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ANOVA Calculator

Enter data for up to six groups to perform a one-way analysis of variance. The calculator shows the full ANOVA table, box plots, F-statistic, p-value, and effect size with step-by-step formulas.

Box Plots

16.620.82529.233.4Grand meanGroup 1n=5Group 2n=5Group 3n=5Value

Group Data

ANOVA Table

SourceSSdfMSFp-value
Between250.52125.340.410.00000467
Within37.2123.1--
Total287.714---
Decision (alpha = 0.05): Reject the null hypothesis. There is a statistically significant difference between group means.
Effect Size (eta squared)
0.8707 (large)

Step-by-Step Calculation

1. Grand Mean

Xˉgrand=∑allxijN\bar{X}_{\text{grand}} = \frac{\sum_{\text{all}} x_{ij}}{N}
Xˉgrand=sum of all values15\bar{X}_{\text{grand}} = \frac{\text{sum of all values}}{15}
Xˉgrand=24.87\bar{X}_{\text{grand}} = 24.87

2. Sum of Squares Between (SS_B)

SSB=∑i=1kni(Xˉi−Xˉgrand)2SS_B = \sum_{i=1}^{k} n_i (\bar{X}_i - \bar{X}_{\text{grand}})^2
SSB=5(24.6−24.87)2+5(30−24.87)2+5(20−24.87)2SS_B = 5(24.6 - 24.87)^2 + 5(30 - 24.87)^2 + 5(20 - 24.87)^2
SSB=250.5SS_B = 250.5

3. Sum of Squares Within (SS_W)

SSW=∑i=1k∑j=1ni(xij−Xˉi)2SS_W = \sum_{i=1}^{k} \sum_{j=1}^{n_i} (x_{ij} - \bar{X}_i)^2
SSW=sum of squared deviations within each groupSS_W = \text{sum of squared deviations within each group}
SSW=37.2SS_W = 37.2

4. Degrees of Freedom

dfB=k−1,dfW=N−kdf_B = k - 1, \quad df_W = N - k
dfB=3−1=2,dfW=15−3=12df_B = 3 - 1 = 2, \quad df_W = 15 - 3 = 12
dftotal=14df_{\text{total}} = 14

5. Mean Squares and F-statistic

MSB=SSBdfB,MSW=SSWdfW,F=MSBMSWMS_B = \frac{SS_B}{df_B}, \quad MS_W = \frac{SS_W}{df_W}, \quad F = \frac{MS_B}{MS_W}
MSB=250.52,MSW=37.212MS_B = \frac{250.5}{2}, \quad MS_W = \frac{37.2}{12}
F=125.33.1=40.41F = \frac{125.3}{3.1} = 40.41

6. P-value and Decision

p=P(F2,12>40.41)p = P(F_{2,12} > 40.41)
p=0.00000467p = 0.00000467
p<α=0.05  ⟹  Reject H0p < \alpha = 0.05 \implies \text{Reject } H_0

7. Effect Size (Eta Squared)

η2=SSBSST\eta^2 = \frac{SS_B}{SS_T}
η2=250.5287.7\eta^2 = \frac{250.5}{287.7}
η2=0.8707\eta^2 = 0.8707

Reference Guide

What is ANOVA?

Analysis of Variance (ANOVA) tests whether the means of three or more groups are equal. It does this by comparing the variation between groups to the variation within groups.

The null hypothesis is that all group means are equal. If the F-statistic is large enough, we reject the null.

The F-Statistic

The F-statistic is the ratio of between-group variance to within-group variance.

F=MSbetweenMSwithin=SSB/(k−1)SSW/(N−k)F = \frac{MS_{\text{between}}}{MS_{\text{within}}} = \frac{SS_B / (k-1)}{SS_W / (N-k)}

A large F means the group means differ more than you would expect from random variation alone.

Effect Size

Eta squared measures how much of the total variance is explained by the grouping variable.

η2=SSbetweenSStotal\eta^2 = \frac{SS_{\text{between}}}{SS_{\text{total}}}

Guidelines: below 0.01 is negligible, 0.01 to 0.06 is small, 0.06 to 0.14 is medium, and above 0.14 is large.

Assumptions

One-way ANOVA assumes that the observations are independent, each group is approximately normally distributed, and all groups have similar variances (homogeneity of variance).

ANOVA is fairly robust to moderate violations of normality, especially with larger sample sizes.