The Minus Sign Never Reaches the Spread

You'll learnwhere the sampling distribution of a difference between two sample means is centered, why the two variances add even though the means subtract, what each of the three wrong combining routes returns, why the smaller group controls the standard error, and which two degrees-of-freedom answers are accepted.

Comparing two groups means studying one new number: the difference between their sample means. Two teaching methods give 78.4 from 30 students and 71.2 from 28, so this pair of samples says 7.2 points. Another pair would say something else, and this animation draws the distribution those differences live in. Its center is the obvious one. Its spread is not: each group hands over its own s²/n, the two ADD under the minus sign — proven here by listing all nine outcomes of two tiny draws, where the differences and the sums both come out at variance 12 — and one square root is taken at the end. Subtracting the variances instead returns −1.104, which has no real square root; adding the standard errors returns 3.293, which is 40.7 percent too large and rings no alarm at all. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · CENTER AND SPREAD · VARIANCES ADD · DEGREES OF FREEDOM · v2

Each group hands over s² over n, the two add, and one root is taken at the end.

Before you start

What you're looking at

Two teaching methods: mean 78.4 from 30 students against mean 71.2 from 28. The axis shows the difference of the two sample means.

The question

Does the difference of two sample means wander more than one mean alone, or less — and where does its spread come from?

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