🧮 Calculators

Welch's t-test

Computes the Welch's t-test statistic, the two-sample t-test version that does NOT assume equal variances. Unlike the classic Student's t-test, it doesn't pool the variances into a single value, which makes it more reliable when the two groups have different spreads — the default recommendation in modern statistics. The degrees of freedom come from the Welch-Satterthwaite equation, usually fractional. Enter the mean, standard deviation and size of each of the two groups.

Eta-Squared (Effect Size)

Computes eta-squared (η²), an effect-size measure for analysis of variance, from the F-statistic and the degrees of freedom. While the F-test says whether there's a significant difference between groups, η² says how much of that difference the grouping variable explains — the proportion of total variance attributable to the groups. It's essential for reporting ANOVA results beyond the p-value. Enter the F-statistic and the numerator and denominator degrees of freedom.

📋

Cronbach's Alpha

Computes Cronbach's alpha, the most used measure of internal consistency of a questionnaire or scale. It answers: do the items that should measure the same thing actually agree with each other? The formula combines the number of items with the ratio between the sum of each item's variance and the total-score variance. Values above 0.7 usually indicate acceptable reliability. It's indispensable in psychometrics and instrument validation. Enter the item variances and the total-score variance.

Mann-Whitney U Test

Computes the Mann-Whitney U test, the non-parametric alternative to the t-test for comparing two independent groups. Instead of comparing means, it compares the ranks of the combined data, which makes it valid even when the data don't follow a normal distribution or are ordinal. The tool computes the U statistic and the normal approximation z, with a tie correction. Enter the two groups of observations, separated by commas.

One-Way ANOVA

Computes one-way analysis of variance (ANOVA), the test that checks whether three or more groups have different means. It decomposes the total variation into between-group and within-group variation, and the ratio between them (the F-statistic) indicates whether the observed differences are larger than would be expected by chance. It's the generalization of the t-test to multiple groups. Enter the groups: values separated by commas, groups separated by semicolons.

Kruskal-Wallis H Test

Computes the Kruskal-Wallis H test, the non-parametric version of ANOVA for comparing three or more independent groups. Just as Mann-Whitney does for two groups, it works with the ranks of the data instead of the raw values, dispensing with the normality assumption. The H statistic approximately follows a chi-square distribution. It's the choice when the data are ordinal or clearly non-normal. Enter the groups: values by comma, groups by semicolon.

📏

Levene's Test (Equality of Variances)

Computes Levene's test, which checks whether several groups share the same variance — the homogeneity assumption that ANOVA and the t-test rely on. Instead of working with the original values, it transforms each data point into the absolute deviation from its group mean and runs an ANOVA on those deviations, gauging whether the spread differs across groups. It's more robust to non-normality than Bartlett's test. Enter the groups: values by comma, groups by semicolon.

Bartlett's Test (Equality of Variances)

Computes Bartlett's test, the classic test for checking whether several groups share the same variance. It compares the pooled variance of all groups against the individual variances through a statistic that approximately follows a chi-square distribution, with a bias correction that improves the approximation in small samples. It's more powerful than Levene's test when the data are genuinely normal, but more sensitive to departures from normality. Enter the groups: values by comma, groups by semicolon.

Wilson Interval (Proportion)

Computes the Wilson confidence interval for a proportion, the modern alternative to the Wald interval (the p̂ ± z·SE formula you learn first). The Wald interval fails badly with small samples or proportions near 0 or 1 — it can even produce negative bounds. Wilson's fixes this by inverting the score test, which keeps it inside [0, 1] with much better coverage. It's the recommended interval for proportions. Enter the successes, the total and the confidence level.

One-Proportion z-Test

Computes the one-proportion z-test statistic, used to check whether the proportion observed in a sample differs from a hypothesized value. For example: out of 100 voters, 45 approve a measure — is that compatible with 50% in the population? The statistic divides the difference between the observed and hypothesized proportions by the standard error computed under the null hypothesis. It applies when the sample size is large enough for the normal approximation. Enter the successes, the total and the hypothesized proportion.

⚖️

Two-Proportion z-Test

Computes the two-proportion z-test statistic, which compares the success rates of two independent groups — the backbone of any A/B test. For example: variant A converted 45 of 100 and B converted 30 of 100; is the difference real or noise? The statistic uses the pooled proportion of the two groups to estimate the standard error under the hypothesis that the proportions are equal. Enter the successes and the total of each of the two groups.

🔁

McNemar's Test (Paired)

Computes McNemar's test, used to compare two paired proportions — when the same subjects are measured twice (before and after, or by two methods). Unlike the two-proportion test, which treats the groups as independent, McNemar looks only at the cases that switched classification (the discordant cells b and c of the 2×2 table), ignoring those that stayed the same. The continuity-corrected version follows a chi-square with 1 degree of freedom. Enter the two discordant counts.

Wilcoxon Signed-Rank Test

Computes the Wilcoxon signed-rank test, the non-parametric alternative to the paired t-test. When you measure the same subjects twice — before and after a treatment — and the data don't follow a normal distribution, this is the test to use. It works with the differences between pairs: it ranks the magnitudes of those differences, assigns ranks and separately sums the ranks of the positive and negative differences. The tool computes W+, W− and the normal approximation z, with a tie correction. Enter the two paired series.

➕

Sign Test (Paired)

Computes the sign test, the simplest and most robust paired test of all. It completely ignores the magnitude of the differences between pairs and looks only at the sign: how many times the value went up and how many it went down. Under the hypothesis of no effect, ups and downs should balance out, like heads and tails, and the p-value comes straight from a binomial distribution. Because it uses so little information, it's less powerful than Wilcoxon, but it makes almost no assumptions. Enter the two paired series.

🔬

Fisher's Exact Test (2×2)

Computes Fisher's exact test for a 2×2 contingency table, the right choice when the counts are small and the chi-square approximation no longer holds. Instead of approximating, it computes the exact probability of observing the current table — and tables even more extreme — holding the margin totals fixed, via the hypergeometric distribution. The tool returns the one-sided and two-sided p-values (the latter by summing the probabilities of tables as likely or less likely than the observed one). Enter the four cells a, b, c and d.

🎲

Runs Test (Wald-Wolfowitz)

Computes the Wald-Wolfowitz runs test, which checks whether a sequence of two values is random or has a pattern. A run is a stretch of consecutive equal values; too few runs indicate clustering (a trend), and too many indicate systematic alternation. By comparing the observed number of runs with the number expected under randomness, the z statistic reveals whether the order of the data is compatible with pure chance. It's useful for checking residuals, time series and number generators. Enter the sequence with two distinct symbols.

📚

Friedman Test

Computes the Friedman test, the non-parametric version of repeated-measures ANOVA. It's used when the same subjects (or blocks) go through three or more conditions and you want to know whether the conditions differ, without assuming normality. Within each block, the treatments are ranked; if one treatment is consistently better, it accumulates low ranks across all blocks, and the Friedman chi-square statistic grows. Enter the blocks: treatments separated by commas, blocks by semicolons.

Point-Biserial Correlation

Computes the point-biserial correlation coefficient, which measures the association between a continuous variable and a dichotomous (two-group) variable. It is, in fact, the Pearson correlation applied to the case where one variable takes only two values — so it ranges from −1 to +1 and carries the same interpretation. It shows up often in psychometrics, to assess how well a test item (right/wrong) discriminates between high- and low-scoring students. Enter the continuous values of each of the two groups.

📉

Ljung-Box Test (Autocorrelation)

Computes the Ljung-Box test statistic, which checks whether a time series exhibits autocorrelation — that is, whether the values carry memory of past values instead of being independent. Rather than looking at one lag at a time, it combines the autocorrelations of several lags into a single statistic that approximately follows a chi-square. It's the standard test for checking whether an ARIMA model's residuals have become white noise. Enter the series and the number of lags.

Mann-Kendall Trend Test

Computes the Mann-Kendall test, used to detect a monotonic trend (upward or downward) in a time series without assuming the data are normal or the trend is linear. It sums the signs of all pairwise comparisons between values: if the series tends to rise over time, positive signs prevail. It's widely used in hydrology, climatology and environmental science to assess trends in long series. The S statistic becomes a z via the normal approximation, with a tie correction. Enter the series in chronological order.

Theil-Sen Estimator (Robust Regression)

Computes the Theil-Sen line, a robust alternative to least squares for fitting a linear trend. Instead of minimizing squared errors (which makes the classic line very sensitive to outliers), it takes the median of all slopes between pairs of points. Because it uses the median, it tolerates up to about 29% of outlying points without being dragged off course. It's the natural companion to the Mann-Kendall test for quantifying the detected trend. Enter the X and Y values.

☑️

Cochran's Q Test

Computes Cochran's Q test, the extension of McNemar's test to three or more binary treatments measured on the same subjects. Imagine asking each person whether they approve of three different products (yes/no): the test checks whether the approval proportion differs across products, respecting the fact that they're the same respondents. It's effectively the dichotomous-response version of the Friedman test. The Q statistic approximately follows a chi-square. Enter the subjects with 0/1 treatments.

Kolmogorov-Smirnov Test (Two-Sample)

Computes the two-sample Kolmogorov-Smirnov test, which checks whether two samples come from the same distribution — without assuming what that distribution is. It compares the empirical cumulative distribution functions of the two samples and takes the largest vertical distance between them, the D statistic. Unlike the t-test, which only compares means, the KS test is sensitive to any difference in shape, spread or location. Enter the two samples of values.

🔔

Anderson-Darling Test (Normality)

Computes the Anderson-Darling test, one of the most powerful normality tests available. It measures how far the empirical distribution of your data departs from the normal curve, giving extra weight to the tails — precisely where other tests often miss departures. The tool returns the A² statistic, the adjusted A²* version (corrected for sample size) and an approximate p-value. The smaller the A², the more compatible with normality. Enter at least 8 values.

📏

CI for Difference of Two Means (t)

Computes the confidence interval for the difference between the means of two independent groups, using the t distribution with pooled variances. It's the natural companion to the two-sample t-test: instead of only saying whether the difference is significant, it shows the plausible range for the real size of that difference. If the interval doesn't contain zero, there's a significant difference at the chosen level. Enter the two samples of values and the confidence level.

Prediction Interval (New Observation)

Computes the prediction interval for a new individual observation, from a sample. Be careful not to confuse it with the confidence interval for the mean: that one estimates where the population mean lies; this one estimates where the next individual value you observe will fall — which is why it's always wider, carrying the uncertainty of the mean plus the natural variability of the data. The decisive factor is the √(1 + 1/n) under the t distribution. Enter the sample and the confidence level.

Paired t-Test

Computes the paired t-test, used when you measure the same subjects twice — before and after a treatment, or by two methods. It works with the differences of each pair and tests whether the mean difference is compatible with zero. By removing the variability between subjects, it's usually far more powerful than comparing two independent groups. The tool returns the t statistic, the degrees of freedom and the two-sided p-value. Enter the two paired series.

Student's t Critical Value

Computes the critical value of Student's t distribution for the degrees of freedom and confidence level you choose, replacing the lookup in the t table at the back of statistics books. It returns both the two-sided value (for confidence intervals and two-tailed tests) and the one-sided value (for one-tailed tests). As the degrees of freedom grow, the value converges to that of the normal distribution. Enter the degrees of freedom and the confidence level.

Z-Test for a Mean (Known σ)

Computes the z-test for a mean when the population standard deviation is known. It's the simplest case of hypothesis testing about a mean: the statistic measures how many standard errors separate the observed mean from the hypothesized value. When σ is known (or the sample is large), the normal distribution is used instead of the t. The tool returns the z statistic and the two-sided p-value. Enter the sample mean, the hypothesized mean, the population standard deviation and the sample size.

🚩

Grubbs' Test (Outlier)

Computes Grubbs' test, used to identify whether the most extreme value in a sample is a statistical outlier, under the assumption that the data would be normal without it. It compares the deviation of the farthest point from the mean, in standard-deviation units, against a critical value derived from the t distribution. If the G statistic exceeds the critical value, the point is flagged as an outlier. It's the classic test for cleaning data before an analysis. Enter the data and the confidence level.

🔲

Chi-Square Test of Independence

Computes the chi-square test of independence for a contingency table, checking whether two categorical variables are associated or independent. For example: does preference for a product depend on age group? The test compares the observed frequencies with those that would be expected if there were no relationship at all, and the chi-square statistic measures the size of that mismatch. The tool returns the statistic, the degrees of freedom and the p-value. Enter the table: columns by comma, rows by semicolon.

Chi-Square Critical Value

Computes the critical value of the chi-square distribution for the degrees of freedom and confidence level you choose, dispensing with the printed table. It's the right-tail cutoff used in goodness-of-fit, independence and variance tests: if the computed statistic exceeds this value, the null hypothesis is rejected. Unlike the normal and the t, the chi-square is asymmetric and takes only positive values. Enter the degrees of freedom and the confidence level.

F Critical Value (Fisher-Snedecor)

Computes the critical value of the Fisher-Snedecor F distribution for the numerator and denominator degrees of freedom and the chosen confidence level, replacing the bulky F tables. It's the cutoff used in ANOVA and in variance-ratio tests: if the observed F statistic exceeds it, the conclusion is that the variances or means differ. The F distribution depends on two degrees of freedom, which made the printed tables enormous. Enter the two degrees of freedom and the confidence level.

CI for Variance and Standard Deviation

Computes the confidence interval for the variance and standard deviation of a normal population, from a sample. While most tools estimate the mean, this one estimates the spread — useful in quality control, where consistency matters as much as the central value. The calculation uses the chi-square distribution, which is asymmetric, so the interval is not centered on the sample variance. Enter the sample and the confidence level.

Clopper-Pearson Binomial CI (Exact)

Computes the exact Clopper-Pearson confidence interval for a binomial proportion. Unlike the Wald formula and even Wilson's, which are approximations, Clopper-Pearson is built directly on the binomial distribution, guaranteeing coverage of at least the nominal level — which is why it's regarded as the reference conservative interval. It's the choice when you need a rigorous guarantee, even with small samples. Enter the successes, the total and the confidence level.

Exact CI for a Poisson Mean

Computes the exact confidence interval for the mean of a Poisson distribution, from an observed event count. It's the right tool for count data: number of defects per batch, accidents per month, particles detected per interval. The calculation uses the exact relationship between the Poisson and the chi-square distribution (Garwood's method), giving reliable bounds even when very few events were observed. Enter the number of events and the confidence level.

F-Test for Equality of Variances

Computes the F-test for comparing the variances of two independent groups, checking whether they have the same spread. It's a common step before choosing between the classic t-test (which assumes equal variances) and Welch's (which doesn't). The statistic is the ratio of the two sample variances, and the F distribution, with its two degrees of freedom, provides the p-value. The tool takes the two samples and computes F, the degrees of freedom and the two-sided p-value. Enter the two groups of values.

CI for the Ratio of Two Variances

Computes the confidence interval for the ratio between the variances of two normal populations, σ₁²/σ₂². Instead of just testing whether the variances differ, it gives a plausible range for how many times one is larger than the other. If the interval contains the number 1, there's no evidence the spreads are different. The calculation uses the F distribution, and because it's asymmetric, the bounds come from F quantiles with the degrees of freedom swapped. Enter the two samples and the confidence level.

Agresti-Coull Interval (Proportion)

Computes the Agresti-Coull confidence interval for a proportion, an elegant middle ground between the simplicity of the Wald interval and the accuracy of Wilson's. The almost folkloric idea is to add a few fictitious successes and failures to the sample before applying the Wald formula — which fixes Wald's terrible behavior in small samples or extreme proportions. For 95% confidence, it's like adding two successes and two failures. Enter the successes, the total and the confidence level.

🪙

Exact Binomial Test

Computes the exact binomial test, which checks whether the observed proportion of successes is compatible with a hypothesized probability, without resorting to the normal approximation. For small samples, where the z-test fails, this is the correct test: it sums the exact probabilities, given by the binomial distribution, of all outcomes as likely or less likely than the observed one. For example: are 8 heads in 10 tosses compatible with a fair coin? Enter the successes, the total and the hypothesized probability.

⚖️

CI for the Difference of Two Proportions

Computes the confidence interval for the difference between two proportions, using the Wald method. It's the companion to the two-proportion z-test and the basis of any A/B test reading: beyond saying whether the difference is significant, it shows the plausible range for its real size. If the interval doesn't contain zero, there's a significant difference. For example: variant A converted 45 of 100 and B, 30 of 100 — what's the interval for the difference? Enter the successes and the total of each group and the confidence level.

📉

One-Sample Variance Test (Chi-Square)

Computes the chi-square test for the variance of a single sample, which checks whether the population variance equals a reference value. It's the dispersion counterpart of the one-sample t-test for the mean. It shows up often in quality control: is a process's variability within the specified limit, or has it increased? The statistic compares the sample variance with the hypothesized one and follows a chi-square distribution. Enter the sample and the hypothesized variance (σ₀²).

Correlation Significance Test

Computes the significance test for the Pearson correlation coefficient, answering whether the observed correlation between two variables is real or could have arisen by chance. A correlation of 0.5 in a sample of 6 points is unimpressive; the same correlation in 600 points is very strong. The test converts r into a t value, with n−2 degrees of freedom, and returns the p-value. The tool computes r, the t statistic and the two-sided p-value. Enter the two series of paired values.

CI for Correlation (Fisher z Transform)

Computes the confidence interval for a correlation coefficient using the Fisher z transformation. The sampling distribution of r is skewed, especially near −1 or +1, which prevents applying the usual symmetric margin. Fisher solved this with a transformation that makes the distribution approximately normal; you build the interval on that scale and then map back to r. The result is an asymmetric interval, narrower on the side near the extremes. Enter the coefficient r and the sample size.

🔔

D'Agostino-Pearson Normality Test (K²)

Computes the D'Agostino-Pearson omnibus normality test, which combines two clues of non-normality into a single verdict: skewness (do the data lean to one side?) and kurtosis (are the tails too heavy or too light?). Each becomes a standardized z score, and the sum of their squares, the K² statistic, follows a chi-square with 2 degrees of freedom. It's more informative than looking at skewness and kurtosis separately. Enter the sample (at least 8 values).

Kolmogorov-Smirnov Test (vs Normal)

Computes the one-sample Kolmogorov-Smirnov test against a normal distribution with specified mean and standard deviation. It measures the largest vertical distance between your data's empirical cumulative distribution and the theoretical normal curve — the D statistic. Unlike tests that look only at skewness or kurtosis, the KS compares the entire distributions, point by point. Enter the sample and the normal parameters (mean and standard deviation) to test.

Mood's Median Test

Computes Mood's median test, a non-parametric test that checks whether several groups share the same median. The idea is simple and robust: you compute the grand median of all the data together and count, in each group, how many values fall above it. If the groups had the same median, those counts would be proportional to the sizes; an imbalance produces a large chi-square statistic. It's quite resistant to outliers, working only with counts. Enter the groups separated by semicolons.

🎛️

Partial Correlation

Computes the partial correlation between two variables while controlling for the effect of a third. It's the tool against spurious correlations: ice cream sales correlate with drownings, but only because both rise in hot weather — controlling for temperature, the partial correlation drops to near zero. It removes the influence of the control variable from both and measures the direct association that remains. The tool returns the partial coefficient and its significance test. Enter the three paired series.

📏

Brown-Forsythe Test (Variances)

Computes the Brown-Forsythe test, a robust version of Levene's test for checking whether several groups share the same variance. The difference lies in one decisive detail: instead of measuring how far each point sits from the group mean, it uses the distance from the median. Because the median resists extreme values and skewed distributions, the test becomes much more reliable when the data aren't normal. Enter the groups: values by comma, groups by semicolon.

Goodman-Kruskal Gamma

Computes the Goodman-Kruskal gamma coefficient, a measure of association between two ordinal variables from a contingency table. It compares the number of concordant pairs (pointing in the same direction) with discordant ones, and ranges from −1 to +1: near +1, the variables rise together; near −1, one rises while the other falls. Unlike other measures, gamma completely ignores ties. Enter the table: columns by comma, rows by semicolon.

Cohen's h (Effect Size for Proportions)

Computes Cohen's h, the effect-size measure for the difference between two proportions. Comparing proportions directly is misleading, because the same arithmetic difference weighs differently near the middle (0.5) and near the ends (0 or 1). Cohen's h fixes this by applying the arcsine transformation, which stabilizes the variance, before measuring the distance. Usual convention: 0.2 is small, 0.5 medium and 0.8 large. Enter the two proportions (between 0 and 1).

Hodges-Lehmann Estimator

Computes the Hodges-Lehmann estimator of a sample's location, a robust alternative to both the mean and the median. It's defined as the median of all pairwise averages (the so-called Walsh averages) among the values, including each value with itself. The result combines the median's resistance to outliers with greater efficiency when the data are nearly normal. It's the estimator associated with the Wilcoxon signed-rank test. Enter the sample of values.

Cliff's Delta (Effect Size)

Computes Cliff's delta, a non-parametric effect-size measure between two groups. It answers a direct question: if I draw a value at random from each group, how much more likely is one to be larger than the other? It ranges from −1 to +1; zero means complete overlap. Because it works only with order comparisons, it requires no assumptions about the distribution and is robust to outliers. It's the ideal companion to the Mann-Whitney test. Enter the two groups of values.

🎲

Common-Language Effect Size (CLES)

Computes the common-language effect size (CLES), also called the probability of superiority. It's the most intuitive way to communicate a difference between groups: the probability that a value drawn from the first group is larger than a value drawn from the second. A CLES of 0.70 means that, in 70% of random comparisons, the first group wins. Ties count as half. It's the friendly version of Cliff's delta. Enter the two groups of values.