---
title: "Two groups: best, consistent, SE of difference"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:two-groups-best-consistent-se"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/two-groups-best-consistent-se
site: BCT Engineering
---

# Two groups: best, consistent, SE of difference

Probability and Statistics · Chapter 1: Descriptive Statistics and Basic Probability

- Asked in: 2076 Chaitra, 3+3+2 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/two-groups-best-consistent-se
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

The scores of randomly selected 32 students of two groups on Probability and Statistics are:
Group A: 50 37 13 37 45 9 11 34 24 6 13 24 32 32 24 40 33 45 37 38 33 32 46 32 46 21 32 4 45 16 43 32
Group B: 56 74 50 43 39 24 55 35 72 32 45 59 47 36 34 32 40 53 55 18 32 42 49 32 41 33 14 60 60 34 38 48
a) Which group is best?
b) Which group is more consistent?
c) Find standard error of difference of their means.

## Short answer

Group B mean 43.19, SD 13.98; Group A mean 30.19, SD 12.88. Group B is best (higher mean) and more consistent (CV 32.38% vs 42.66%). Standard error of difference = √(165.83/32 + 195.58/32) ≈ 3.36.

## Answer

Compare the groups by their means (which is better) and by the coefficient of variation CV = (s / x̄) × 100 (which is more consistent). Both groups have n = 32.

#### Calculations

| | Group A | Group B |
|---|---|---|
| Σx | 966 | 1382 |
| Σx² | 34,302 | 65,748 |
| Mean x̄ | 30.19 | 43.19 |
| Sxx = Σx² − (Σx)²/n | 5140.9 | 6062.9 |
| s² = Sxx/31 | 165.83 | 195.58 |
| s | 12.88 | 13.98 |
| CV | 42.66 % | 32.38 % |

Working for Group A: x̄ = 966/32 = 30.1875; Sxx = 34,302 − 966²/32 = 5140.875; s² = 5140.875/31 = 165.83.

#### a) Best group
Group **B** is better because its average score (43.19) is higher than that of A (30.19).

#### b) More consistent group
Group **B** is more consistent because its CV (32.38 %) is smaller than the CV of A (42.66 %).

#### c) Standard error of the difference of means

```text
SE(x̄A − x̄B) = √( sA²/nA + sB²/nB )
             = √( 165.83/32 + 195.58/32 )
             = √( 5.182 + 6.112 )
             = √11.294
             = 3.36
```

**Answer:** (a) Group B, (b) Group B, (c) SE ≈ 3.36.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. On what basis is Group B better?
   A. Its SD is larger
   B. Its CV is larger
   C. Its n is larger
   D. Its mean (43.19) is higher than A's (30.19)
   Answer: D. Average score decides which group is better. (Key point: Which group is best)

2. Why is Group B more consistent even though its SD is larger?
   A. Its CV (32.38%) is smaller than A's (42.66%)
   B. Its mean is smaller
   C. Its SD is smaller
   D. Its Σx^2 is larger
   Answer: A. CV relates SD to the mean; B's SD is a smaller share of its mean. (Key point: Consistency)

3. Which formula gives the SE of the difference of means?
   A. sqrt(sA^2 + sB^2)
   B. sqrt(sA^2/nA + sB^2/nB)
   C. (sA + sB)/n
   D. sqrt(sA^2/nA - sB^2/nB)
   Answer: B. Variances of independent means add, not subtract. (Key point: SE of difference)

4. What is the SE of the difference?
   A. About 13
   B. About 11.29
   C. About 3.36
   D. About 0.36
   Answer: C. sqrt(5.182 + 6.112) = sqrt(11.294) = 3.36. (Key point: SE value)

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