Equal Counts, a 25-Point Gap, and Two Summaries That Disagree

You'll learnto turn one table of counts into the three tables of proportions it supports, and to report a comparison as a difference and as a ratio without letting either one do the arguing.

500 employees, cross-classified by whether they took an optional training and whether they later passed a certification. 150 of the trained passed. 150 of the untrained passed. The two counts are identical, and the pass rates are 25.0 percentage points apart, because one 150 came out of 200 people and the other out of 300. Divide every cell by the grand total and the four numbers sum to 1; divide by each row's own total and every row sums to 1; divide by each column's and every column does. Then the comparison itself is a choice: 75.0% against 50.0% is a gap of 25.0 points and a ratio of 1.5, and a side effect striking 2 and 6 per 1,000 is a gap of 0.4 points and a ratio of 3.0. One summary ranks those studies one way and the other ranks them the other way round.

8 STEPS · 6 QUICK CHECKS · JOINT · MARGINAL · CONDITIONAL · PERCENTAGE POINTS · RATIO · v1

150 / 200 = 75.0% · 150 / 300 = 50.0% · difference 25.0 points · ratio 1.5

Before you start

What you're looking at

A two-way table of 500 employees. Rows are whether they took an optional training, columns are whether they passed a certification, and the edges carry the totals.

The question

Both pass counts are 150. Is that a table with no association in it — and once you know it is not, how large is the effect?

Watch for

The direction that sums to 1. It changes every time the divisor does, and it is what names the family of proportions you are reading.

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