---
title: "Breaking strength: average, median, SD"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:breaking-strength-three-materials"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/breaking-strength-three-materials
site: BCT Engineering
---

# Breaking strength: average, median, SD

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

- Asked in: 2069 Chaitra, 2×3 marks
- Video: [Measures of Dispersion | Variance, Standard Deviation, Range,  IQR, Quartiles, Percentiles | Part 8](https://www.youtube.com/watch?v=TLRKawtvC1Q) — Tech Classes
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/breaking-strength-three-materials
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

The following are data on the breaking strength (in pounds) of 3 kinds of material:
| Material 1 | 144 | 181 | 200 | 187 | 169 | 171 |
|---|---|---|---|---|---|---|
| Material 2 | 186 | 194 | 176 | 182 | 133 | 183 |
| Material 3 | 197 | 165 | 180 | 198 | 175 | 164 |
i) Calculate the average breaking strength and the median breaking strength for each material.
ii) Calculate standard deviation and variance for each material.

## Short answer

Material 1: mean 175.33 lb, median 176 lb, variance 363.47 lb², SD 19.06 lb. Material 2: mean 175.67 lb, median 182.5 lb, variance 471.47 lb², SD 21.71 lb. Material 3: mean 179.83 lb, median 177.5 lb, variance 223.77 lb², SD 14.96 lb.

## Answer

For each material n = 6. Mean = Σx/n, median is the average of the 3rd and 4th ordered values, sample variance s² = [Σx² − (Σx)²/n]/(n − 1), and s = √s².

#### i) Average and median

| Material | Ordered values | Σx | Mean | Median |
|---|---|---|---|---|
| 1 | 144, 169, 171, 181, 187, 200 | 1052 | 175.33 | (171 + 181)/2 = 176 |
| 2 | 133, 176, 182, 183, 186, 194 | 1054 | 175.67 | (182 + 183)/2 = 182.5 |
| 3 | 164, 165, 175, 180, 197, 198 | 1079 | 179.83 | (175 + 180)/2 = 177.5 |

#### ii) Variance and standard deviation

| Material | Σx² | Sxx | s² (÷ 5) | s |
|---|---|---|---|---|
| 1 | 186,268 | 1817.33 | 363.47 | 19.06 |
| 2 | 187,510 | 2357.33 | 471.47 | 21.71 |
| 3 | 195,159 | 1118.83 | 223.77 | 14.96 |

Working for Material 1:

```text
Sxx = 186,268 − (1052)²/6 = 186,268 − 184,450.67 = 1817.33
s²  = 1817.33 / 5 = 363.47
s   = 19.06
```

**Answer:** averages 175.33, 175.67, 179.83 lb; medians 176, 182.5, 177.5 lb; variances 363.47, 471.47, 223.77 lb²; SDs 19.06, 21.71, 14.96 lb. Material 3 has the highest average and the smallest spread. Material 2 has the greatest spread.

## Check yourself

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

1. How is the median found for 6 values?
   A. Sum divided by 6
   B. Average of the 3rd and 4th ordered values
   C. The 6th value
   D. The 3rd value
   Answer: B. With an even number of values, take the mean of the two middle ordered values. Material 1 gives (171+181)/2 = 176. (Key point: Median for n = 6)

2. Which formula gives the sample variance?
   A. [Sum x^2 - (Sum x)^2/n]/(n-1)
   B. Sxx x (n-1)
   C. Sum x^2/n^2
   D. Sum x/n
   Answer: A. s^2 = Sxx/(n-1); for material 1, 1817.33/5 = 363.47. (Key point: Variance formula)

3. Which material has the highest mean and smallest spread?
   A. All equal
   B. Material 3
   C. Material 2
   D. Material 1
   Answer: B. Material 3 has mean 179.83 and SD 14.96, the smallest. (Key point: Comparison)

## Related in Probability and Statistics

- [Five number summary and box plot of guests](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Afive-number-summary-guests) — Probability and Statistics, chapter 1
- [ANOVA: breaking strength of three cable brands](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aanova-three-brands-cables) — Probability and Statistics, chapter 6, asked 2×
- [Central tendency; mean, median, SD of marks](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Amean-median-sd-marks-100-students) — Probability and Statistics, chapter 1
- [Paired t-test: dietary product weight loss](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apaired-t-test-dietary-weight-loss) — Probability and Statistics, chapter 6
- [Central tendency; middle 70% income, median](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acentral-tendency-middle-70-income) — Probability and Statistics, chapter 1
- [Central tendency and dispersion; median, pass marks](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acentral-tendency-median-pass-marks) — Probability and Statistics, chapter 1

## Related in other subjects

- [Rankine's data, 8° curve, 187.62 m long chord](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asimple-curve-8deg-18762-long-chord) — Surveying II, chapter 6
- [Insulator discs per string, 180 MW over 200 km](https://www.bctengineer.com/question.md?key=semester_8%2Ftransmission_and_distribution_system_design%3Ainsulator-discs-180mw-200km) — Transmission and Distribution System Design, chapter 3
- [Brick and mortar for 23 cm wall (165 kN/m)](https://www.bctengineer.com/question.md?key=semester_5%2Fconcrete_technology_and_masonry_structure%3Awall-eccentric-23cm-165kn) — Concrete Technology and Masonry Structure, chapter 8, asked 7×
- [Demagnetizing via B-H curve; data storage material](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Ademagnetize-bh-curve-storage) — Electrical Engineering Material, chapter 4