The numbers of books read by five students in a month are 4, 6, 8, 10, and 12. If each number is increased by 3, what is the variance of the new data set?
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Correct answer
A. 8
Principle or equation
Adding a constant to every data value shifts the mean but does not change the variance. The original data set is arithmetic with common difference 2, so its variance is the variance of {4,6,8,10,12}.
Why this answer is correct
Compute variance of original data: mean = (4+6+8+10+12)/5 = 40/5 = 8. Deviations: -4, -2, 0, 2, 4. Squares: 16,4,0,4,16. Sum = 40. Variance = 40/5 = 8. Adding 3 to each value does not change deviations, so variance remains 8.
Example
For data 1,2,3, variance is 2/3? Actually mean=2, deviations -1,0,1 squares 1,0,1 sum=2, variance=2/3. Adding 5 gives 6,7,8, variance still 2/3.
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