One-Way ANOVA Test
One-way analysis of variance (ANOVA) compares the population means of
\(k\) independent groups using sample data from each group. It generalizes
the two-sample \(t\)-test to more than two groups while controlling the
overall Type I error rate.
Hypotheses
Suppose the populations have means \(\mu_1, \mu_2, \dots, \mu_k\). The
hypotheses of the one-way ANOVA test are:
\[
\begin{aligned}
H_0 &: \mu_1 = \mu_2 = \dotsb = \mu_k \\
H_1 &: \text{At least one population mean is different.}
\end{aligned}
\]
Decomposing variability
Let \(x_{ij}\) be the \(j\)-th observation in group \(i\), where
\(i = 1, \dots, k\) and \(j = 1, \dots, n_i\).
Denote the sample mean of group \(i\) by \(\bar{x}_i\),
the grand mean by \(\bar{x}\), the total sample size by
\(n = n_1 + \dotsb + n_k\), and the number of groups by \(k\).
Between-groups and within-groups sums of squares.
\[
\begin{aligned}
\text{SSB} &= \sum_{i=1}^{k} n_i\left(\bar{x}_i - \bar{x}\right)^{2} \\
\text{SSW} &= \sum_{i=1}^{k} \sum_{j=1}^{n_i}
\left(x_{ij} - \bar{x}_i\right)^{2}
\end{aligned}
\]
The total variability is split into these two components:
\[
\begin{aligned}
\text{SST} &= \sum_{i=1}^{k} \sum_{j=1}^{n_i}
\left(x_{ij} - \bar{x}\right)^{2} \\
&= \text{SSB} + \text{SSW}.
\end{aligned}
\]
Mean squares and F statistic
Each sum of squares is divided by its degrees of freedom to obtain a
mean square (an estimate of variance).
\[
\begin{aligned}
df_{\text{between}} &= k - 1, \\
df_{\text{within}} &= n - k, \\
\text{MSB} &= \dfrac{\text{SSB}}{df_{\text{between}}}, \\
\text{MSW} &= \dfrac{\text{SSW}}{df_{\text{within}}}.
\end{aligned}
\]
The one-way ANOVA test statistic compares the variability between
groups with the variability within groups:
\[
\begin{aligned}
F &= \frac{\text{MSB}}{\text{MSW}} .
\end{aligned}
\]
Decision rule
Under the null hypothesis \(H_0\) and the standard ANOVA assumptions
(independent random samples, normal populations, and equal variances),
the test statistic follows an F distribution with
\(df_1 = k - 1\) and \(df_2 = n - k\).
Right-tailed rejection region.
\[
\begin{aligned}
F_{\text{crit}}
&= F_{1-\alpha}\!\left(df_1 = k - 1,\ df_2 = n - k\right), \\
\text{Reject } H_0
&\quad \text{if } F \geq F_{\text{crit}}, \\
\text{Fail to reject } H_0
&\quad \text{if } F < F_{\text{crit}}.
\end{aligned}
\]
Equivalently, you may compute a right-tailed p-value
\(p = P(F_{df_1,df_2} \geq F)\) and compare it to the chosen
significance level \(\alpha\). The calculator uses these ideas to build
the ANOVA table, draw the F distribution graph, and report the test
decision.
