Reference · 48 tables
Statistical Tables
Critical values and probabilities for the distributions and tests used in clinical research. Every value is computed exactly, and every table comes with a calculator for values it does not list, a worked example and a CSV download.
Normal Distribution
Areas under the standard normal curve and the critical z-values behind confidence intervals and z-tests.
- Z-Score TableThe area under the standard normal curve to the left of every z-score from −3.99 to 3.99. Use it to turn a z-score into a probability, a percentile or a p-value.z from −3.99 to 3.99 · 4 decimal places
- Z Critical ValuesThe z-values that cut off a given area in one or both tails of the standard normal distribution: the multipliers behind confidence intervals and z-tests.50% to 99.9% confidence · One- and two-tailed
t, Chi-Square and F Distributions
Critical values for t-tests, chi-square tests, ANOVA and tests on variances.
- Student's t TableUpper-tail critical values of Student's t-distribution, for t-tests and confidence intervals when the standard deviation is estimated from the sample.df 1 to 1000 and ∞ · 11 significance levels
- Chi-Square TableCritical values of the chi-square distribution for tests of independence, goodness of fit and variances, with both upper-tail and lower-tail areas.Every df from 1 to 100 · 11 tail areas
- F Table, α = 0.10Upper 10% critical values of the F distribution, used for ANOVA F-tests, regression F-tests and comparing two variances.α = 0.10 · df₁ 1 to ∞, df₂ 1 to ∞
- F Table, α = 0.05Upper 5% critical values of the F distribution, used for ANOVA F-tests, regression F-tests and comparing two variances.α = 0.05 · df₁ 1 to ∞, df₂ 1 to ∞
- F Table, α = 0.025Upper 2.5% critical values of the F distribution, used for ANOVA F-tests, regression F-tests and comparing two variances.α = 0.025 · df₁ 1 to ∞, df₂ 1 to ∞
- F Table, α = 0.01Upper 1% critical values of the F distribution, used for ANOVA F-tests, regression F-tests and comparing two variances.α = 0.01 · df₁ 1 to ∞, df₂ 1 to ∞
- F Table, α = 0.001Upper 0.1% critical values of the F distribution, used for ANOVA F-tests, regression F-tests and comparing two variances.α = 0.001 · df₁ 1 to ∞, df₂ 1 to ∞
- t Probability TableThe area to the right of t for Student's t distribution: turn a t-statistic into a p-value directly.t = 0 to 5 · df 1 to ∞
Discrete Distributions
Exact binomial, Poisson, hypergeometric and negative binomial probabilities.
- Binomial TableThe probability of exactly x successes in n independent trials, for n from 1 to 20.n = 1 to 20 · 15 values of p
- Cumulative Binomial TableThe probability of at most x successes in n independent trials, for n from 1 to 20.n = 1 to 20 · 15 values of p
- Poisson TableThe probability of exactly x events when events occur independently at an average rate of λ per interval.λ = 0.1 to 20 · All non-negligible x
- Cumulative Poisson TableThe probability of at most x events when events occur at an average rate of λ per interval.λ = 0.1 to 20 · All non-negligible x
- Hypergeometric TableProbabilities for the number of successes in a sample drawn without replacement from a finite population.N = 10 and 20 · Every K and n
- Negative Binomial TableProbabilities for the number of failures before the r-th success, including the geometric distribution (r = 1).r = 1 to 5 · 10 values of p
Nonparametric Tests
Exact critical values for rank-based and distribution-free tests, including trend and runs tests.
- Wilcoxon Signed-Rank TableExact critical values for the Wilcoxon signed-rank test, the nonparametric alternative to the paired t-test.n = 5 to 50 · Exact values
- Mann-Whitney U, α = 0.05Exact critical values for the Mann-Whitney U test (Wilcoxon rank-sum test), the nonparametric alternative to the two-sample t-test.n₁, n₂ = 2 to 20 · Exact values
- Mann-Whitney U, α = 0.01Exact critical values for the Mann-Whitney U test (Wilcoxon rank-sum test), the nonparametric alternative to the two-sample t-test.n₁, n₂ = 2 to 20 · Exact values
- Sign Test TableCritical values for the sign test, the simplest test for paired data, based only on the direction of each difference.n = 5 to 100 · Exact binomial values
- Kolmogorov-Smirnov TableExact critical values for the one-sample Kolmogorov-Smirnov test of whether data follow a fully specified distribution.n = 1 to 100 · Exact values
- Kruskal-Wallis TableExact critical values for the Kruskal-Wallis test with small samples, where the chi-square approximation is unreliable.k = 3 (nᵢ ≤ 7) and k = 4 (nᵢ ≤ 4) · Exact values
- Friedman Test TableExact critical values for the Friedman test, the nonparametric alternative to repeated-measures ANOVA.k = 3 to 6 treatments · Exact values
- Jonckheere-Terpstra TableExact critical values for the Jonckheere-Terpstra test for an ordered trend across groups, such as increasing dose.k = 3 to 6 groups · Exact values
- Page's L TableExact critical values for Page's trend test, used for ordered alternatives in repeated-measures designs.k = 3 to 8 · n = 2 to 20 blocks
- Runs Test TableCritical numbers of runs for testing whether a sequence of two kinds of item is random.n₁, n₂ = 2 to 20 · Exact values
- Two-Sample KS TableExact critical values for the two-sample Kolmogorov-Smirnov test of whether two samples come from the same distribution.Equal n to 40 · Unequal n₁, n₂ to 20
Normality and Goodness of Fit
Critical values for Shapiro-Wilk, Lilliefors and Anderson-Darling tests.
- Shapiro-Wilk TableCritical values and coefficients for the Shapiro-Wilk test of normality, consistent with the W statistic reported by R, SAS, Stata and SciPy.n = 3 to 50 · Coefficients included
- Lilliefors Test TableCritical values for the Kolmogorov-Smirnov test of normality when the mean and standard deviation are estimated from the data.n = 4 to 100 · 10 million simulations per n
- Anderson-Darling TableCritical values of the Anderson-Darling statistic, which gives extra weight to the tails of the distribution.Normality and specified distributions · n = 4 to 100
Multiple Comparisons and Outliers
Tukey, Dunnett and Bonferroni critical values, and Grubbs' and Dixon's outlier tests.
- Tukey q Table, α = 0.05Critical values of the studentized range distribution for Tukey's honestly significant difference test at α = 0.05.k = 2 to 20 groups · df 1 to ∞
- Tukey q Table, α = 0.01Critical values of the studentized range distribution for Tukey's honestly significant difference test at α = 0.01.k = 2 to 20 groups · df 1 to ∞
- Grubbs' Test TableCritical values for Grubbs' test, which checks whether the most extreme value in a normally distributed sample is an outlier.n = 3 to 500 · One- and two-sided
- Dunnett's Test TableCritical values of Dunnett's t for comparing several treatment groups with one control group while controlling the familywise error rate.k = 1 to 12 treatments · One- and two-sided
- Bonferroni t TableCritical t-values that keep the familywise error rate at α when you run m tests, each at level α/m.m = 2 to 30 comparisons · df 1 to ∞
- Dixon's Q Test TableCritical values for Dixon's gap-to-range tests for a single outlier in small samples.n = 3 to 30 · Four Dixon ratios
Correlation
Critical values for Pearson, Spearman and Kendall correlations, and the Fisher z-transformation.
- Pearson r Critical ValuesThe smallest correlation coefficient that is statistically significant for a given sample size and significance level.df 1 to 1000 · 5 significance levels
- Fisher z-TransformationFisher's transformation makes the sampling distribution of r approximately normal, so you can build confidence intervals and compare correlations.r = 0.000 to 0.999 · 4 decimal places
- Spearman's rho TableThe smallest Spearman correlation that is statistically significant for a given number of pairs.n = 4 to 100 · Exact for n ≤ 18
- Kendall's tau TableExact critical values of Kendall's tau, the rank correlation based on concordant and discordant pairs.n = 4 to 100 · Exact values
Variances and Regression Residuals
Tests for equal variances and the Durbin-Watson test for autocorrelated residuals.
- Cochran's C TableCritical values for Cochran's test of whether the largest of several group variances is too large.k = 2 to 120 groups · α = 0.05 and 0.01
- Hartley's Fmax TableCritical values of the ratio of the largest to the smallest sample variance, a quick check for equal variances.k = 2 to 12 groups · Exact integration
- Durbin-Watson TableLower and upper bounds for the Durbin-Watson test for first-order autocorrelation in regression residuals.n = 6 to 200 · k′ = 1 to 10 regressors
Confidence and Tolerance Intervals
Exact confidence limits for proportions and Poisson counts, and normal tolerance factors.
- Clopper-Pearson Exact CIExact confidence limits for a proportion, the standard choice for response rates in small single-arm trials.n = 1 to 30 · 90%, 95% and 99%
- Exact Poisson CIExact confidence limits for the expected number of events. Divide by the exposure time to get a confidence interval for a rate.x = 0 to 100 events · 90%, 95% and 99%
- Tolerance Factors (k)Factors k for tolerance intervals x̄ ± k·s that contain a stated proportion of the population with a stated confidence.n = 2 to 1000 · Exact values
Clinical Trial Safety and Dose Finding
The chance of detecting rare adverse events, and the operating characteristics of the 3+3 design.
- Adverse Event DetectionHow likely a trial is to see a rare adverse event, how many patients it needs, and what zero events really rules out.Incidence 0.01% to 20% · n = 10 to 10,000
- 3+3 Design ProbabilitiesExact operating characteristics of the 3+3 design at a single dose, and a calculator for a whole escalation scheme.DLT rate 1% to 70% · Exact probabilities
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