One Way Analysis of Variance, Calculating the Test Statistic F

Statistics • Analysis of Variance

Written by STEM Calculators Team Published December 26, 2025 Updated February 24, 2026

Paste your grouped data and compute the one-way ANOVA quantities: SSB, SSW, MSB, MSW, and the test statistic F = MSB / MSW. A right-tail p-value is also reported.

Input format

Long format example: Method I,48. Wide format example header: Method I,Method II,Method III.

Significance level (α)

Used to show a reference critical value on the F curve (optional).

Graph scale (suggested percentile)

Controls how far right the F-curve panel extends.

Paste data or upload CSV

Data (CSV)

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Output: —

Visualization

Group means (top) and the F right-tail region (bottom) will appear after calculation.

k = — · n = — · df = (—, —)

The top graph shows group means (with animated jittered points). The bottom graph shows the F distribution with the right-tail p-value shaded.

Paste data and click “Calculate”.

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One-Way ANOVA: Calculating the test statistic

One-way ANOVA tests whether the means of k independent groups are all equal. The test is right-tailed and uses the F distribution.

How to use the calculator

Paste data in Long format (group,value) or Wide format (each column is a group).
Set α to show a reference critical value on the F curve.
Click Calculate to compute SSB, SSW, MSB, MSW, F, and the right-tail p-value.
Use Copy results CSV / Download results CSV to export the ANOVA summary.

Core formulas

Let k be the number of groups, n the total sample size, T_i the sum in group i, and n_i the size of group i.

\[ \begin{aligned} \mathrm{SSB} &= \left(\sum_{i=1}^{k}\frac{T_i^2}{n_i}\right)-\frac{(\sum x)^2}{n} \\ \mathrm{SSW} &= \sum x^2-\left(\sum_{i=1}^{k}\frac{T_i^2}{n_i}\right) \\ \mathrm{SST} &= \mathrm{SSB}+\mathrm{SSW} \end{aligned} \] \[ \begin{aligned} df_1 &= k-1 \\ df_2 &= n-k \\ \mathrm{MSB} &= \frac{\mathrm{SSB}}{df_1} \\ \mathrm{MSW} &= \frac{\mathrm{SSW}}{df_2} \end{aligned} \] \[ \begin{aligned} F &= \frac{\mathrm{MSB}}{\mathrm{MSW}} \end{aligned} \]

Assumptions (brief): independent random samples, approximately normal populations, and equal population variances.

Frequently Asked Questions

What does one-way ANOVA test?

One-way ANOVA tests whether the population means of three or more independent groups are all equal. It does this by comparing between-group variability to within-group variability using an F statistic.

How is the ANOVA F statistic computed?

The calculator computes MSB = SSB/df1 and MSW = SSW/df2, then forms F = MSB/MSW. Here df1 = k - 1 and df2 = N - k, where k is the number of groups and N is the total number of observations.

What are SSB and SSW in one-way ANOVA?

SSB (between groups) measures how far group means are from the grand mean, weighted by group sizes. SSW (within groups) measures variation of observations around their group means.

What assumptions are important for one-way ANOVA?

The groups should be independent random samples, the data in each group should be approximately normal for small samples, and the population variances are assumed to be roughly equal across groups. Strong violations can affect the reliability of the F test.

If the ANOVA p-value is small, which groups are different?

A small p-value indicates at least one group mean differs, but it does not identify which groups are different. Follow-up procedures such as multiple comparisons (for example Tukey) are used to determine which means differ.

Calculation steps

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Frequently Asked Questions

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