---
title: "Corrected mean and SD after item error"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:corrected-mean-sd-20-items"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/corrected-mean-sd-20-items
site: BCT Engineering
---

# Corrected mean and SD after item error

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

- Asked in: 2072 Chaitra, 6 marks
- Video: [Corrected Mean | Corrected Standard Deviation | corrected Mean and standard deviation](https://www.youtube.com/watch?v=EDw9NCq4xA4) — Prof. Evneet's COMMERCE CLASSES
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/corrected-mean-sd-20-items
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

What are the differences between measures of central tendency and measures of dispersion? The mean and standard deviation of 20 items is found to be 10 and 2 respectively. At the time of checking it was found that one item 8 was incorrect. Calculate the mean and standard deviation if: (i) the wrong item is omitted (ii) it is replaced by 12.

## Short answer

Corrected sums: (i) omitting 8: Σx=192, Σx²=2016; mean=10.105, σ≈1.997. (ii) replacing with 12: Σx=204, Σx²=2160; mean=10.2, σ≈1.990.

## Answer

#### Central tendency vs dispersion

| Basis | Measures of central tendency | Measures of dispersion |
|---|---|---|
| Purpose | Give the central or typical value | Show how far values are spread about the centre |
| Examples | Mean, median, mode | Range, quartile deviation, mean deviation, standard deviation |
| Answers | "Where is the middle?" | "How scattered is the data?" |
| Units | Same units as data | Absolute measures in the same units; relative measures have none |
| Use | Summarise a data set by one value | Judge reliability of the average and compare consistency |

#### Numerical
Given n = 20, x̄ = 10, σ = 2. The wrong item is 8. The SD is taken with divisor n (σ² = Σx²/n − x̄²).

```text
Σx  = 20 × 10 = 200
Σx² = 20 (4 + 100) = 2080
```

**(i) Wrong item omitted (n = 19)**

```text
Σx  = 200 − 8 = 192
Σx² = 2080 − 64 = 2016
Mean = 192/19 = 10.105
σ² = 2016/19 − (10.105)² = 106.105 − 102.116 = 3.989
σ = 1.997
```

**(ii) Replaced by 12 (n = 20)**

```text
Σx  = 200 − 8 + 12 = 204
Σx² = 2080 − 64 + 144 = 2160
Mean = 204/20 = 10.2
σ² = 2160/20 − (10.2)² = 108 − 104.04 = 3.96
σ = 1.990
```

**Answer:** (i) mean ≈ 10.11, SD ≈ 2.00; (ii) mean = 10.2, SD ≈ 1.99.

## Check yourself

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

1. What do measures of dispersion show?
   A. How far values spread about the centre
   B. The centre
   C. Most frequent value
   D. The sum
   Answer: A. Unlike mean/median/mode. (Key point: Dispersion vs central tendency)

2. After omitting 8, what is n and Σx?
   A. 20 and 192
   B. 19 and 192
   C. 19 and 200
   D. 20 and 204
   Answer: B. Remove 8 from the sum. (Key point: Omit item)

3. What is Σx^2 after omitting 8?
   A. 2080 - 8
   B. 2080
   C. 2080 - 64 = 2016
   D. 2016 - 64
   Answer: C. Subtract the square. (Key point: Σx^2 corrected)

4. What is mean when 8 replaced by 12?
   A. 10
   B. 10.4
   C. 10.1
   D. 204/20 = 10.2
   Answer: D. Σx rises by 4. (Key point: Replace)

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