---
title: "Central tendency; mean, median, SD of marks"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:mean-median-sd-marks-100-students"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/mean-median-sd-marks-100-students
site: BCT Engineering
---

# Central tendency; mean, median, SD of marks

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

- Asked in: 2073 Chaitra, 6 marks
- Video: [Statistics - Mean, Median & Mode for a grouped frequency data](https://www.youtube.com/watch?v=RALOqYe9JEI) — Vectors Academy
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/mean-median-sd-marks-100-students
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Describe the various measures of central tendency and its application. The following table represents the marks of 100 students.
| Marks | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|---|---|---|---|---|
| No. of students | 14 | 18 | 27 | 26 | 15 |
Find the mean, median and standard deviation of all 100 students.

## Short answer

Using midpoints: mean = Σfx / 100 = 52 marks. Median class 40–60, median = 40 + (18/27)×20 ≈ 53.33 marks. Standard deviation = √(636) ≈ 25.22 marks.

## Answer

#### Measures of central tendency and their use
- **Arithmetic mean:** the sum divided by the number of values. Used for general averages such as average marks or income, and in further statistical work.
- **Median:** the middle value of the ordered data. Used for skewed data and open-ended classes (for example, median income).
- **Mode:** the most frequent value. Used when the most common item matters, such as the most popular shoe size.
- **Geometric mean:** used for growth rates, ratios and index numbers.
- **Harmonic mean:** used for averages of rates such as speed over equal distances.

#### Calculation
Class width h = 20 and N = 100.

| Marks | Mid x | f | fx | f(x − 52)² | cf |
|---|---|---|---|---|---|
| 0-20 | 10 | 14 | 140 | 24,696 | 14 |
| 20-40 | 30 | 18 | 540 | 8,712 | 32 |
| 40-60 | 50 | 27 | 1350 | 108 | 59 |
| 60-80 | 70 | 26 | 1820 | 8,424 | 85 |
| 80-100 | 90 | 15 | 1350 | 21,660 | 100 |
| Total | | 100 | 5200 | 63,600 | |

**Mean**

```text
x̄ = Σfx / N = 5200 / 100 = 52
```

**Median:** N/2 = 50, class 40-60 (cf before = 32, f = 27)

```text
Median = 40 + (50 − 32)/27 × 20 = 40 + 13.33 = 53.33
```

**Standard deviation**

```text
σ² = Σf(x − x̄)² / N = 63,600 / 100 = 636
σ  = √636 = 25.22
```

(With divisor N − 1 the SD would be 25.35.)

**Answer:** mean = 52, median ≈ 53.33, standard deviation ≈ 25.22 marks.

## Check yourself

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

1. How is the mean found?
   A. Σf/N
   B. Midpoint of the first and last class
   C. Σfx/N using mid-values = 52
   D. Σx/5
   Answer: C. Mid-values weighted by frequency. (Key point: Mean from grouped data)

2. What is the median?
   A. 40 + 50/27 x 20
   B. 60
   C. 52
   D. 40 + (50-32)/27 x 20 = 53.33
   Answer: D. N/2 = 50 is in 40-60. (Key point: Median)

3. How is SD computed?
   A. sqrt[Σf(x - x̄)^2/N] = 25.22
   B. Σf(x - x̄)^2/N = 636
   C. sqrt(Σfx/N)
   D. sqrt(636)/100
   Answer: A. Variance 636, SD is its square root. (Key point: SD)

4. When is median preferred?
   A. Ratios and growth
   B. Skewed data and open-ended classes
   C. Rates like speed
   D. Qualitative data
   Answer: B. Unaffected by extremes. (Key point: Use of median)

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