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 quantity: the difference between their sample means. Its center is the obvious one — the true difference. Its spread is not: each group hands its own s²/n to the comparison, the two ADD under the minus sign, and one square root is taken at the end, giving √(2.187 + 3.291) ≈ 2.341 for the study on this page. Subtracting instead returns −1.104, a number with no real square root — an alarm. Adding the standard errors returns 3.293, which is 40.7% too large and rings no alarm at all, which is worse. And the degrees of freedom stop being a whole number: technology computes 53.0 here, the by-hand rule says 27, and the habitual 56 is neither of the accepted answers.

8 STEPS · 6 QUICK CHECKS · CENTER AND SPREAD · VARIANCES ADD · DEGREES OF FREEDOM · v1

2.187 + 3.291 = 5.478, one root: 2.341 · subtracting gives −1.104 · adding SEs gives 3.293

Before you start

What you're looking at

Two teaching methods: mean 78.4 from 30 students against mean 71.2 from 28. One new statistic — the difference of the two sample means — with its own sampling distribution.

The question

The center is obviously the true difference. How much does the difference of two sample means wander around it — more than one mean alone, or less?

Watch for

The plus sign inside the square root. Each group contributes its own s²/n, and the two ADD even though the means subtract. Every error in this topic is a different way of dodging that plus sign.

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