Weighted Average Lab

Set the values and weights for each group. Watch how the weighted mean acts as a physical balance point while the simple average ignores group sizes.

Worked example — Weighted Average Lab

Weighted mean	63.00
Average of averages	75.00
Error from simple average	-12.00
Largest group share	90.0 %

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The average of the averages is not the average 🖖

Averaging two group percentages gives each group equal billing regardless of how many people are in them. When one group holds 90% of the population and the other holds 10%, giving them equal weight is an arithmetic error. The true overall rate is found by counting the actual totals, not by averaging the summary numbers.

The mean is a physical balance point 🖖

In mechanics, the center of mass is the point where weighted distances balance to zero: Σ wᵢ(xᵢ − x̄) = 0. A mean acts as exactly the same fulcrum. A value carrying ten times the weight exerts ten times the leverage on the average, which is why the fulcrum slides toward the larger group.

With two groups the error cannot pass half the spread 🖖

With two groups, however extreme the imbalance, the gap between the simple average and the weighted average cannot exceed half the distance between the smaller and the larger value. At equal sizes the gap is zero; at a 1/99 split it is 49% of the spread, and it reaches half only in the limit. Then add a third group and the bound goes with it, which is worth doing in the tool rather than taking on trust: four groups at 0 and one at 100 put the simple average at 20 and the weighted average just under 100, a gap of four fifths of the spread. The half only ever came from the simple average of TWO numbers sitting exactly between them.

Problem solved in full

Average raise for twelve people on 45,000 getting 6% and forty-eight on 72,000 7 steps
Twelve people on 45,000 get a 6% raise. Forty-eight on 72,000 get 2%. What was the average raise? There are two defensible answers and one wrong one, and the wrong one is the one that usually gets quoted.
1. The tempting move is to average the two rates. It gives a department of twelve the same say as a department of forty-eight, which is the error the tool above exists to show — but write it down anyway, because 4% is the figure that ends up in the announcement.
  
  $\frac{6% + 2%}{2} = 4.00%$
2. Weight by headcount instead and each person counts once. This is the honest answer to "what did the average employee get", and it is already more than a point below the number in the announcement.
  
  $\frac{12 ( 6% ) + 48 ( 2% )}{12 + 48} = \frac{72 + 96}{60} = 2.80%$
3. Money, though, is not paid per head. The second department is four times the size and each of its people earns more, so it holds 86% of the payroll. Work out what each group actually costs before the raise.
  
  $12 \times 45, 000 = 540, 00048 \times 72, 000 = 3, 456, 000$
4. Weight by payroll and you get the number the budget feels — the percentage the wage bill actually grew by. It is lower again, because the larger raise landed on the smaller pot.
  
  $\frac{540 , 000 ( 6% ) + 3 , 456 , 000 ( 2% )}{3 , 996 , 000} = \frac{32 , 400 + 69 , 120}{3 , 996 , 000} = 2.54%$
5. Both answers are correct. They answer different questions, and which one you need depends on whether you are counting people or money. The cash is the same either way, and it does not care which average anyone quotes.
  
  $32, 400 + 69, 120 = 101, 520$
6. Now the part this tool cannot show you. Every weighted average above assumes the same people are there before and after. Take twenty employees, ten on 30,000 and ten on 60,000 — a mean of 45,000.
  
  $10 \times 30, 000 + 10 \times 60, 000 = 900, 000 \Rightarrow \overset{x}{ˉ} = 45, 000$
7. Lay off the ten lowest paid, and cut everyone who remains by 10%. The mean is now 54,000. It rose by a fifth while every single person still employed lost a tenth, and ten more lost everything. No weighting error was made anywhere; the set changed underneath the average. That is why "average wage rose" has to be read alongside who was still being counted.
  
  $10 \times (60, 000 \times 0.9) = 540, 000 \Rightarrow \overset{x}{ˉ} = 54, 000 (+ 20%)$
Answer

2.80% weighted by headcount, 2.54% weighted by payroll, and about 101,500 a year in cash. The 4% is the one number that is simply wrong, and it is the one that gets announced.

Learning path

One number for many: summary statistics

Leads to Mean vs median the mean as a physical balance point where group weights pull on the fulcrum.

References (2)

The weighted mean, the balance-point property, and why averaging averages misleads Freedman, D., Pisani, R. and Purves, R. (2007). Statistics (4th ed.). W. W. Norton — Part II, on averages and the pitfalls of pooling groups of unequal size.
The reversal this tool deliberately does NOT claim Simpson, E. H. (1951). The Interpretation of Interaction in Contingency Tables. Journal of the Royal Statistical Society: Series B, 13(2), 238–241.

By the 42Math team · Updated 2026-08-25

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

10 at 90% and 90 at 60% - Class A (10 students) gets 90%; Class B (90 students) gets 60%. The average of 90% and 60% is 75.00%, but the true weighted average is 63.00% — twelve points lower.
Equal group sizes - When groups are equal (50 and 50), the simple average (75.00%) equals the weighted average (75.00%) exactly.
Pay rise: 10% vs 2% - 10 people get a 10% raise, 90 people get 2%. Averaging 10% and 2% gives 6.00%, but the actual payroll increase across all 100 people is 2.80%.
Extreme split: 1 vs 99 - 1 person gets 90%, 99 get 60%. The weighted average is 60.30%, while the plain average says 75.00% — a gap of -14.70 points.