Hypothesis Tests

One-Way ANOVA

Compares the means of k independent, normally distributed samples.

Statusvalidated

Version1.0.0

Minitab equivalentStat > ANOVA > One-Way

Description

One-way analysis of variance (One-Way ANOVA) tests whether the means of k ≥ 2 independent groups are all equal. The total variation SST is decomposed into the between-group variation SSB and the within-group variation SSW. The F-statistic compares the mean square between groups to the mean square within groups. A significant F-test (p < α) indicates that at least one group mean differs from the others — but does not say which.

Formulas

F = \frac{M S _{B}}{M S _{W}} = \frac{S S _{B} / ( k - 1 )}{S S _{W} / ( N - k )}

F-statistic as ratio of mean squares

SSB

S S_{B} = i = 1 \sum k n_{i} (\overset{x}{ˉ}_{i} - \overset{x}{ˉ})^{2}

Between-group sum of squares

SSW

S S_{W} = i = 1 \sum k j = 1 \sum n_{i} (x_{ij} - \overset{x}{ˉ}_{i})^{2}

Within-group sum of squares

H₀

H_{0} : μ_{1} = μ_{2} = \dots = μ_{k}

Null hypothesis: All group means are equal

Assumptions

Observations are independent
Data within each group are normally distributed
Variances are equal across groups (homogeneity of variance)

Limitations

Use Welch ANOVA for substantially unequal variances
Use Kruskal-Wallis for non-normal data
Omnibus test only — does not identify which group differs (post-hoc tests required)

References

Fisher, R. A. (1925), Statistical Methods for Research Workers
NIST/SEMATECH e-Handbook of Statistical Methods, Section 7.4.3