---
title: "Entrance scores of three institutes"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:entrance-scores-three-institutes"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/entrance-scores-three-institutes
site: BCT Engineering
---

# Entrance scores of three institutes

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

- Asked in: 2073 Chaitra, 8 marks; 2069 Chaitra, 1+2+2+3 marks (asked 2 times)
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/entrance-scores-three-institutes
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

The entrance score of three engineering institutes are as follows:
| Institute | Entrance scores |
|---|---|
| A | 740 800 830 840 860 890 830 930 1070 |
| B | 655 775 825 978 989 1025 950 980 1100 |
| C | 850 825 749 870 565 978 925 950 1000 |
a) Calculate mean and standard deviation (and coefficient of variation) for institute A, B and C.
b) Which institute is good?
c) Which institute is consistent?

## Short answer

Institute A: mean 865.56, SD 93.42, CV 10.79%. B: mean 919.67, SD 139.79, CV 15.20%. C: mean 856.89, SD 135.34, CV 15.79%. Institute B is best (highest mean). Institute A is most consistent (lowest CV).

## Answer

Use mean x̄ = Σx/n, sample SD s = √[(Σx² − n·x̄²)/(n − 1)] and CV = (s/x̄) × 100. Each institute has n = 9 scores. A higher mean means better performance. A lower CV means more consistent scores.

#### a) Mean, SD and CV

| Institute | Σx | Σx² | Mean | Sxx | s | CV |
|---|---|---|---|---|---|---|
| A | 7790 | 6,812,500 | 865.56 | 69,822.2 | 93.42 | 10.79 % |
| B | 8277 | 7,768,405 | 919.67 | 156,324.0 | 139.79 | 15.20 % |
| C | 7712 | 6,754,860 | 856.89 | 146,532.9 | 135.34 | 15.79 % |

Working for A:

```text
x̄ = 7790 / 9 = 865.56
Sxx = 6,812,500 − 9 × (865.56)² = 69,822.2
s² = 69,822.2 / 8 = 8727.8
s = √8727.8 = 93.42
CV = 93.42 / 865.56 × 100 = 10.79 %
```

Institutes B and C are worked the same way (s² = 19,540.5 and 18,316.6).

#### b) Which institute is good?
Institute **B** is the best because it has the highest mean entrance score (919.67).

#### c) Which institute is consistent?
Institute **A** is the most consistent because it has the smallest CV (10.79 %), against 15.20 % for B and 15.79 % for C.

If the population SD (divisor n) is used instead, the values are 88.08, 131.79 and 127.60 and the conclusions stay the same.

## Check yourself

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

1. On what basis is institute B declared the best?
   A. It has the highest mean (919.67)
   B. It has the lowest SD
   C. It has the lowest CV
   D. It has the smallest Σx^2
   Answer: A. A higher mean means better performance; B's mean is the highest. (Key point: Judging which is good)

2. Which measure is used to decide the most consistent institute?
   A. Largest mean
   B. Smallest CV
   C. Largest SD
   D. Largest range
   Answer: B. A lower CV means more consistent scores. (Key point: Judging consistency)

3. Which institute is most consistent and what are the CVs?
   A. B, with the highest mean
   B. C, with 15.79%
   C. A, with 10.79% against 15.20% (B) and 15.79% (C)
   D. A, with 15.20%
   Answer: C. A has the smallest CV (10.79%). (Key point: Most consistent institute)

4. Which formula is used for the sample SD in this answer?
   A. s = sqrt[(Σx^2 - n x̄^2)/n^2]
   B. s = (Σx^2 - n x̄^2)/(n - 1)
   C. s = sqrt[Σx^2/n]
   D. s = sqrt[(Σx^2 - n x̄^2)/(n - 1)]
   Answer: D. The corrected sum of squares is divided by n - 1 = 8, then square-rooted. (Key point: SD formula)

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