Weighted averages, and why averaging averages fails
An ordinary mean gives every value one vote. That is the right thing to do exactly when every value deserves one vote, and it is wrong the rest of the time.
Grades where courses carry different credit, prices where you bought different quantities, ratings where one figure came from five reviews and another from two hundred — in all of these, some numbers stand for more than others, and a plain mean pretends they do not.
The method
Multiply each value by its weight, add up those products, then divide by the total of the weights.
weighted average = sum of (value × weight) ÷ sum of weights
The weights do not have to add to 100, or to anything in particular. Dividing by their total is what makes the scale cancel out, so credits, quantities and review counts can all be used raw.
A worked example with course credits
Three hypothetical courses, invented for this example:
- 4 credits, scored 92
- 3 credits, scored 78
- 1 credit, scored 60
The plain mean of the three scores: (92 + 78 + 60) ÷ 3 = 230 ÷ 3 = 76.67.
The weighted average: (4 × 92) + (3 × 78) + (1 × 60) = 368 + 234 + 60 = 662, then 662 ÷ (4 + 3 + 1) = 662 ÷ 8 = 82.75.
76.67 against 82.75. Six points apart, from the same three scores. The plain mean let the one-credit course count as much as the four-credit one, so the low score got four times the influence it was entitled to.
Averaging the averages, which is the same mistake wearing a suit
The version that catches people is a summary of summaries. Two groups report their own averages, and someone averages those two numbers.
Two hypothetical groups:
- Group A: 12 items, averaging 80
- Group B: 3 items, averaging 50
Average the averages: (80 + 50) ÷ 2 = 65.
Now do it properly. Recover each group's total from its average, because mean × count = total:
- Group A total:
80 × 12 = 960 - Group B total:
50 × 3 = 150 - Combined:
(960 + 150) ÷ (12 + 3) = 1,110 ÷ 15 = 74
65 against 74. Nine apart, and the wrong figure is the one that takes less work. Averaging the averages silently treats a group of 3 as equal in weight to a group of 12.
The two methods agree in exactly one case: when the groups are the same size. Equal weights cancel, and the weighted average collapses back into the plain mean. That is why the mistake goes unnoticed in testing on tidy example data and then goes wrong on real inputs.
The same shape, twice more
Prices by quantity. Buy 10 units at $2.00 and 90 units at $4.00. The mean of the two prices is $3.00. What you actually paid per unit is ((10 × 2) + (90 × 4)) ÷ 100 = (20 + 360) ÷ 100 = $3.80. Most of what you bought was the dear one.
Ratings by review count. A hypothetical 4.8 from 5 reviews and a 4.2 from 200. The mean of the two ratings is 4.5. Weighted by reviews: ((4.8 × 5) + (4.2 × 200)) ÷ 205 = (24 + 840) ÷ 205 = 4.21. Five reviews barely move a figure backed by two hundred, and they should not.
Picking the weights is the judgement
The arithmetic above is fixed. The weights are a decision, and it is the only decision in the calculation. Credits, headcount, quantity, review count, days in the period — whichever you choose, say so where the number is published, because a weighted average without its weights stated cannot be checked by anybody.
Two things to watch. A weight of zero removes an item entirely rather than including it neutrally. And if every weight is zero there is no answer at all, since the divisor is the total of the weights.
Doing it without the paper
The free Weighted Average tool takes a list of values and a matching list of weights, pairs them up by position — first value with first weight, and so on — and divides by the total weight, all in your browser with nothing sent anywhere. Two consequences worth knowing: a value with no weight opposite it is counted with a weight of zero, so a short weight list quietly drops the tail of your data; and if all your weights are equal you will simply get the plain mean back, which is a handy way to check you pasted the lists in the right order.