Which of the following assertions are CORRECT? P: Adding 7 to each entry in a…
2012
Which of the following assertions are CORRECT?
P: Adding 7 to each entry in a list adds 7 to the mean of the list
Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list
R: Doubling each entry in a list doubles the mean of the list
S: Doubling each entry in a list leaves the standard deviation of the list unchanged
- A.
P, Q
- B.
Q, R
- C.
P, R
- D.
R, S
Attempted by 50 students.
Show answer & explanation
Correct answer: C
Answer: P and R are correct.
Explanation:
Adding a constant to every entry (statement P and Q): If each data value x_i is replaced by x_i + c, the mean increases by c because the average shifts by that constant. However, each deviation from the mean (x_i + c) - (mean + c) = x_i - mean, so the deviations — and therefore the standard deviation — do not change.
Multiplying every entry by a factor (statement R and S): If each x_i is replaced by k·x_i, the mean is multiplied by k (so the mean doubles when k = 2). The deviations become k·(x_i - mean), so the standard deviation is multiplied by |k| (it doubles when k = 2).
Conclusion: Adding 7 to every entry changes the mean by +7 but does not change the standard deviation (so P true, Q false). Doubling every entry doubles the mean and also doubles the standard deviation (so R true, S false).
Quick numeric check: For [1,2,3], mean = 2 and standard deviation ≈ 0.816. Adding 7 gives [8,9,10] with mean = 9 (2+7) and the same standard deviation ≈ 0.816. Doubling gives [2,4,6] with mean = 4 (2×2) and standard deviation ≈ 1.633 (≈2×0.816).