One-way ANOVA
Separated means are only half the story
Three groups have means 2, 3 and 5. ANOVA compares their separation with the variability of observations inside each group.
Keep all three means fixed but triple every deviation from its group mean. Watch within-group variation grow and the F statistic fall.
Individual observations & sample group means
Dots show individual observations in each group. Squares mark the sample means. Compare the separation between means with the variation within groups.
Null F density & observed F
This curve describes F statistics from repeated studies with equal population means. The vertical line marks the observed F. The shaded upper tail is its p-value.
Worked comparison
Follow the calculation
What the symbols mean
SS means sum of squares. k is the number of groups and N is the total sample size. Divide each sum of squares by its degrees of freedom to get a mean square. F is the between-group mean square divided by the within-group mean square.
Assumptions and full calculation
The simulated groups have population means of −spacing, 0 and +spacing. They share a population standard deviation, and observations are independent normal values. ANOVA divides the total sum of squares into between-group and within-group parts. Exact F-test p-values assume independent observations, normal errors and equal population variances. Even with equal population means, random samples usually give different sample means and an F above zero. A significant result does not identify which groups differ. Manual input allows unequal group sizes. Enter at least two observations per group.