Weighted Mean of Two Groups

You cannot average two averages unless the groups are the same size. Twenty students scoring a mean of 80 and thirty scoring 70 do not combine to 75; the correct answer weights each mean by its count: (20 × 80 + 30 × 70)/50 = 3700/50 = 74. The pull is always toward the larger group. This is the scalar form of the general weighted mean, and it covers most of what people actually need — merging two class sections, two shifts, two lab runs — without requiring a full data list.

Ignoring the weights is one of the most common quantitative errors in circulation, and it has a famous escalation: Simpson's paradox, where a treatment can look better in every subgroup yet worse overall once unequal group sizes are folded in. The 1973 Berkeley graduate-admissions case is the textbook example — the university appeared to favour men overall, but almost every individual department favoured women; the aggregate simply reflected that women applied more often to departments that admitted few of anyone. Reversed, the formula does mixture arithmetic: if 20 students averaging 80 must be blended down to a combined 74 with a second group averaging 70, that second group needs 20 × (74 − 80)/(70 − 74) = 30 students.