Statistics Online

Statistical tests, charts, probabilities and clear results. Automatically checks assumptions, interprets results and outputs graphs, histograms and other charts. The calculators support not only the test statistic and the p-value, but more results like effect size, test power, and the normality level. If one of the validations fails, the tool recommends a solution.

Checks assumptions Step-by-step results R code included ▶ Video tutorial

Z-test Calculator The general Z-test — one or two samples, means and proportions, when σ is known. Open calculator T-test Calculator The general T-test — one sample, two samples or paired, when σ is unknown. Open calculator

Test statistic calculators 26 tests

Test				Assumptions						Required sample data
#	Test name	Check	Statistic		Ind	σ	σ₁=σ₂	d	μ₀/p₀	x/p̂	n	S
1	One Sample Z-Test	Mean	$z = \frac{\bar{x} - μ_{0}}{σ / \sqrt{n}}$	✔	✔	✔			✔	✔	✔
2	One Sample T-Test	Mean	$t = \frac{\bar{x} - μ_{0}}{S / \sqrt{n}}$	✔	✔	✘			✔	✔	✔	✔
3	Two Sample Z-Test	Mean	$z = \frac{x_{1} - x_{2} - d}{\sqrt{\frac{σ^{2}}{n_{1}} + \frac{σ^{2}}{n_{2}}}}$	✔	✔	✔		✔		✔	✔
4	Two Sample T-Test (Pooled variance)	Mean	$t = \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - d}{s \sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}}$	✔	✔	✘	✔	✔		✔	✔	✔
5	Two Sample T-Test (Welch's)	Mean	$t = \frac{{\bar{x}}_{1} - {\bar{x}}_{2} - d}{\sqrt{\frac{{s_{1}}^{2}}{n_{1}} + \frac{{s_{2}}^{2}}{n_{2}}}}$	✔	✔	✘	✘	✔		✔	✔	✔
6	Two Sample Mann-Whitney U Test	Rank	$z = \frac{U - μ + c}{σ}$		✔			✔					✔
7	Paired T-Test	Mean	$t = \frac{{\bar{x}}_{d} - μ_{0}}{S_{d} / \sqrt{n}}$	✔	Paired								✔
8	Paired Wilcoxon Sign Rank Test	Rank	$z = \frac{W - μ + c}{σ}$		Paired								✔
9	One Way ANOVA Test	Mean	$F = \frac{{MS}_{Between}}{{MS}_{Within}}$	✔	✔		✔						✔
10	Repeated Measures ANOVA Test	Mean	$F = \frac{{MS}_{Between}}{{MS}_{Within}}$	✔	Dependent groups		Sphericity						✔
11	Friedman Test	Rank	$χ²(df) = \frac{12 \sum_{j = 1}^{k} {R_{j}}^{2} - 3 n^{2} k {(k + 1)}^{2}}{n k (k + 1) - C_{t i e s}}$		Dependent groups								✔
12	Kruskal-Wallis Test	Rank	$H = \frac{12}{n (n + 1)} \sum_{j = 1}^{k} \frac{{R_{j}}^{2}}{n_{j}} - 3 (n + 1)$		✔								✔
13	Two Way ANOVA Test Fixed model, Mixed model, Random mode, Mixed model with repeats	Mean	$F = \frac{{MS}_{Between}}{{MS}_{Within}}$	✔	✔		✔						✔
14	One Way MANOVA Test	Mean	$F = \frac{V}{s - V} . \frac{d f_{2}}{d f_{1}}$	✔	✔		✔						✔
15	One Sample Proportion Test	Proportion	$z = \frac{(\hat{p} - p_{0}) + c}{\sqrt{\frac{p_{0} (1 - p_{0})}{n}}}$	Binomial	✔				✔	✔	✔
16	Two Sample Proportion Test	Proportion	$z = \frac{{\hat{p}}_{1} - {\hat{p}}_{2} + Cont}{\sqrt{\hat{p} (1 - \hat{p}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}}$	Binomial	✔					✔	✔
17	Chi-Squared Test For Variance	σ	$χ^{2} = \frac{(n - 1) S^{2}}{{σ_{0}}^{2}}$	✔	✔						✔	✔
18	F Test For Variances	σ	$F = \frac{{S_{1}}^{2}}{{S_{2}}^{2}}$	✔	✔						✔	✔
19	Levene's Test For Variances	σ	$F = \frac{{MS}_{Between}}{{MS}_{Within}}$	✘	✔							✔
20	Chi-Squared Test For Goodness Of Fit McNemar test, Chi-squared test for association, Fisher test, Binomial test	Fit	$χ^{2} = \sum_{i = 1}^{n} \frac{{(o_{i} - e_{i})}^{2}}{e_{i}}$		✔								✔
21	Shapiro-Wilk Test	Fit	$W = \frac{{(\sum_{i = 1}^{⌊ n / 2 ⌋} (a_{i} (x_{n+1-i} - x_{i})))}^{2}}{\sum_{i = 1}^{n} {((x_{i} - \bar{x}))}^{2}}$	✔	✔								✔
22	Kolmogorov Smirnov Test	Fit	D = Max_1≤i≤n(D_i⁺,D_i^-)		✔								✔
23	Two-Sample Kolmogorov Smirnov Test	Fit	D = Max_1≤i≤n(D_i⁺,D_i^-)		✔								✔
24	Kaplan Meier Survival Analysis	Survival	$χ^{2} = \sum_{i = 1}^{n} \frac{{(o_{i} - e_{i})}^{2}}{e_{i}}$		✔								✔
25	DeLong test - Independent curves Compares the AUCs of two independent ROC curves	AUC	$D = \frac{{AUC}_{1} - {AUC}_{2}}{\sqrt{{Var}_{1} + {Var}_{1}}}$		✔								✔
26	DeLong test - Paired curves Compares the AUCs of two dependent ROC curves	AUC	$Z = \frac{{AUC}_{1} - {AUC}_{2}}{\sqrt{Var ₁ + Var ₂ - 2 * Cov (AUC ₁, AUC ₂)}}$		Paired								✔

** ✔ - Yes, ✘ - No, empty - irrelevant

Regression calculators 6 calculators

	Regression	Statistic
	Simple Linear Regression	$F = \frac{{MS}_{Regression}}{{MS}_{Error}}$
	Multiple Linear Regression	$F = \frac{{MS}_{Regression}}{{MS}_{Error}}$
	Binary Logistic Regression	χ² = 2(LL₁−LL₀)
	Multinomial Logistic Regression	χ² = 2(LL₁−LL₀)
	Propensity Score Matching	$SMD = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{s_{pooled}}$
	Neural Network Calculator	feed-forward · RNN · LSTM · GRU

Confidence interval, Sample size, Effect Size 4 calculators

	Calculator	Notes
	Sample size calculators	plan the study — n for means, proportions and more
	Test power calculators	1−β — the probability to detect a real effect
	Effect size calculators	Cohen's d, η², f²
	Confidence intervals calculators	mean, means difference, σ, proportion, correlation

Distribution calculators 10 calculators

	Distribution	Notes
	P-value	test statistic ⇄ p-value, for every common distribution
	Distributions	probabilities, quantiles and charts for all distributions
	Normal Distribution	z, probabilities and quantiles
	Binomial Distribution	exact and cumulative probabilities, B(n, p)
	T Distribution (Student's)	heavier tails than the normal
	Chi-Squared Distribution	χ²(df)
	F Distribution (Fisher-Snedecor)	ratio of variances
	Poisson Distribution	event counts, λ
	Weibull Distribution	lifetimes and reliability
	Exponential Distribution	time between events

Statistical tables 2 tables

	Table
	Z table
	T table

Basic statistics calculators 22 calculators

Each statistics calculator contains a step by step calculation.

Calculator	Statistic calculator	Statistic
Statistics calculator	Computes statistics such as count, average and standard deviation for each combination of the categorical variables.
Descriptive statistics calculator	Computes statistics such as count, average, and standard deviation for each variable.
Percentile calculator	A percentile tells you the percentage of values below a given value in a data set.
Mean Median Mode calculator	Q₁, Q₃, IQR, Range
Measures Of Dispersion calculator	Range, IQR, variance, standard deviation and more.
Standard Deviation calculator	Sample and population, APA style.	$S = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - x̄)}^{2}}{n - 1}}$
Variance calculator	Variance, Mean, SEM, MAD	$S^{2} = \frac{\sum_{i = 1}^{n} {(x_{i} - x̄)}^{2}}{n - 1}$
Coefficient of Variation calculator	Relative variability — the standard deviation divided by the mean.	$CV = \frac{s}{\bar{x}}$
Average calculator	Average, Sum, Count	$\bar{x} = \frac{x_{1} + \dots + x_{n}}{n}$
Sample Mean calculator	Sample mean, Sum, Count	$\bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}$
Geometric Mean calculator	Product, n-th root	$G M = \sqrt[n]{x_{1} x_{2} \dots x_{n}}$
Harmonic Mean calculator	An average for rates (e.g. average speed) — n divided by the sum of the reciprocals.	$H_{n} = \frac{n}{\frac{1}{x_{1}} + \dots + \frac{1}{x_{n}}}$
Probability calculator	Probability formula, Dependent events, Bayes' Theorem, Independent events	P(A), P(A\|B), P(A∩B), P(A∪B)
Combinations calculator	With/Without Repetitions	$C_{r}^{n} = \frac{n!}{r! (n - r)!}$
Permutation calculator	With/Without Repetitions	$P_{r}^{n} = \frac{n!}{(n - r)!}$
Correlation calculator	Pearson correlation, Spearman's rank correlation	r, ρ (rho)
Outliers	Tukey's Fences (Boxplot), Z-Score
Markov chain calculator	nth step probability vector, steady-state vector
Odds calculator	Odds to probability, probability to odds
Betting odds calculator	Fractional odds, decimal odds, American odds
Mean absolute deviation calculator	Average absolute distance from the mean.	\|x_i − μ\|
Median absolute deviation calculator	Median absolute distance from the median — robust to outliers.	\|x_i − Mdn\|

Visualization 18 makers

	Chart
	Advanced histogram maker
	Advanced box plot maker
	Violin plot maker
	Percentile calculator
	Venn diagram maker
	Venn diagram calculator
	Stem and Leaf plot
	ROC Calculator
	Cluster analysis
	Principal component analysis
	Bar graph maker
	Pie chart maker
	Line chart maker
	Scatter chart maker
	Area chart maker
	Bubble chart maker
	3D bubble chart maker
	3D chart maker

Guide

Goal

The goal of this website is to make statistics accessible to anyone, converting complex statistical jargon into common everyday language.
We help everybody to calculate statistics.
The calculators also provide R codes for your convenience.
We started adding a step-by-step work-through for education purposes, an easy guide on how to calculate test statistics.

Terms

We put in a lot of effort to provide accurate results and we automatically test every new page, The older pages were tested manually, but we slowly also automate the old pages.
We usually compare the results to the R results.
The information and tools on this website are given AS IS WITHOUT ANY WARRANTY OF ANY KIND.

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Statistics Online

Test statistic calculators 26 tests

Regression calculators 6 calculators

Confidence interval, Sample size, Effect Size 4 calculators

Distribution calculators 10 calculators

Statistical tables 2 tables

Basic statistics calculators 22 calculators

Visualization 18 makers

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