For four different populations, the number of positive xi values in the…
2026
For four different populations, the number of positive xi values in the population (n), the sum of xi values (∑xi), and the sum of squares of xi values (∑xi²) are given as A–D below. Compute the standard deviation of each population and arrange them in ascending order.
A. n = 10, ∑xi = 450, ∑xi² = 24250
B. n = 8, ∑xi = 104, ∑xi² = 1424
C. n = 5, ∑xi = 120, ∑xi² = 3600
D. n = 7, ∑xi = 28, ∑xi² = 140
Choose the correct answer from the options given below:
- A.
A, B, C, D
- B.
B, D, A, C
- C.
D, B, C, A
- D.
C, A, B, D
Show answer & explanation
Correct answer: C
For a population of n values with sum Σx and sum of squares Σx2, the population variance is σ2 = Σx2/n − (Σx/n)2, and the population standard deviation σ is simply its square root — one number that summarizes how spread out the values are around the mean. Because σ is always the non-negative square root of σ2, ranking populations by variance gives exactly the same order as ranking them by standard deviation.
Population A (n = 10, Σx = 450, Σx2 = 24250): mean = 450/10 = 45; variance = 24250/10 − 452 = 2425 − 2025 = 400; σ = √400 = 20.
Population B (n = 8, Σx = 104, Σx2 = 1424): mean = 104/8 = 13; variance = 1424/8 − 132 = 178 − 169 = 9; σ = √9 = 3.
Population C (n = 5, Σx = 120, Σx2 = 3600): mean = 120/5 = 24; variance = 3600/5 − 242 = 720 − 576 = 144; σ = √144 = 12.
Population D (n = 7, Σx = 28, Σx2 = 140): mean = 28/7 = 4; variance = 140/7 − 42 = 20 − 16 = 4; σ = √4 = 2.
As an independent check, comparing the four variances directly (400, 9, 144, 4 for A, B, C, D) gives the same ranking as comparing their square roots, since the square-root function preserves order for non-negative numbers — so the variance ranking alone already confirms the standard-deviation ranking.
Arranging the four standard deviations in ascending order: D (2) < B (3) < C (12) < A (20) — that is, the sequence D, B, C, A.