---
title: "Two companies: wages and combined SD"
subject: "Probability and Statistics"
chapter: 1
key: "semester_5/probability_and_statistics:two-companies-wage-combined-sd"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/two-companies-wage-combined-sd
site: BCT Engineering
---

# Two companies: wages and combined SD

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

- Asked in: 2073 Shrawan, 6 marks
- Video: [Introduction to Standard Deviation of Combined Group|BBA Maths|BCA Maths|BCOM Maths](https://www.youtube.com/watch?v=H6zzQX8XUj0) — Dream Maths
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/two-companies-wage-combined-sd
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-1

## Question

In two companies A and B engaged in similar type of industry, the average weekly wage and standard deviation are given below:
| | Company A | Company B |
|---|---|---|
| Average weekly wage (Rs) | 460 | 490 |
| Standard deviation | 50 | 40 |
| No. of wage earners | 100 | 80 |
i) Which company pays larger amount as weekly wages?
ii) Which company show greater variability in the distribution weekly wages?
iii) What is the mean and standard deviation of all the workers in two companies taken together?

## Short answer

Company A pays larger total wages (Rs 46,000 vs 39,200). Company A shows greater variability (CV 10.87% vs 8.16%). Combined mean = Rs 473.33, combined SD = Rs 48.19.

## Answer

Let n₁ = 100, x̄₁ = 460, σ₁ = 50 for Company A, and n₂ = 80, x̄₂ = 490, σ₂ = 40 for Company B.

#### i) Which company pays the larger amount?
Total weekly wages = number of workers × mean wage.

```text
Company A: 100 × 460 = Rs 46,000
Company B:  80 × 490 = Rs 39,200
```

Company **A** pays the larger total weekly wage bill (Rs 46,000). Company B pays the higher average wage per worker (Rs 490).

#### ii) Which shows greater variability?
Compare using the coefficient of variation, CV = σ / x̄ × 100.

```text
CV(A) = 50/460 × 100 = 10.87 %
CV(B) = 40/490 × 100 =  8.16 %
```

Company **A** has greater variability (higher CV). Its SD is also higher (50 against 40).

#### iii) Combined mean and SD

```text
Combined mean = (n1 x̄1 + n2 x̄2) / (n1 + n2)
              = (46,000 + 39,200) / 180
              = 473.33
```

Deviations of each mean from the combined mean:

```text
d1 = 460 − 473.33 = −13.33,  d1² = 177.78
d2 = 490 − 473.33 =  16.67,  d2² = 277.78
```

```text
σ² = [ n1(σ1² + d1²) + n2(σ2² + d2²) ] / (n1 + n2)
   = [ 100(2500 + 177.78) + 80(1600 + 277.78) ] / 180
   = (267,777.8 + 150,222.2) / 180
   = 418,000 / 180
   = 2322.2
σ  = √2322.2 = 48.19
```

**Answer:** (i) Company A has the larger wage bill (Rs 46,000 against Rs 39,200). (ii) Company A is more variable (CV 10.87 % against 8.16 %). (iii) Combined mean = Rs 473.33 and combined SD = Rs 48.19.

## Check yourself

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

1. Which company pays the larger total weekly wages, and how is it found?
   A. B, because its mean wage 490 is higher
   B. A, because 100 x 460 = Rs 46,000 exceeds 80 x 490 = Rs 39,200
   C. B, because its CV is smaller
   D. A, because its standard deviation is larger
   Answer: B. Total wages = workers x mean wage. A gives 46,000 and B gives 39,200, so A pays more in total even though B pays more per worker. (Key point: Total wage bill)

2. Why is the coefficient of variation used to compare the variability of the two companies?
   A. It is a unit-free relative measure, SD/mean x 100
   B. It is always equal to the variance
   C. It adds the two means together
   D. It uses the number of workers only
   Answer: A. CV = SD/mean x 100 expresses spread relative to the mean, so groups with different means can be compared. A has CV 10.87% against 8.16% for B. (Key point: Coefficient of variation)

3. What is the combined mean of all 180 workers?
   A. 476.67
   B. 470
   C. 473.33
   D. 475
   Answer: C. Combined mean = (46,000 + 39,200)/180 = 473.33. (Key point: Combined mean)

4. In the combined variance formula, why are d1^2 and d2^2 added to the group variances?
   A. Because the SDs are the same in both groups
   B. Because the group means differ from the combined mean, which adds extra spread
   C. To convert variance into SD
   D. To remove the effect of group sizes
   Answer: B. Each group's mean lies away from the overall mean, so n(sigma^2 + d^2) accounts for both within-group spread and between-group difference. The result is SD = 48.19. (Key point: Combined SD)

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