The Proportions Subtract and the Uncertainties Add, and Only One of Those Is Obvious

You'll learnwhere the sampling distribution of a difference between two sample proportions is centered, why its spread comes from ADDING two variances even though the statistic subtracts, what the three wrong standard errors return, and why independence between the samples is the assumption the plus sign rests on.

A store compares checkout systems: 138 of 300 customers finish on the new one, 90 of 250 on the old — sample proportions of 0.46 and 0.36, a difference of 10 percentage points. To judge whether that gap means anything, you need the distribution of the gap itself, and it answers the same three questions as any sampling distribution. Center: the true difference, 0.10. Shape: approximately normal once all four counts clear 10. Spread: the one that surprises. Each group carries its own uncertainty into the comparison and neither cancels the other, so the variances add even though the proportions subtract, and the difference ends up wider than either proportion alone — 0.0418, against 0.0288 and 0.0304. This animation enumerates that claim outcome by outcome rather than quoting it, draws what the three tempting shortcuts return, and finishes on the assumption the whole formula rests on. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · VARIANCES ADD · FOUR COUNTS · INDEPENDENCE · v2

The gap has its own curve: centered at 10 points, wider than either group alone.

Before you start

What you're looking at

A store's two checkout systems — 138 of 300 finish on the new one, 90 of 250 on the old — and then the gap between those rates, as a quantity of its own.

The question

Where is that gap centered, and how wide is it? The center is the easy half; the width is where the standard error goes wrong.

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