Three Fertilizers, One F-Test

Plots of wheat get one of three fertilizers. Yields vary within each group anyway, so is the difference between the groups bigger than that everyday variation? ANOVA's F statistic is the ratio of the two, and a large F says the fertilizers really differ.

How it works

One-way ANOVA: between-group vs within-group variation

In one sentence

One-way ANOVA tests $H_0: \mu_1 = \dots = \mu_k$ with $F = \text{MS}_{\text{between}}/\text{MS}_{\text{within}}$ , where $\text{MS}_{\text{between}} = \sum n_j(\bar x_j - \bar x)^2/(k-1)$ and $\text{MS}_{\text{within}} = \sum\sum(x_{ij} - \bar x_j)^2/(N-k)$ . Under $H_0$ , with normal groups of equal variance, $F$ follows the $F$ distribution with $k-1$ and $N-k$ degrees of freedom.

General formula

{\htmlData{sym=0}{F}} = \frac{\text{MS}_{\text{between}}}{\text{MS}_{\text{within}}} = \frac{\sum_j {\htmlData{sym=1}{n_j}}({\htmlData{sym=2}{\bar x_j}} - {\htmlData{sym=3}{\bar x}})^2/({\htmlData{sym=4}{k}}-1)}{\sum_j\sum_i ({\htmlData{sym=5}{x_{ij}}} - {\htmlData{sym=2}{\bar x_j}})^2/({\htmlData{sym=6}{N}}-{\htmlData{sym=4}{k}})}

Compare how far apart the groups are with how spread out each group is. Groups that are far apart relative to their own spread really differ.

What the symbols mean

$F$
F statistic
Between-group variation divided by within-group variation.
$n_j$
Group size
How many plots got fertilizer j (8 here).
$\bar x_j$
Group mean
The average yield of the plots that got fertilizer j.
$\bar x$
Grand mean
The average yield of all the plots.
$k$
Number of groups
Three fertilizers.
$x_{ij}$
One plot's yield
The yield of plot i in group j.
$N$
Total plots
All 24 plots together.

Within-group variation is how much plots that got the same fertilizer still differ; $\text{MS}_{\text{within}}$ pools it over all $k$ groups.
Between-group variation is how far each group mean $\bar x_j$ sits from the grand mean $\bar x$ , weighted by group size; $\text{MS}_{\text{between}}$ averages it over $k-1$ degrees of freedom.
If all the fertilizers were equal, both would estimate the same natural variance and $F$ would hover near 1. A big $F$ means the group means differ by more than chance.
ANOVA tests all the means at once. Running many separate t-tests instead would inflate the chance of a false alarm.

Learn the math

One-Way ANOVAStats · Lesson 41, Analysis of Variance Randomization, Replication and BlockingEng Stats · Lesson 24, Design of Experiments