---
title: "SD of marks; effect of adding one"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:sd-marks-increase-by-one"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sd-marks-increase-by-one
site: BCT Engineering
---

# SD of marks; effect of adding one

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

- Asked in: 2070 Chaitra, 3+3 marks
- Video: [Adding Vs. Multiplying Effect on Median and Standard Deviation](https://www.youtube.com/watch?v=raoTRsdAc3Y) — Mario's Math Tutoring
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/sd-marks-increase-by-one
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Calculate the standard deviation from the following data regarding marks obtained by students in a test:
| Marks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| No. of Students | 32 | 41 | 57 | 98 | 123 | 83 | 46 | 17 | 3 |
What will be the value of standard deviation if the marks obtained by each of the students are increased by one?

## Short answer

SD = √(Σfx²/N − (Σfx/N)²) = 1.774 marks. Adding 1 to every mark shifts the mean but does not alter deviations; the SD stays 1.774.

## Answer

Formula for a frequency distribution: σ = √[ Σfx²/N − (Σfx/N)² ].

| Marks x | f | fx | fx² |
|---|---|---|---|
| 1 | 32 | 32 | 32 |
| 2 | 41 | 82 | 164 |
| 3 | 57 | 171 | 513 |
| 4 | 98 | 392 | 1568 |
| 5 | 123 | 615 | 3075 |
| 6 | 83 | 498 | 2988 |
| 7 | 46 | 322 | 2254 |
| 8 | 17 | 136 | 1088 |
| 9 | 3 | 27 | 243 |
| Total | 500 | 2275 | 11,925 |

```text
Mean = 2275/500 = 4.55
σ² = 11,925/500 − (4.55)² = 23.85 − 20.7025 = 3.1475
σ  = √3.1475 = 1.774
```

(With divisor N − 1 the SD would be 1.776.)

#### Effect of adding one to every mark
Adding a constant to every observation shifts the mean by the same constant but does not change the distances from the mean. So the SD is **unchanged**: σ(x + 1) = σ(x). The new mean would be 5.55.

**Answer:** SD = 1.774 marks; after increasing each mark by one, the SD is still 1.774.

## Check yourself

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

1. Which formula is used for the SD of a frequency distribution here?
   A. Sum f/N^2
   B. (Sum fx)^2/N
   C. sqrt[Sum fx^2/N - (Sum fx/N)^2]
   D. Sum fx/N
   Answer: C. Variance is mean of squares minus square of mean: 23.85 - 4.55^2 = 3.1475, so SD = 1.774. (Key point: SD formula)

2. What is the mean mark?
   A. 23.85
   B. 5.55
   C. 4.55
   D. 3.15
   Answer: C. Sum fx = 2275 and N = 500, so mean = 4.55. (Key point: Mean)

3. What happens to the SD if every mark is increased by one?
   A. It rises by 1
   B. It doubles
   C. It is unchanged at 1.774
   D. It becomes 5.55
   Answer: C. Adding a constant shifts the mean by the same amount, so deviations from the mean do not change. The SD stays 1.774. (Key point: Adding a constant)

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