Question 1 (b)

Mean deviation about the median of workers' wages

6 marks

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

Question

Question 1 (b) · 6 marks

The frequency distribution of wages (in 000 rupees) paid to the workers of a factory is given below:

Wages (in 000' ₹)	24 - 32	32 - 40	40 - 48	48 - 56	56 - 64	64 - 72	72 - 80
No. of Workers	5	10	19	33	15	13	5

Compute the Mean Deviation about the Median.

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

ComputeFormula 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 Cumulative frequencies and the median 2 marks 3 min
2 Deviations from the median and their total 3 marks 5 min
3 Mean deviation and coefficient 1 mark 2 min
Check: arithmetic, units (₹, %), final answer underlined 1 min

123

Short on time? Step 2 carries 3 of the 6 marks: guard its minutes.

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. Mean deviation about the median = Σf |x − Me| / N, where x is the class mid-point; the mean deviation is least when deviations are taken from the median. The DU guideline asks for the direct method, so deviations are taken from the median itself. The median of grouped data is found from the cumulative frequencies, Me = L + [(N/2 − c.f.) / f] × h. Wages are in ₹'000.

Median

N / 2 = 100 / 2 = 50; the 50th worker falls in the class 48–56 (c.f. 34 before it, f = 33):

Me = 48 + [(50 − 34) / 33] × 8 = 48 + 3.88 = 51.88 (₹51,880).

Mean deviation about the median

Wages (₹'000)	f	c.f.	Mid-point x	\|x − 51.88\|	f \|x − 51.88\|
24–32	5	5	28	23.88	119.4
32–40	10	15	36	15.88	158.8
40–48	19	34	44	7.88	149.72
48–56	33	67	52	0.12	3.96
56–64	15	82	60	8.12	121.8
64–72	13	95	68	16.12	209.56
72–80	5	100	76	24.12	120.6
Total	100				883.84

MD about the median = 883.84 / 100 = 8.84, that is ₹8,840.

Coefficient of mean deviation = MD / Me = 8.84 / 51.88 = 0.1704.

Comment. On average a worker's wage differs from the median wage of about ₹51,880 by about ₹8,840, or 17% of the median: wages in the factory are moderately spread.

Where the 6 marks sit

#	Component	Marks
1	Cumulative frequencies and the median	2
2	Deviations from the median and their total	3
3	Mean deviation (and coefficient)	1
	Total	6

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