Variance of the distribution $73, 77, 81, 85,..., 113$ is
- $10$
- $160$
- $161$
-
None of these
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C
Correct answer
AI explanation
The given sequence 73, 77, 81, 85, ..., 113 is an arithmetic progression with 11 terms, a first term of 73, and a common difference of 4. To simplify the calculation, subtract 73 from each term to create a new sequence: 0, 4, 8, 12, ..., 40, which does not change the variance. The sum of this new sequence is (11 / 2) times (0 + 40) = 220, making its mean 220 / 11 = 20. The variance is the average of the squared terms minus the square of the mean: (0^2 + 4^2 + 8^2 + ... + 40^2) / 11 - 20^2 = 16(0^2 + 1^2 + 2^2 + ... + 10^2) / 11 - 400. Using the formula for the sum of the first n squares, this equals 16(385) / 11 - 400 = 16(35) - 400 = 560 - 400. The result is 161.