Reconciling Three Office Payrolls After New Hires
A firm has three offices. Office A has 40 employees with a mean annual salary of $52,000. Office B has 25 employees with a mean annual salary of $68,000. Office C has 15 employees, and the mean salary across all 80 employees in the three offices is $58,000. A consultant's draft report states: 'The average of the office averages is about $59,000, which we use as the company-wide figure.' Separately, management plans to hire 5 additional people into Office C and wants the company-wide mean salary to remain exactly $58,000 afterward. Salaries are treated as exact values, not rounded.
1. Find the mean salary in Office C, and explain why the consultant's figure misstates the company-wide mean.
Read the worked answer
Company total salary = 80 × $58,000 = $4,640,000.
- Office A total: 40 × $52,000 = $2,080,000
- Office B total: 25 × $68,000 = $1,700,000
- Office C total: $4,640,000 − $2,080,000 − $1,700,000 = $860,000
- Office C mean: $860,000 ÷ 15 = $57,333.33
The consultant computed (52,000 + 68,000 + 57,333.33) ÷ 3 ≈ $59,111, which treats the 15-person office as equally important as the 40-person office. A company mean is a weighted mean, weighted by headcount: (40(52,000) + 25(68,000) + 15(57,333.33)) ÷ 80 = $58,000. An unweighted average of group means is only correct when all groups have equal size - here they are 40, 25 and 15, so it overweights the high-paying, small office B.
2. What salary must each of the 5 new Office C hires receive so that the company-wide mean stays exactly $58,000?
Read the worked answer
After hiring, there are 80 + 5 = 85 employees. For a mean of $58,000: required total = 85 × $58,000 = $4,930,000.
Current total = $4,640,000, so the 5 new hires must account for $4,930,000 − $4,640,000 = $290,000.
Per hire: $290,000 ÷ 5 = $58,000.
This is the general invariance principle: adding observations does not change the overall mean exactly when the new observations average the old mean. It holds for any number of new hires - 5, 50, or 500 - and does not depend on which office they join. Any pay above $58,000 pushes the company mean up; any pay below pulls it down, in proportion to the number hired.
3. Suppose instead the 5 hires each receive $45,000. Compute the new company-wide mean and the new Office C mean, and say which of the two changed by more.
Read the worked answer
New hires' total: 5 × $45,000 = $225,000.
Company mean: (4,640,000 + 225,000) ÷ 85 = 4,865,000 ÷ 85 = $57,235.29 (a drop of about $765).
Office C mean: (860,000 + 225,000) ÷ (15 + 5) = 1,085,000 ÷ 20 = $54,250 (a drop of about $3,083).
Office C's mean moved about four times as much as the company mean, because the new hires are 5 of 20 in Office C but only 5 of 85 company-wide - the same dollar shock is diluted by the larger population. A small group's mean is sensitive to a few atypical members; a large aggregate is not.