Problem 20 · 1990 AJHSME
Hard
Arithmetic & Operations
mean-error
The annual incomes of 1,000 families range from 8200 dollars to 98,000 dollars. In error, the largest income was entered on the computer as 980,000 dollars. The difference between the mean of the incorrect data and the mean of the actual data is
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Answer: A — 882 dollars.
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Hint 1 of 2
Don't compute either mean — you can't, you don't know the other 999 incomes! Only ONE number changed, so ask: by how much did the *total* change, and how does that one change ripple into the average?
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Hint 2 of 2
The 999 unchanged incomes cancel out completely when you subtract the two means. The mean shifts by (size of the error) spread evenly over all 1000 families.
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Approach: subtract the means — everything cancels but the one error, spread over 1000
- The two data sets differ in *one* entry: 980,000 instead of 98,000. Every other family is identical, so when you subtract (wrong mean) − (right mean), all those matching incomes cancel.
- What's left is just the single error spread over everyone: the total was overstated by 980,000 − 98,000 = 882,000.
- A mean is total ÷ 1000, so the mean is off by 882,000 ÷ 1000 = $882.
- *Why this transfers:* changing one value in a set of n shifts the mean by (the change) ÷ n — you never need the other values, and the giant 'range 8200–98,000' detail is a red herring.
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