Hypothesis Tests

kstats-hypothesis provides statistical tests organized by the question they answer. Most functions return a TestResult; oneWayAnova() returns an AnovaResult with the full ANOVA table.

Reading a Test Result

Every test produces a result with a consistent shape. The statistic is the computed test value, pValue is the probability of observing a result at least as extreme under the null hypothesis, and isSignificant() compares the p-value to a threshold.

val sample = doubleArrayOf(5.0, 6.0, 7.0, 5.5, 6.5)
val result = tTest(sample, mu = 5.0)

result.testName          // "One-Sample T-Test"
result.statistic         // 2.8284
result.pValue            // 0.0474
result.degreesOfFreedom  // 4.0
result.isSignificant()   // true (p < 0.05)
result.confidenceInterval // (5.02, 6.98)

isSignificant() defaults to

\alpha = 0.05

. Pass a different threshold explicitly: result.isSignificant(alpha = 0.01).

Set alternative explicitly for one-sided tests. The default is Alternative.TWO_SIDED. Using Alternative.GREATER tests whether the sample mean exceeds the reference value; Alternative.LESS tests whether it falls below.

Is my sample mean different from a reference value?

The one-sample t-test compares the mean of a single sample against a known or hypothesized value.

val sample = doubleArrayOf(5.1, 4.9, 5.3, 5.0, 4.8)

// Two-sided: is the mean different from 5.0?
val two = tTest(sample, mu = 5.0)
two.statistic            // t value
two.pValue               // two-sided p-value

// One-sided: is the mean greater than 5.0?
val one = tTest(sample, mu = 5.0, alternative = Alternative.GREATER)
one.pValue               // one-sided p-value

Use Alternative.GREATER or Alternative.LESS for directional hypotheses instead of halving a two-sided p-value. The confidence interval adjusts accordingly.

Are two groups different?

Two-sample t-test

Compares the means of two independent samples. Welch’s t-test (unequal variances) is the default.

val group1 = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val group2 = doubleArrayOf(6.0, 7.0, 8.0, 9.0, 10.0)

val result = tTest(group1, group2)
result.statistic         // -5.0
result.pValue            // 0.0011
result.isSignificant()   // true

equalVariances is false by default, meaning Welch’s t-test is used. Set equalVariances = true only after confirming equal variances with leveneTest() or bartlettTest().

Paired t-test

Compares two related measurements (before/after, left/right) on the same subjects.

val before = doubleArrayOf(200.0, 190.0, 210.0, 180.0, 195.0)
val after = doubleArrayOf(190.0, 180.0, 195.0, 170.0, 185.0)

val result = pairedTTest(before, after)
result.statistic         // positive t (before > after)
result.pValue            // p-value for the difference
result.isSignificant()   // true if the change is significant

Mann-Whitney U test

Non-parametric alternative to the two-sample t-test. Compares ranks instead of means.

val group1 = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val group2 = doubleArrayOf(6.0, 7.0, 8.0, 9.0, 10.0)

val result = mannWhitneyUTest(group1, group2)
result.statistic         // U = 0.0
result.pValue            // < 0.02
result.isSignificant()   // true

Wilcoxon signed-rank test

Non-parametric alternative to the paired t-test. Tests whether paired differences are symmetrically distributed around zero.

val before = doubleArrayOf(10.0, 12.0, 14.0, 16.0, 18.0)
val after = doubleArrayOf(8.0, 9.0, 11.0, 12.0, 13.0)

val result = wilcoxonSignedRankTest(before, after)
result.statistic         // W+ = 15.0
result.pValue            // p-value with continuity correction

Are three or more groups different?

One-way ANOVA

Tests whether the means of three or more groups differ. Returns AnovaResult with the full ANOVA table.

val g1 = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val g2 = doubleArrayOf(6.0, 7.0, 8.0, 9.0, 10.0)
val g3 = doubleArrayOf(11.0, 12.0, 13.0, 14.0, 15.0)

val anova = oneWayAnova(g1, g2, g3)
anova.fStatistic         // 50.0
anova.pValue             // < 0.00001
anova.dfBetween          // 2
anova.dfWithin           // 12
anova.ssBetween          // 250.0
anova.ssWithin           // 30.0
anova.msBetween          // 125.0
anova.msWithin           // 2.5

ANOVA assumes normality within each group and equal variances across groups. Check normality with shapiroWilkTest() and equal variances with leveneTest() or bartlettTest() before running ANOVA.

Friedman test

Non-parametric alternative to repeated-measures ANOVA. Compares ranks across matched groups.

val treatment1 = doubleArrayOf(5.0, 6.0, 7.0, 5.5, 6.5)
val treatment2 = doubleArrayOf(4.0, 5.0, 6.0, 4.5, 5.5)
val treatment3 = doubleArrayOf(7.0, 8.0, 9.0, 7.5, 8.5)

val result = friedmanTest(treatment1, treatment2, treatment3)
result.statistic         // Friedman chi-squared
result.pValue            // p-value

Is my data normally distributed?

Four normality tests are available. Each returns a TestResult — a significant result (low p-value) indicates evidence against normality.

val sample = doubleArrayOf(
    -1.2, -0.5, 0.0, 0.5, 1.2, 0.3, -0.1, 0.8, -0.4, 0.6,
    -0.8, 0.2, 0.9, -0.3, 0.4, -0.6, 1.0, -0.9, 0.1, 0.7
)

shapiroWilkTest(sample).pValue       // high p → consistent with normality
andersonDarlingTest(sample).pValue   // high p → consistent with normality
dagostinoPearsonTest(sample).pValue  // combines skewness and kurtosis tests
jarqueBeraTest(sample).pValue        // asymptotic test based on skewness and kurtosis

Shapiro-Wilk is the most common choice and generally the most powerful for small-to-moderate samples (n < 5000). Anderson-Darling is more sensitive to deviations in the tails. D’Agostino-Pearson and Jarque-Bera are faster for large samples.

Do my observed counts match expected proportions?

Chi-squared goodness-of-fit

Tests whether observed counts match expected frequencies.

val observed = intArrayOf(50, 30, 20)
val expected = doubleArrayOf(40.0, 40.0, 20.0)

val result = chiSquaredTest(observed, expected)
result.statistic         // 5.0
result.pValue            // 0.0821
result.isSignificant()   // false at α = 0.05

G-test

Likelihood-ratio alternative to the chi-squared test. Often preferred for small expected counts.

val observed = intArrayOf(50, 30, 20)
val expected = doubleArrayOf(40.0, 40.0, 20.0)

val result = gTest(observed, expected)
result.statistic         // G statistic
result.pValue            // p-value

Binomial test

Tests whether the observed proportion of successes matches a hypothesized probability.

val result = binomialTest(successes = 60, trials = 100, probability = 0.5)
result.pValue            // p-value for H₀: p = 0.5
result.isSignificant()   // true if 60/100 is significantly different from 0.5

Fisher exact test

Exact test for 2×2 contingency tables. Preferred over chi-squared for small samples.

// 2×2 table: [[10, 30], [20, 40]]
val result = fisherExactTest(arrayOf(intArrayOf(10, 30), intArrayOf(20, 40)))
result.pValue            // exact p-value
result.isSignificant()   // true or false

Fisher exact test is designed for small contingency tables. For larger tables, use chiSquaredTest() with a chi-squared test of independence.

Do my groups have equal variances?

Variance homogeneity is an assumption of ANOVA and some t-test variants. Three tests are available:

val g1 = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val g2 = doubleArrayOf(2.0, 4.0, 6.0, 8.0, 10.0)
val g3 = doubleArrayOf(1.0, 3.0, 5.0, 7.0, 9.0)

leveneTest(g1, g2, g3).pValue         // robust to non-normality
bartlettTest(g1, g2, g3).pValue       // most powerful when normality holds
flignerKilleenTest(g1, g2, g3).pValue // rank-based, robust to outliers

Levene’s test is the safest default — it is robust to non-normality. Use Bartlett’s test only when the data within each group is approximately normal (Bartlett is more powerful in that case). Fligner-Killeen is the most robust to outliers.

Does my sample match a reference distribution?

Kolmogorov-Smirnov test

The two-sample KS test compares whether two samples come from the same continuous distribution.

val sample1 = doubleArrayOf(1.0, 2.0, 3.0, 4.0, 5.0)
val sample2 = doubleArrayOf(1.5, 2.5, 3.5, 4.5, 5.5)

val result = kolmogorovSmirnovTest(sample1, sample2)
result.statistic         // D = max difference between ECDFs
result.pValue            // p-value

Are any observations outliers?

Grubbs’ test

Grubbs’ test (the extreme studentized deviate test) formally checks whether the observation farthest from the mean is an outlier, assuming the remaining data is approximately normal. The test statistic is

G = \frac{\max_{i} |x_i - \bar{x}|}{s},

converted to a Student-

t

statistic on

N - 2

degrees of freedom and Bonferroni-corrected for having tested every observation.

// Response times (ms) with a suspected outlier
val latencies = doubleArrayOf(12.0, 14.0, 11.0, 13.0, 15.0, 98.0, 12.0)

val result = grubbsTest(latencies)
result.statistic                       // G statistic
result.pValue                          // Bonferroni-corrected p-value
result.additionalInfo["outlierIndex"]  // index of the suspected outlier
result.additionalInfo["outlierValue"]  // the suspected outlier's value
result.isSignificant()                 // true if outlier is significant at α = 0.05

Use Alternative.GREATER or Alternative.LESS to test a single tail when you only care about a suspiciously large or small value:

// Only test for a suspiciously large value (upper tail)
val data = doubleArrayOf(2.1, 2.5, 2.3, 2.8, 10.0, 2.4, 2.2)
val upper = grubbsTest(data, alternative = Alternative.GREATER)
upper.additionalInfo["outlierValue"] // 10.0 — the maximum

// Only test for a suspiciously small value (lower tail)
val dataLow = doubleArrayOf(2.1, 2.5, 2.3, 2.8, -5.0, 2.4, 2.2)
val lower = grubbsTest(dataLow, alternative = Alternative.LESS)
lower.additionalInfo["outlierValue"] // -5.0 — the minimum

For multiple outliers, grubbsTestIterative() reapplies the test and removes one significant outlier at a time until none remain or the sample shrinks below three observations.

// Remove multiple outliers by repeatedly applying the test
val data = doubleArrayOf(10.0, 11.0, 12.0, 13.0, 14.0, 80.0, 90.0)
val cleaned = grubbsTestIterative(data, alpha = 0.05)

cleaned.outlierIndices // indices (in the original array) that were removed
cleaned.cleanedData    // observations after removing all detected outliers
cleaned.iterations     // TestResult from each round (last one is non-significant)

Grubbs’ test assumes the data is approximately normal apart from the outlier. Validate with shapiroWilkTest() on the data with the suspected extreme removed before reporting a significant result.

The iterative procedure can mask outliers when several extremes cluster together — each test may be diluted by its peers. For large clusters prefer a dedicated multiple-outlier test (e.g. generalized ESD) or a robust estimator.

The Alternative Enum

Alternative controls the direction of the test:

Value	Meaning
`TWO_SIDED`	Test whether the parameter differs from the reference in either direction (default)
`GREATER`	Test whether the parameter exceeds the reference
`LESS`	Test whether the parameter is below the reference

Directional alternatives are supported by tTest, pairedTTest, mannWhitneyUTest, wilcoxonSignedRankTest, and binomialTest.

API Reference

Full API Reference

Browse all test functions, result types, and parameter overloads in the Dokka-generated reference.

Getting Started

Modules

Hypothesis Tests

Reading a Test Result

Is my sample mean different from a reference value?

Are two groups different?

Two-sample t-test

Paired t-test

Mann-Whitney U test

Wilcoxon signed-rank test

Are three or more groups different?

One-way ANOVA

Friedman test

Is my data normally distributed?

Do my observed counts match expected proportions?

Chi-squared goodness-of-fit

G-test

Binomial test

Fisher exact test

Do my groups have equal variances?

Does my sample match a reference distribution?

Kolmogorov-Smirnov test

Are any observations outliers?

Grubbs’ test

The Alternative Enum

API Reference

Full API Reference

​Reading a Test Result

​Is my sample mean different from a reference value?

​Are two groups different?

​Two-sample t-test

​Paired t-test

​Mann-Whitney U test

​Wilcoxon signed-rank test

​Are three or more groups different?

​One-way ANOVA

​Friedman test

​Is my data normally distributed?

​Do my observed counts match expected proportions?

​Chi-squared goodness-of-fit

​G-test

​Binomial test

​Fisher exact test

​Do my groups have equal variances?

​Does my sample match a reference distribution?

​Kolmogorov-Smirnov test

​Are any observations outliers?

​Grubbs’ test

​The Alternative Enum

​API Reference

Full API Reference

Reading a Test Result

Is my sample mean different from a reference value?

Are two groups different?

Two-sample t-test

Paired t-test

Mann-Whitney U test

Wilcoxon signed-rank test

Are three or more groups different?

One-way ANOVA

Friedman test

Is my data normally distributed?

Do my observed counts match expected proportions?

Chi-squared goodness-of-fit

G-test

Binomial test

Fisher exact test

Do my groups have equal variances?

Does my sample match a reference distribution?

Kolmogorov-Smirnov test

Are any observations outliers?

Grubbs’ test

The Alternative Enum

API Reference