Question 1 (c)
Wages before and after a settlement: bill, mode, variation and skewness
9 marks
- Time to write
- About 18 minutes
Question
Question 1 (c) · 9 marksA car manufacturing company, facing a dispute with its labour, decided after deliberation with the labour union to change its compensation policies. The table below shows information before and after the settlement:
| Before dispute | After dispute | |
|---|---|---|
| No. of workers | 1800 | 1700 |
| Mean wages | 620 | 665 |
| Median Wages | 650 | 620 |
| Standard Deviation | 48 | 38 |
Compare the position before and after the settlement of the dispute in terms of:
The total wages bill,
The modal wages,
The coefficient of variation,
Skewness, and
Also comment on whether the employees gained from this settlement or not.
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
Read from the question
- Compare…Comment…Two asks in one question: answer both.
- comment onThe comment carries marks too: say what the figures show, in two or three lines.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Total wages bill 1½ marks 3 min
- 2 Modal wages 1½ marks 3 min
- 3 Coefficient of variation 1½ marks 3 min
- 4 Skewness 2 marks 3 min
- 5 Comment on the settlement 2½ marks 4 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
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.
Measures used. Total wages = number of workers × mean wage. The mode is estimated from Karl Pearson's empirical relation, Mode = 3 Median − 2 Mean. CV = (σ / x̄) × 100. Skewness is measured by Karl Pearson's coefficient, Sk = (Mean − Mode) / σ = 3(Mean − Median) / σ. (Wages are in ₹, as given.)
| Measure | Before the dispute | After the dispute |
|---|---|---|
| (i) Total wages bill | 1,800 × 620 = ₹11,16,000 | 1,700 × 665 = ₹11,30,500 |
| (ii) Modal wage | 3 × 650 − 2 × 620 = ₹710 | 3 × 620 − 2 × 665 = ₹530 |
| (iii) Coefficient of variation | (48 / 620) × 100 = 7.74% | (38 / 665) × 100 = 5.71% |
| (iv) Skewness, 3(Mean − Median) / σ | 3(620 − 650) / 48 = −1.875 | 3(665 − 620) / 38 = +3.553 |
(i) Total wages bill. The bill rises by ₹14,500 (from ₹11,16,000 to ₹11,30,500), although 100 fewer workers are employed. The settlement costs the company more in total and spends more per worker.
(ii) Modal wages. The most common wage falls sharply, from ₹710 to ₹530. This is the empirical mode (3 × median − 2 × mean), an estimate of the most common wage: about ₹710 before and about ₹530 after, which points to a lower typical wage.
(iii) Coefficient of variation. The CV falls from 7.74% to 5.71%, so wages are more uniform (less unequal) after the settlement.
(iv) Skewness. Before the settlement the distribution is negatively skewed (−1.875): mean < median < mode, with most workers above the mean and a tail of low earners. After it, the distribution is strongly positively skewed (+3.553): mean > median > mode, so most workers are below the mean and a smaller group of high earners pulls the mean up. Pearson's coefficient normally lies between −3 and +3; a value of 3.553 shows that the after-settlement figures describe a more than moderately skewed distribution, so the empirical mode is only a rough estimate there.
Note: the coefficient beyond ±3 comes from the paper's own figures (the printed data give a value above +3 as well); it is not an error in the working.
(v) Did the employees gain? Only partly.
- Gains: the mean wage rises from ₹620 to ₹665 (by 7.3%) and the total bill rises, so the workers retained are paid more on average; wages are also more uniform (lower CV).
- Losses: 100 fewer workers are employed (1,800 to 1,700; the data do not say whether the settlement caused this); the median wage falls from ₹650 to ₹620 and the modal wage from ₹710 to ₹530, so the typical worker appears to earn less, while the positive skew suggests that the rise in the mean comes from a smaller group of better-paid workers.
On these summary figures the gain looks concentrated: the mean rises while the median and the estimated mode fall, so the typical worker does not seem to have gained, although aggregate figures cannot show how each worker fared.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Total wages bill | 1.5 |
| 2 | Modal wages | 1.5 |
| 3 | Coefficient of variation | 1.5 |
| 4 | Skewness | 2 |
| 5 | Comment on the settlement | 2.5 |
| Total | 9 |
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