---
title: "Ideal central tendency; correct mean and SD"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:characteristics-ideal-central-tendency"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/characteristics-ideal-central-tendency
site: BCT Engineering
---

# Ideal central tendency; correct mean and SD

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

- Asked in: 2068 Chaitra, 2+4 marks
- Video: [Mode, Median, Mean, Range, and Standard Deviation (1.3)](https://www.youtube.com/watch?v=mk8tOD0t8M0) — Simple Learning Pro
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/characteristics-ideal-central-tendency
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Write any four characteristics of ideal measure of central tendency. For a group of 16 candidates, the mean and standard deviation were found to be 20 and 5 respectively. Later it is discovered that the score 32 was measured as 23. Find the correct mean and correct standard deviation.

## Short answer

Ideal measure: rigidly defined, easy to understand/calculate, based on all observations, not unduly affected by extremes. Corrected mean = 20.56, corrected SD = 5.76 (using n=16, wrong score 23→32).

## Answer

#### Characteristics of an ideal measure of central tendency
1. It should be **rigidly defined**, so that everyone gets the same value from the same data.
2. It should be **easy to understand and calculate**.
3. It should be **based on all observations**.
4. It should be **not unduly affected by extreme values** (sampling stability).
5. It should be **suitable for further algebraic treatment**.

#### Corrected mean and SD
Given n = 16, x̄ = 20, σ = 5. The wrong score 23 should be 32. The SD is taken with divisor n.

```text
Σx  = 16 × 20 = 320
Σx² = 16 (σ² + x̄²) = 16 (25 + 400) = 6800
```

Corrected totals:

```text
Σx  = 320 − 23 + 32 = 329
Σx² = 6800 − 23² + 32² = 6800 − 529 + 1024 = 7295
```

Corrected mean and SD:

```text
Mean = 329 / 16 = 20.5625
σ²   = 7295/16 − (20.5625)² = 455.9375 − 422.8164 = 33.1211
σ    = √33.1211 = 5.755
```

**Answer:** correct mean ≈ 20.56 and correct SD ≈ 5.76.

## Check yourself

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

1. Which is a characteristic of an ideal measure of central tendency?
   A. Based on all observations
   B. Defined by judgement
   C. Easily distorted by extremes
   D. Open-ended
   Answer: A. An ideal average is rigidly defined, simple, based on all observations, stable and algebraically workable. (Key point: Ideal average)

2. How are the corrected Sum x and Sum x^2 found?
   A. Add 23 and 32 to both
   B. Subtract 32 from sum only
   C. Leave Sum x^2 unchanged
   D. Subtract 23 and add 32; subtract 23^2 and add 32^2
   Answer: D. Sum x = 329, Sum x^2 = 7295. (Key point: Corrected sums)

3. What is the correct mean?
   A. 20.5625
   B. 21.0
   C. 19.5
   D. 20.00
   Answer: A. 329/16 = 20.5625. (Key point: Correct mean)

4. What is the correct SD?
   A. 5.00
   B. 6.5
   C. 5.76
   D. 33.12
   Answer: C. Variance = 7295/16 - 20.5625^2 = 33.12, SD = 5.755. (Key point: Correct SD)

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