Question 1 OR (e)

First four central moments and the moment coefficients of skewness and kurtosis

9 marks

Time to write: About 18 minutes
Topic: Moments, skewness and kurtosis

Question

Question 1 OR (e) · 9 marks

A survey gives data on the monthly income (in thousand rupees) of some consumers, as below:

Monthly Income (000 ₹)	45 - 47	48 - 50	51 - 53	54 - 56	57 - 59
Number of Consumers	30	26	28	40	26

Calculate the following statistical measures:

The first four moments about the mean.
The moment coefficient of skewness.
The moment coefficient of kurtosis.

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

18 minfor 9 marks

Read from the question

CalculateFormula 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 Mean and the working table 2 marks 4 min
2 The four central moments 3 marks 5 min
3 Moment coefficient of skewness, with comment 2 marks 4 min
4 Moment coefficient of kurtosis, with comment 2 marks 3 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

1234

Try it first. About 18 minutes for 9 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. Moments, skewness and kurtosis are in Unit 1 of the syllabus; the DU guideline restricts their computation to the direct method through the actual mean. So d = x − x̄ and μᵣ = Σf dʳ / N. The classes are inclusive (45–47, 48–50, …); their true limits would be 44.5–47.5 and so on, which leaves the mid-points (46, 49, …) unchanged. Incomes are in ₹'000.

Working table

x̄ = Σfx / N = 7,818 / 150 = 52.12.

Income (₹'000)	x	f	fx	d = x − 52.12	fd²	fd³	fd⁴
45–47	46	30	1,380	−6.12	1,123.63	−6,876.63	42,084.96
48–50	49	26	1,274	−3.12	253.09	−789.65	2,463.72
51–53	52	28	1,456	−0.12	0.4	−0.05	0.01
54–56	55	40	2,200	2.88	331.78	955.51	2,751.88
57–59	58	26	1,508	5.88	898.93	5,285.73	31,080.12
Total		150	7,818		2,607.83	−1,425.09	78,380.69

(Σfd = 0, since deviations are taken from the actual mean.)

(i) The first four moments about the mean

μ₁ = 0.
μ₂ = 2,607.83 / 150 = 17.3855 (σ = √17.3855 = 4.1696).
μ₃ = −1,425.09 / 150 = −9.5006.
μ₄ = 78,380.69 / 150 = 522.5379.

(ii) Moment coefficient of skewness

γ₁ = μ₃ / σ³ = −9.5006 / (4.1696)³ = −9.5006 / 72.49 = −0.1311 (equivalently β₁ = μ₃² / μ₂³ = 0.0172, with the sign of μ₃ negative).

The distribution is slightly negatively skewed: the tail towards the lower incomes is a little longer, while the heaviest class (₹54,000–56,000) lies above the mean. Against the zero skewness of a symmetrical curve such as the normal, the departure is small.

(iii) Moment coefficient of kurtosis

β₂ = μ₄ / μ₂² = 522.5379 / (17.3855)² = 522.5379 / 302.26 = 1.7288 (γ₂ = β₂ − 3 = −1.27).

β₂ is well below 3, the value for the normal curve, so the distribution is platykurtic: flat-topped, with the consumers spread fairly evenly across the income classes rather than concentrated near the mean.

Where the 9 marks sit

#	Component	Marks
1	Mean and the working table	2
2	The four central moments	3
3	Moment coefficient of skewness, with comment	2
4	Moment coefficient of kurtosis, with comment	2
	Total	9

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