Question 1 (b)

Revised mean, standard deviation and coefficient of variation after late returns

6 marks

Time to write: About 12 minutes
Topic: Measures of dispersion

Question

Question 1 (b) · 6 marks

For an industry, the mean profit of 80 firms is found to be ₹55 crores with a coefficient of variation of 40%. During the survey, 5 firms could not report profits before the data were compiled. Their profits were reported later as ₹20, 26, 48, 36, and 52 crores. Find the revised mean, standard deviation and coefficients variation.

Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.

Before you write

Read off the question and its marks

12 minfor 6 marks

Read from the question

FindFormula first, then the working, then the answer with its unit (₹, %, units).

Write it in this order

Read the data; note exactly what is asked 1 min
1 SD and the two totals of the original group 2 marks 3 min
2 Totals of the late returns and the revised totals 1 mark 2 min
3 Revised mean 1 mark 2 min
4 Revised SD and CV, with comment 2 marks 3 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

1234

Try it first. About 12 minutes for 6 marks. Write it out, then open the solution.

Solution

Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.

Method. When observations are added to a group whose mean and standard deviation are known, rebuild the two totals Σx and Σx² of the original group from its mean and SD, add the new observations to each total, and work the revised measures from the new totals. The coefficient of variation is the standard deviation expressed as a percentage of the arithmetic mean. Totals of groups can be pooled in this way because the mean and the variance are both built from sums over all observations; the same idea gives the combined mean and combined SD of two groups.

Step 1: totals of the original 80 firms (profits in ₹ crore)

CV = (σ / x̄) × 100, so 40 = (σ / 55) × 100 and σ = 22.
Σx = n × x̄ = 80 × 55 = 4,400.
σ² = Σx²/n − x̄², so Σx² = n(σ² + x̄²) = 80 × (484 + 3,025) = 80 × 3,509 = 2,80,720.

Step 2: the five late returns

Firm	1	2	3	4	5	Total
Profit x (₹ crore)	20	26	48	36	52	182
x²	400	676	2,304	1,296	2,704	7,380

Step 3: revised totals

Group	Firms (N)	Σx	Σx²
Original firms	80	4,400	2,80,720
Late returns	5	182	7,380
All firms	85	4,582	2,88,100

Revised measures

Revised mean: x̄ = Σx / N = 4,582 / 85 = ₹53.91 crore.
Revised variance: σ² = Σx²/N − x̄² = 2,88,100 / 85 − (53.9059)² = 3,389.4118 − 2,905.8442 = 483.57.
Revised standard deviation: σ = √483.57 = ₹21.99 crore.
Revised coefficient of variation: CV = (21.99 / 53.91) × 100 = 40.79%.

Comment. The five late firms earn less on average (₹182 / 5 = ₹36.4 crore) than the first 80, so the mean falls from ₹55 crore to ₹53.91 crore. The SD hardly moves (22 to 21.99), but because the mean is now smaller the relative variability rises slightly, from 40% to 40.79%: profits in the industry are a little less uniform than the first survey suggested.

Where the 6 marks sit

#	Component	Marks
1	SD and the two totals of the original group	2
2	Totals of the late returns and the revised totals	1
3	Revised mean	1
4	Revised SD and CV, with comment	2
	Total	6

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