Decomposing Variance

Statistics · Axiom Academy

Discover the key insight behind ANOVA: where does variance come from? Step 1: Observe the Total Spread Here are three groups of data points. Notice how all the points are scattered around. This overall spread is the total variance . Now let's distinguish the three groups with different colors. Move the group means to see how this affects the variance between groups. Step 3: Look Within Each Group Even within each group, points don't all sit exactly on the group mean. They scatter around it. This is the within-group variance . Step 4: The Variance Decomposition Here's the magic: the total variance can be split into these two components. Adjust both sliders and watch how the pieces add up! Measures how spread out all data points are from the grand mean, regardless of which group they belong to. Captures how different the group means are from each other. Large between-group variance means groups are clearly separated. Measures the natural scatter of points around their own group mean. This is the "noise" or "error" within each group. Every bit of variance in your data comes from one of these two sources. ANOVA uses this decomposition to test whether groups are truly different by comparing the ratio of between-group to within-group variance.

This is the written version of the interactive lesson above. See the full Statistics course.