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.