlation with a permutation test and Pearson's r on the same data, detecting curved and variance-based dependence that a correlation scores as zero.
Multiple Correlation CalculatorMultiple R with adjusted R-squared, an F test, and a per-predictor table of simple, partial and unique contributions beside the variance inflation factors.
Partial Correlation CalculatorPartial and semi-partial correlation with any number of controls, computed from residuals, with the raw correlation and the share each control already explains.
Hotelling T-Squared CalculatorOne-sample and two-sample multivariate mean tests with every single-variable t test printed alongside, so the joint answer can be compared against the alternative.
Mahalanobis Distance CalculatorMultivariate outlier distance with the chi-square cutoff, each row's largest z-score beside it, and a count of the points only the joint screen finds.
Gauge R&R CalculatorANOVA gauge repeatability and reproducibility with both acceptance criteria, the operator-by-part interaction tested, and the full variance decomposition.
Theil-Sen CalculatorThe median-of-slopes estimator with a distribution-free interval, least squares beside it, and a leave-one-out table naming what each fit depends on.
Concordance Correlation CalculatorLin's concordance correlation with its exact decomposition into precision and accuracy, the scale and location shifts separated, and a bounded interval.
Passing-Bablok CalculatorRobust method-comparison regression with rank-based intervals, the symmetry shift reported, and the pairwise-slope distribution shown.
Deming Regression CalculatorRegression when both variables carry error, with a settable error-variance ratio, jackknife intervals and least squares shown as the limiting case.
Bland-Altman CalculatorLimits of agreement with confidence intervals on the limits themselves, a proportional-bias test and the correlation printed beside them for contrast.
Cliff's Delta CalculatorNon-parametric effect size from the full pairwise win–loss–tie count, with a DeLong interval, magnitude bands and Cohen's d for comparison.
Hedges' g CalculatorBias-corrected standardised mean difference using the exact gamma correction, with Cohen's d beside it and a confidence interval.
Fleiss' Kappa CalculatorAgreement among any number of raters, with a per-category breakdown, a bootstrap interval over subjects and both agreement figures shown.
Egger's Test CalculatorEgger's regression test for funnel-plot asymmetry, with the intercept, its interval, the plotted regression and I² beside it.
I² CalculatorHeterogeneity across studies: I² with a confidence interval, Cochran's Q, τ², H and every study's share of Q.
Meta-Analysis CalculatorPool study effects by inverse variance under both fixed-effect and random-effects models, with τ², a prediction interval, per-study weights and leave-one-out influence.
Mutual Information CalculatorDependence a correlation misses: y = x squared gives a Pearson r of exactly 0 and 1.358298 bits — with the binning bias measured on unrelated data rather than glossed over.
Cosine Similarity CalculatorCosine with the angle, both norms and Pearson's r beside it — on [1,0,1,0] against [0,1,0,1] the cosine is exactly 0 and Pearson exactly -1, and centring is the only difference.
Jaccard Index CalculatorJaccard with Dice and the overlap coefficient beside it, and the exact transform linking them: Dice = 2J/(1+J), so the two rank every pair of sets identically.
Herfindahl Index CalculatorHHI with each firm's contribution, the numbers-equivalent and the exact merger delta: merging the top two adds precisely 2 x s1 x s2, whatever else is in the market.
Gini Coefficient CalculatorGini with the Lorenz curve, top and bottom shares and the Palma ratio — because two societies with a Gini of exactly 0.50 can have bottom-50% shares of 0% and 22%.
Shannon Entropy CalculatorEntropy in bits, nats and bans with per-symbol contributions, efficiency and perplexity — and the demonstration that a sorted sequence and its shuffle give identical values to the last bit.
Life Table CalculatorThe actuarial estimator with the effective-at-risk column shown and Kaplan-Meier beside it: ten-unit intervals cost 1.3753 percentage points, one-unit intervals cost exactly zero.
Hazard Ratio CalculatorBoth standard formulas from one log-rank output — Peto one-step 3.1899 and O/E 2.6554 — with a note on which of them is guaranteed to agree with the p-value beside it.
Log-Rank Test CalculatorObserved minus expected accumulated at every event time, so the crossing-hazards blind spot is visible: a running total that peaks at +3.2652 and ends at -1.4717 gives p = 0.458065.
Kaplan-Meier CalculatorSurvival with censoring handled, and the naive count printed beside it: five events in twenty subjects give 25.0000% by the plain count and a 30.1202% cumulative incidence by Kaplan-Meier.
Tolerance Interval CalculatorCovers a stated proportion of the population with stated confidence: even at n = 1,000 the 95/95 factor is 2.03608, not the 1.95996 that covers 95% of a known normal.
Prediction Interval CalculatorFor the next single observation, printed beside the confidence interval: the confidence factor goes to zero as n grows while this one converges to 1.95996 and stops.
Permutation Test CalculatorEnumerates every split below 400,000 — two groups of ten is 184,756 of them — so the p-value is a ratio of two integers with no distribution assumed anywhere.
Bootstrap Confidence Interval CalculatorNormal, percentile, basic and BCa intervals from one seeded resample set, with measured coverage: on lognormal data at n = 20 a nominal 95% achieves 82.60% to 87.20% depending on the method.
Games-Howell Test CalculatorEvery pair with its own standard error and Welch degrees of freedom, plus the measured cost of pooling: with all means equal Tukey fires 29.92% of the time in one variance pattern and 3.00% in another.
Dunnett's Test CalculatorMany treatments against one control, with the critical value computed by numerical integration rather than read from a table — at one comparison it collapses to the two-sided t to 3.55e-15.
Omega Squared CalculatorThe bias-corrected effect size from a published F and its degrees of freedom, exactly zero when F is exactly 1 — because the correction subtracts precisely what chance contributes.
Eta Squared CalculatorFour effect sizes with the denominator each one uses, and a measured table: with eight groups of six and no effect at all, 94.7% of datasets still give an eta squared above 0.05.
Repeated Measures ANOVA CalculatorGreenhouse-Geisser, Huynh-Feldt and Mauchly computed together, with the p-value under all four sphericity assumptions: the shipped example goes from 0.031895 to 0.085420.
Two-Way ANOVA CalculatorFull source table with three effect sizes per row and the interaction reported first, because a crossover gives both main effects a sum of squares of exactly zero while the interaction has F = 600.
Standardized Mortality Ratio CalculatorSMR with the expected count built stratum by stratum, and the small-count disagreement shown: 5 observed against 2.07 expected gives p = 0.0417 and an exact interval of 0.7843 to 5.6369.
Incidence Rate CalculatorEvents per person-time with the exact Poisson interval, including at zero events: the exact two-sided upper limit is 3.688879 per unit of person-time, not the 3 the one-sided rule of three gives.
Attributable Risk CalculatorRisk difference, attributable fraction and population attributable fraction, which rank risk factors in opposite orders: RR 9 at 1% exposure gives a PAF of 7.41%, RR 1.2 at 80% gives 13.79%.
Likelihood Ratio CalculatorLR+ and LR- with post-test probability after both results, and the base-rate sweep: a 90/90 test moves a patient 0.79 points at a 0.1% pre-test and 49.41 points at 30%.
ROC Curve & AUC CalculatorBuilds the curve from raw scores with every threshold enumerated, and computes the AUC twice — trapezoid and Mann-Whitney U — which agree to 1.11e-16 across 300 datasets.
Cramer's V CalculatorThe effect size a chi-square test does not give you, with your own table rescaled six ways: the same 3x3 pattern gives V = 0.27136852 at n = 90 and at n = 1,800 while p falls from 0.0101 to 3.6e-56.
Phi Coefficient CalculatorPhi for a 2x2 table printed against the ceiling its marginals impose: on [10, 40, 0, 50] every case is exposed, a complete association, and phi is 0.3333333, which is exactly max phi.
Point-Biserial Correlation Calculatorr with Cohen's d, t and Pearson's r on the same columns computed independently, plus the split table: hold d at 1.9799 and r falls from 0.707107 at 50/50 to 0.269630 at 2/98.
Intraclass Correlation CalculatorAll six ICC forms from one subject-by-rater matrix, with the rater means that drive them apart: one 8x3 matrix gives ICC(1,1) = 0.1277 and ICC(3,1) = 0.9852.
Cronbach's Alpha CalculatorAlpha with alpha-if-item-dropped, item-total correlations and the mean inter-item correlation, plus the length table: at a mean inter-item correlation of 0.05, a 100-item scale still reports 0.8403.
Friedman Test CalculatorRanks within each block and prints the rank matrix beside the raw values: three blocks of [1,2,3] plus one of [1000,2000,3000] give Kendall's W of exactly 1, because only the order is read.
Runs Test CalculatorThe Wald-Wolfowitz test with every run boundary drawn: ten 1s then ten 0s gives z = −3.9055 and perfect alternation +3.9055 — identical counts, mirror-image results, which no frequency test can see.
Goodness of Fit CalculatorChi-square with per-cell contributions and standardised residuals, plus a measured answer to the “expected at least 5” rule — below the floor the test is CONSERVATIVE (3.27% at 2 per cell), not liberal.
Bartlett's Test CalculatorThe most powerful equal-variance test under normality and the worst without it, with both halves measured — 4.20% false positives on normal data and 40.40% on skewed — plus a normality check per group.
Levene's Test CalculatorComputes both centre conventions side by side and shows the median is what makes it robust: on skewed data the mean-centred original rejects 15.00% of true nulls against the median version's 4.47%.
F-Test CalculatorThe variance ratio with Levene and Bartlett computed beside it, and a measured false-positive table: on skewed data the F-test reports a difference in spread 40.40% of the time when there is none.
Laplace Distribution CalculatorFits by maximum likelihood and shows the location estimate IS the sample median, with the log-likelihood at the mean beside it — the fact that least-absolute-deviations regression is built on.
Gamma Distribution CalculatorComputes the Erlang identity alongside the CDF: for an integer shape the gamma CDF equals a Poisson tail exactly, verified to 5.6e-17 across 126 combinations — waiting time and event count are one question.
Sensitivity and Specificity CalculatorSensitivity, specificity, PPV, NPV, likelihood ratios and MCC from a 2×2 table, with predictive values recomputed across the prevalence range — a 99%/99% test has a PPV of 50% at 1% prevalence and 9% at 0.1%.
Number Needed to Treat CalculatorNNT from the absolute risk reduction, with the relative figure beside it — two trials reporting the identical “50% reduction” have NNTs of 7 and 1,000, and the common shortcut says 2 for both.
Relative Risk CalculatorRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
Two-Proportion Z-Test CalculatorCompares two rates with the pooled z-test and a Newcombe interval for the difference, and shows that z² equals the 2×2 chi-square statistic exactly — so the “z-test or chi-square” question has no content.
One-Proportion Z-Test CalculatorThe score z-test with the exact binomial test beside it: 60 of 100 against 0.5 gives p = 0.0455 by one and 0.0569 by the other — opposite verdicts at the conventional threshold, on identical data.
Wilson Score Interval CalculatorWilson, Wald, Agresti-Coull and Clopper-Pearson on one set of counts, with the EXACT coverage each delivers at your sample size — a "95%" Wald interval covers the truth 80.85% of the time at n = 30, p = 0.1.
Midrange CalculatorReports how many of your values actually entered the answer — always two — and simulates where the midrange wins: for a uniform population it beats the sample mean by (n+1)(n+2)/6n, which is 17x at n = 100.
Exponential Smoothing CalculatorSimple smoothing and Holt's linear trend on one series, with the SSE-optimal alpha found for you — and the flat-forecast problem shown rather than described: no alpha makes a level-only model slope.
Multiple Regression CalculatorFits several predictors with a VIF on every term, and names the configuration people misread: a model significant at p = 0.0103 where neither predictor reaches 0.05, at a VIF of only 7.11.
Sign Test CalculatorThe exact binomial sign test with the paired t-test on the same numbers beside it, and every difference listed next to the sign it was reduced to — so the cost of discarding the magnitudes is visible.
Normality Test CalculatorShapiro-Wilk, Anderson-Darling and Jarque-Bera with a Q-Q plot, plus a resampled sweep answering the question the tests cannot: was your sample size big enough to detect anything?
Conditional Probability CalculatorPrints P(A|B) and P(B|A) together, because they are different numbers — on the worked example one is 100 times the other — and checks that your three probabilities describe a world that can exist.
Cpk CalculatorCp, Cpk, Pp and Ppk with the defect rates they predict and the rate actually observed — including the built-in case where Cp is 2.05, Cpk is 0.57 and a quarter of the sample is already out of spec.
Equivalence Test Calculator (TOST)Two one-sided tests for equivalence, with the ordinary “is there a difference” p-value beside it — because a p of 0.80 with equivalence unproven means the study detected nothing and demonstrated nothing.
McNemar's Test CalculatorChi-square and the exact binomial p for a paired 2×2 table, with the fact that makes the test surprising: the agreeing cells contribute nothing, so a study of 1,425 and one of 25 can give the identical result.
Trimmed Mean CalculatorTrimmed and Winsorized means at any fraction, plotted across the full sweep — because a 0% trim is the arithmetic mean and a 50% trim is exactly the median, so the trim fraction is a dial between them.
Moving Average CalculatorSimple, exponential, weighted and centred moving averages with the lag each one carries — exactly (n−1)/2 periods for a simple average, which is also the reason α = 2/(n+1) is in every EMA formula.
Bayes' Theorem CalculatorPosterior probability from a prior, sensitivity and specificity — plus the true and false positives per 100,000, because a 99% accurate test for a 1-in-10,000 condition is right 0.98% of the time and the percentage alone does not make that believable.
Cohen's Kappa CalculatorKappa with the two figures that explain it: the maximum the marginals permit, and PABAK. Two built-in tables with identical 85% agreement give kappas of 0.6995 and 0.3219, and a third with 94.4% agreement gives −0.0234.
Kendall Tau CalculatorComputes τ-a, τ-b and τ-c with the concordant and discordant pair counts, because the three definitions disagree — 0.7143, 0.8729 and 0.9375 on the built-in tied preset, from the same eight pairs.
Spearman Correlation CalculatorComputes rho correctly as Pearson on the midranks, and beside it the 6Σd²/(n(n²−1)) shortcut every textbook teaches — which is exact only when no two values tie, and overstates the correlation when they do.
Bonferroni Correction CalculatorAdjusts p-values by Bonferroni, Holm, Šidák, Holm-Šidák and Benjamini-Hochberg at once — and shows that Holm controls exactly what Bonferroni controls while never rejecting fewer, which makes plain Bonferroni dominated.
Tukey HSD CalculatorEvery pairwise comparison after an ANOVA with simultaneous confidence intervals — and the studentized range integrated rather than interpolated from a table, so any group count, df and level works. At two groups the critical q is exactly √2 times the critical t.
Frequency Distribution CalculatorA frequency table with relative and cumulative columns, and all six standard bin-count rules computed at once — they disagree by a factor of 6.67 at n = 10,000, where Sturges asks for 15 bins and the square-root rule asks for 100.
Mean Absolute Deviation CalculatorAll three statistics the acronym MAD names — about the mean, about the median, and the median absolute deviation — plus the ratio to the standard deviation, which is exactly 0.7979 for a normal distribution and so reads as a free shape check.
F-Distribution CalculatorUpper and lower critical values at any degrees of freedom — including the lower tail printed tables omitted, because F(a,b;p) = 1/F(b,a;1−p) — plus F(1,v) against t(v)² computed both ways, which agree exactly.
t-Distribution CalculatorProbabilities and critical values at any degrees of freedom including fractional ones, with the percentage gap from the normal printed — 4.199% at df = 30, where the textbooks say to switch to z, and df must reach 4,746 before it rounds to 1.960.
Fisher's Exact Test CalculatorThe exact p for a 2×2 table under all three two-sided conventions, because they disagree — 0.0406 against 0.0699 on the built-in table, across the 5% line — plus the test's actual size by enumeration, which is 2.30% at a nominal 5%.
Statistical Power CalculatorPower and sample size from the non-central t rather than a normal approximation, with the gap shown — plus a live demonstration that post-hoc power is a function of the p-value alone, and 0.500044 at p = 0.05 for every study ever run.
Sum of Squares CalculatorComputes Σx² and Σ(x − x̄)² side by side, because the term means both and they differ by 94.7 times on the built-in preset — plus SST, SSR and SSE with the intercept condition the textbook identity depends on.
Residual CalculatorResiduals, standardised and studentised residuals, leverage and Cook's distance, with Anscombe's quartet built in — including the dataset every residual rule scores cleanest and which one point controls entirely.
Power Regression CalculatorFits y = ax^b and reads b as what it is — a dimensionless elasticity. Change the units in one click and watch the coefficient move by a million while the exponent holds; fed the eight planets it returns Kepler's 1.4997.
Logarithmic Regression CalculatorFits y = a + b·ln x and reports what one doubling of x adds to y — the constant increment that defines the shape, and the only log-transformed fit on this site whose R² needs no caveat.
Quadratic Regression CalculatorFits y = a + bx + cx² and prints the vertex, which is the optimum a dose-response curve is fitted to find — plus whether that turning point falls inside your data or is being assumed by the shape.
Exponential Regression CalculatorFits y = ae^(bx) by true least squares as well as by the usual log-transform, and shows the gap — on the built-in preset the standard method is out by 8% on the doubling time and leaves 4.79 times the squared error.
Kruskal-Wallis CalculatorApplies the tie correction and shows it against the uncorrected value, because on ordinal data it moves p from 0.054 to 0.027 — across the conventional threshold, on identical data.
Wilcoxon Signed-Rank CalculatorReports how many zero differences it dropped and gives the Hodges-Lehmann shift, because the classical and Pratt variants disagree on the same data and the median of the differences is not what this test estimates.
Expected Value CalculatorReports the spread, the chance of any gain and the single most likely outcome beside the average — because on a roulette number bet the expected value is −2.70% and the modal result is losing, 36 times in 37.
Monty Hall CalculatorGeneralises the puzzle so the famous 100-door explanation stops hiding its own assumption — open one door instead of 98 and switching is worth 1.0102% against 1%, still right and barely.
Birthday Paradox CalculatorAnswers both questions the paradox conflates — 50.73% that some pair shares a birthday in a room of 23, and 5.86% that anyone shares yours — with the pair count that explains the gap.
Simpson's Paradox CalculatorDetects when a pooled comparison contradicts every subgroup, then repairs it by standardising — which takes the real Berkeley admissions gap from +14.16 points to −4.26 and changes its sign.
Mann-Whitney U CalculatorMann-Whitney U with the tie correction and both medians shown, because it is not a test of medians — two samples with identical medians of 4 are separated at p = 0.01.
One-Way ANOVA CalculatorThe full F table with eta and omega squared, plus every pairwise gap — because a significant F says something differs and never says which, and ten groups tested pairwise carry a 90% false-positive rate.
Odds Ratio CalculatorOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
Effect Size CalculatorCohen d, Hedges g and the overlap between groups, with a sample-size control that moves the p-value while leaving the effect size fixed — the same d gives t = 1.29 at n=30 and 23.57 at n=10,000.
Coefficient of Determination CalculatorR-squared across five models at once with adjusted R-squared beside it, because adding a term can never lower it and the value most tools report for a curve fit was computed in log space.
Kurtosis CalculatorExcess, raw and Excel KURT side by side, with the tail-weight reading made explicit — kurtosis measures tails, not peakedness, and "kurtosis 3" means two opposite things.
Skewness CalculatorAll three skewness conventions at once with the standard error beside them, because Excel SKEW runs 18.6% above the textbook formula at n=10 and a normal sample of 30 swings ±0.85 by chance.
Covariance CalculatorSample and population covariance with a unit multiplier that shows why the magnitude is unreadable — rescaling x multiplies the covariance while leaving the correlation untouched.
Outlier CalculatorRuns four outlier rules together and says when each is failing — the 3-sigma rule cannot flag anything below n=11, two outliers hide each other, and the quartile convention alone can decide the answer.
Correlation Coefficient CalculatorReports Pearson, Spearman and Kendall together with the scatter plot, and ships Anscombe's quartet built in — four datasets with an identical r of 0.816 that Spearman tells apart.
Empirical Rule CalculatorGives the exact 68.27/95.45/99.73 at any k, plus the multiplier for a round 95% — and prints Chebyshev's distribution-free floor beside each figure so the cost of assuming normality is a number.
Central Limit Theorem CalculatorComputes exact sampling distributions instead of simulating them, so you can see what a nominal 95% interval really covers — and that the two-sided figure hides a tail 36% too large.
Beta Distribution CalculatorTakes raw successes and failures and runs the conjugate update, so 5 out of 5 returns 6/7 rather than the 100% a plain proportion claims — with a credible interval that stays honest at zero.
Negative Binomial CalculatorShows both conventions at once — failures before the rth success and total trials differ by exactly r, so the same k gives two answers 32% apart and nobody says which they meant.
Uniform Distribution CalculatorHandles the continuous and discrete uniform as the separate distributions they are — P(X = x) is zero in one and 1/n in the other, and the variances differ for identical endpoints.
Lognormal Distribution CalculatorTakes log-scale parameters or your data's own mean and SD and converts between them — because μ describes ln x, not x, and entering the wrong one is wrong by orders of magnitude with nothing to flag it.
Weibull Distribution CalculatorReliability probabilities with the hazard curve drawn beside the density — because the shape parameter is the hazard story: below 1 the risk falls, at 1 it is exactly an exponential, above 1 it rises.
Exponential Distribution CalculatorTime-until-event probabilities from a rate or a mean, always showing both — with a panel proving elapsed time changes nothing, which is a constant hazard rate and false for anything that wears out.
Geometric Distribution CalculatorWaiting until the first success, with both conventions shown — and a panel that computes your remaining wait given past failures and from scratch, proving they are identical. The gambler's fallacy as arithmetic.
Chi-Square CalculatorGoodness of fit and tests of independence with every expected count and per-cell contribution shown — because the validity condition is about expected counts, not observed ones, and most calculators hide them.
Hypergeometric Distribution CalculatorExact probabilities for drawing without replacement, with the binomial answer beside them and the 10% rule evaluated — at a 40% sampling fraction the binomial is 22% wrong, at 0.2% it is 0.06% wrong.
Poisson Distribution CalculatorPoisson probabilities with a dispersion test against your own variance — because a Poisson forces variance to equal the mean, and real count data usually does not, which is exactly where the tail goes wrong.
Binomial Distribution CalculatorExact binomial probabilities at any n — including thousands, where a factorial overflows — with the normal approximation beside them and its error measured, which is 0.6% at the centre and 261% in the tail.
Normal Distribution CalculatorProbabilities under a normal curve in all four directions with the region shaded — and the empirical rule given exactly, because two standard deviations is 95.45% and the 95% everyone quotes sits at 1.96σ.
z-test CalculatorOne-proportion, two-proportion and known-sigma mean z-tests — with the t-test computed beside the mean case, because for a mean the z-test needs a σ almost nobody actually has.
t-test CalculatorOne-sample, two-sample and paired t-tests defaulting to Welch, with Student's pooled version printed beside it — and a warning when the two disagree on the verdict.
Critical Value CalculatorCritical values for z, t, chi-square and F at any alpha and any degrees of freedom — with one- and two-tailed values shown together, because reading the wrong column of a printed table is the classic error.
Standard Error CalculatorStandard error of a mean or proportion, printed beside the standard deviation it gets confused with — the ratio is always √n, and at n = 50 that is a factor of seven.
Margin of Error CalculatorMargin of error for a percentage or an average, shown across seven sample sizes so the square-root law is visible — every doubling buys exactly 29.3%, never more.
p-value CalculatorA p-value from a t or z statistic, one- or two-tailed — with a panel that holds an effect fixed and grows the sample, so you can watch significance appear from nothing but n.
Five Number Summary CalculatorMinimum, Q1, median, Q3 and maximum, drawn as a box plot with every one of your observations plotted underneath it — so you can see what the summary threw away.
IQR CalculatorInterquartile range, the 1.5 × IQR outlier fences, and exactly which values fall outside — with a slider that corrupts one value so you can watch the range move while the IQR does not.
Quartile CalculatorQ1, Q2 and Q3 by all eleven conventions in common use — Excel, R, Tukey hinges and the TI-84 rule — with a warning when they disagree.
Linear Regression CalculatorThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.

Special Functions

Error Function Calculatorerf, erfc and the inverse to full double precision through the incomplete gamma function, with the normal-distribution conversion and a demonstration of why 1 minus erf fails.
Gamma Function CalculatorGamma, log-gamma, digamma and beta with the factorial shift stated every time, values reconstructed past the double overflow, and the poles explained through the reflection formula.

Combinatorics

Binomial Coefficient Calculatorn choose k exactly on big integers, by the multiplicative formula that divides as it goes and never builds a factorial — with Pascal's rule and permutations beside it.

Averages

Weighted Average CalculatorEach value carries the weight you give it, with every item's share of the total shown as a percentage so you can see what is actually driving the answer.
Geometric Mean CalculatorThe average for things that compound. Growth of +50% then -50% averages to zero arithmetically and to a real 13.4% loss geometrically, which is what actually happened.
Harmonic Mean CalculatorSixty there and forty back averages 48, not 50. The page simulates the journey from its legs as well as computing the formula, so the answer arrives with its own proof.
Root Mean Square CalculatorFor values whose sign must not cancel. Mains voltage averages zero and still boils a kettle; squaring, averaging and rooting gives the number that describes the work.

Compare statistics & probability calculators

What each calculator is best for, the main inputs it needs, what it shows, and what to keep in mind.

Comparison of Statistics & Probability calculators: what each is best for, its key inputs, what it shows, and its limitations.
Calculator	Best for	Key inputs	Shows	Keep in mind
Percent Error Calculator	Scoring a measurement against an accepted value	Measured and accepted values, repeated trials, or measured/accepted pairs	Percent error to your significant figures, with the spread and the MAPE	Divide by the accepted value, not your reading; undefined when it is zero
Standard Deviation Calculator	Summarizing spread of a numeric dataset	Dataset values, sample or population basis, decimals	Sample and population SD, variance, mean, median	Sample SD undefined for one value; educational only
Error Function Calculator	erf, erfc and the inverse, to full double precision	One real value	erfc computed directly against 1 − erf, which loses its digits	Past x = 3, computing erfc by subtraction destroys most of the answer
Gamma Function Calculator	The factorial extended to non-integers, plus beta	One real value, and two positives for the beta function	Γ against exact factorials, and the recurrence and reflection formulas	Γ(n) = (n − 1)!, so the argument is one MORE than the factorial you want
Binomial Coefficient Calculator	n choose k exactly, however many digits it takes	Whole numbers n and k, up to n = 2000	Pascal's rule and the row sum, both computed from the entries	n!/(k!(n−k)!) overflows long before the coefficient does; the answer is far smaller

Best calculator path

All math calculators, by sub-hub