---
title: "Central tendency and dispersion; median, pass marks"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:central-tendency-median-pass-marks"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-median-pass-marks
site: BCT Engineering
---

# Central tendency and dispersion; median, pass marks

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

- Asked in: 2075 Asoj, 6 marks
- Video: [Mean, Median & Mode for a Grouped Frequency Data Table | Statistics](https://www.youtube.com/watch?v=TIYN-FFuN6Y) — Brain Station
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/central-tendency-median-pass-marks
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Define measures of central tendency and measures of variance. Following data gives the distribution of marks of 50 students in statistics.
| Marks more than | 10 | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|---|
| No. of students | 50 | 45 | 35 | 20 | 10 | 4 | 1 |
Compute median marks. Also compute minimum marks obtained by a pass candidate if 60% student pass in the test.

## Short answer

Median marks = 36.67. Minimum pass marks for 60% passing (40th percentile) = 33.33. Convert ‘more than’ data to frequencies: 5,10,15,10,6,3,1. Median class 30–40, P₄₀ also in 30–40.

## Answer

#### Measures of central tendency
These give one representative value around which the data cluster. Examples: mean, median, mode.

#### Measures of variance (dispersion)
These show how widely the values are spread about the centre. Examples: range, quartile deviation, mean deviation, variance and standard deviation.

#### Convert "more than" into a frequency table
Number in a class = (more than lower limit) − (more than next limit). The last class is taken as 70-80.

| Marks | f | cf (less than) |
|---|---|---|
| 10-20 | 50 − 45 = 5 | 5 |
| 20-30 | 45 − 35 = 10 | 15 |
| 30-40 | 35 − 20 = 15 | 30 |
| 40-50 | 20 − 10 = 10 | 40 |
| 50-60 | 10 − 4 = 6 | 46 |
| 60-70 | 4 − 1 = 3 | 49 |
| 70-80 | 1 | 50 |

#### Median
N/2 = 25, so the median class is 30-40 (cf before = 15, f = 15).

```text
Median = 30 + (25 − 15)/15 × 10 = 30 + 6.67 = 36.67 marks
```

#### Pass mark
If 60% pass, the lowest 40% fail. The pass mark is the 40th percentile (P₄₀): position 0.40 × 50 = 20, class 30-40.

```text
P40 = 30 + (20 − 15)/15 × 10 = 30 + 3.33 = 33.33 marks
```

**Answer:** median = 36.67 marks; minimum marks of a passing candidate ≈ 33.3 (about 34 in whole marks).

## Check yourself

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

1. How is class frequency obtained from a 'more than' table?
   A. Add consecutive values
   B. Multiply
   C. Use as is
   D. Subtract next 'more than' from current one
   Answer: D. e.g. 10-20: 50 - 45 = 5. (Key point: More-than to frequency)

2. What is the median?
   A. 30 + (25-15)/15 x 10 = 36.67
   B. 30 + 25/15 x 10
   C. 30 + 10/15
   D. 40
   Answer: A. N/2 = 25 in class 30-40 with cf before 15. (Key point: Median)

3. Why is the pass mark P40?
   A. Top 60% fail
   B. Lowest 40% fail when 60% pass
   C. It is the median
   D. It is P60
   Answer: B. P40 = 30 + 5/15 x 10 = 33.33. (Key point: Pass mark)

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