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In a data set of 2n observations, exactly half the values equal some number a and the remaining half equal -a. A constant b is then added to every observation. After this shift, the mean of the new data set is 5 and the standard deviation is 20. Find the value of a² + b².

1. 425
2. 650
3. 250
4. 925

Correct answer: 425

Solution

The original data set has mean 0 and standard deviation a. Adding b to every value shifts the mean to b = 5 and leaves the SD unchanged at a = 20. Hence a² + b² = 400 + 25 = 425.

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