One class has an average score of 60 and another has an average of 80. Is the combined average 70? Only if the two groups contain the same number of scores, or the figures happen to produce that result for another reason. The group sizes belong in the calculation.
Recover the total before combining the groups
Consider an invented example using the same test and marking scale. Class A has 10 pupils and an arithmetic mean of 60. Its total score is therefore 10 × 60 = 600. Class B has 30 pupils with a mean of 80, giving a total of 30 × 80 = 2,400.
Across both classes there are 3,000 points and 40 pupils. Divide 3,000 by 40 to obtain the combined mean: 75. Averaging 60 and 80 directly would give each class equal influence, even though the second class contains three times as many pupils.
Make the weighting visible
Draw four equal boxes, each representing ten pupils. Label one box 60 and the other three 80. The picture explains why the combined result sits closer to 80: more individual scores come from that group. The labels represent group means, not a claim that every pupil earned the same mark.
Try a second example by swapping the class sizes. Thirty pupils averaging 60 contribute 1,800 points; ten averaging 80 contribute 800. The combined mean becomes 2,600 ÷ 40 = 65. The two class averages have stayed the same, but the number of pupils behind each one has changed.
Check what the averages actually describe
This method requires arithmetic means of comparable measurements and the number of observations behind each mean. Do not apply it automatically to medians, percentages with unknown denominators or tests marked on different scales. If published averages have already been rounded, the reconstructed total may also be approximate.
For practice, give a learner two group sizes and two means. Ask them to predict which mean the combined result should be nearer before calculating. Then ask what would happen if the groups were equally large. Explaining whose scores receive weight is the central idea; the multiplication simply keeps that explanation honest.