---
title: "Significance of statistics; missing frequencies"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:plants-missing-frequencies"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/plants-missing-frequencies
site: BCT Engineering
---

# Significance of statistics; missing frequencies

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

- Asked in: 2075 Chaitra, 6 marks
- Video: [#Arithmetic Mean (Missing Frequencies)](https://www.youtube.com/watch?v=1xrUW7BC8Ls) — Dr. Bhupender Kumar Som
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/plants-missing-frequencies
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

Write down the significance of statistics in engineering. An experiment shows the height of 51 plants given below. If average heights of all the 51 plants are 40 cm find the missing frequencies corresponding to the height 30 and 50 cm.
| Height (cm) | 10 | 20 | 30 | 40 | 50 | 60 |
|---|---|---|---|---|---|---|
| No. of plant | 2 | 3 | - | 21 | - | 5 |

## Short answer

Statistics in engineering aids quality control, reliability testing, design of experiments, decision making under uncertainty, error analysis, and modeling. For the plant heights: total 51 and mean 40 give equations a+b=20, 30a+50b=820, so missing frequencies: f30=9, f50=11.

## Answer

#### Significance of statistics in engineering
- **Quality control:** control charts and sampling inspection keep product quality within limits.
- **Reliability and testing:** life of components, failure rates and warranty periods are estimated from sample data.
- **Design of experiments:** helps to choose the best material, process or settings with fewer trials.
- **Decision making under uncertainty:** forecasting load, traffic or demand, and risk analysis.
- **Measurement and error analysis:** handling noise, variation and measurement errors in data.
- **Research and modelling:** regression and hypothesis tests validate models and compare designs.

#### Missing frequencies
Let the missing frequencies be f₃₀ = a and f₅₀ = b.

Total plants: 2 + 3 + a + 21 + b + 5 = 51, so a + b = 51 − 31 = 20 ... (1)

Mean = Σfx / Σf = 40, so Σfx = 40 × 51 = 2040.

| Height x | f | fx |
|---|---|---|
| 10 | 2 | 20 |
| 20 | 3 | 60 |
| 30 | a | 30a |
| 40 | 21 | 840 |
| 50 | b | 50b |
| 60 | 5 | 300 |
| Total | 51 | 1220 + 30a + 50b |

```text
1220 + 30a + 50b = 2040
30a + 50b = 820  →  3a + 5b = 82 ... (2)
```

Substitute a = 20 − b in (2):

```text
3(20 − b) + 5b = 82
60 + 2b = 82
b = 11,  a = 9
```

Check: 20 + 60 + 270 + 840 + 550 + 300 = 2040, and 2040/51 = 40. ✓

**Answer:** the frequency for 30 cm is **9** and for 50 cm is **11**.

## Check yourself

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

1. Which equation comes from the total of 51 plants?
   A. a + b = 51
   B. a + b = 31
   C. a - b = 20
   D. a + b = 20
   Answer: D. 2 + 3 + 21 + 5 = 31, so a + b = 51 - 31 = 20. (Key point: Total frequency equation)

2. Which equation comes from mean = 40?
   A. 30a + 50b = 820
   B. 30a + 50b = 2040
   C. a + b = 40
   D. 30a + 50b = 1220
   Answer: A. Σfx = 40 x 51 = 2040 and the known part is 1220, so 30a + 50b = 820. (Key point: Mean equation)

3. What are the frequencies for 30 cm and 50 cm?
   A. 11 and 9
   B. 9 and 11
   C. 10 and 10
   D. 8 and 12
   Answer: B. 3(20 - b) + 5b = 82 gives b = 11, a = 9. (Key point: Solution)

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