Q.45 Consider the data set 14, 18, 14, 14, 10, 29, 33, 31, 25. If you add 20 to each of the values, then
(A) both mean and variance change (B) both mean and variance are unchanged
(C) the mean is unchanged, variance changes (D) the mean changes, the variance is unchanged
Adding Constant to Dataset: Mean Changes, Variance Stays Same – Statistics MCQ Explained
Original Dataset Stats
The original mean is sum (14+18+14+14+10+29+33+31+25=188) divided by 9, yielding about 20.89. Variance is the average of squared deviations from this mean, roughly 61.11 (population formula). Standard deviation is sqrt(variance) ≈ 7.82.
Effect After Adding 20
New values: 34, 38, 34, 34, 30, 49, 53, 51, 45. New mean = 20.89 + 20 = 40.89 (shifts by constant). New variance remains 61.11, since deviations (e.g., 34-40.89 = -6.89 matches original 14-20.89) square identically.
Why Variance Unchanged
Variance formula: σ² = (1/n) Σ(xi - x̄)². Adding c=20 makes new points yi = xi + c, new mean ȳ = x̄ + c, so yi - ȳ = xi - x̄. Squared differences stay same.
