Question 1 OR (f)
Purpose of the coefficient of variation; variance compared with it
3 marks
- Time to write
- About 6 minutes
Question
Question 1 OR (f) · 3 marksWhat purpose is served by the coefficient of variation? Distinguish between variance and the coefficient of variation.
Reworded from the university paper; 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
- What…?Distinguish…2 asks, each with its own marks below. Answer both.
Write it in this order
- 1 Purpose of the CV 1 mark 2 min
- 2 Distinction between variance and CV 2 marks 4 min
Try it first. About 6 minutes for 3 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.
Purpose of the coefficient of variation. The coefficient of variation (CV) is a relative measure of dispersion: the standard deviation stated as a percentage of the arithmetic mean, CV = (σ / x̄) × 100. Since the mean is the divisor, the CV cannot be found when the mean is 0 and is unstable when the mean is close to 0, so it is used for data with a positive mean, such as wages, prices, output or returns. Because the unit cancels, it serves to:
- compare the variability of two or more series measured in different units (heights in cm and weights in kg) or with very different means (wages of directors and of clerks);
- judge consistency, stability or uniformity: the series with the smaller CV is the more consistent (a batsman's scores, a machine's output, a dealer's daily sales);
- compare risk relative to return, as with two shares whose average returns differ.
Variance and coefficient of variation compared.
| Point | Variance (σ²) | Coefficient of variation |
|---|---|---|
| Nature | Absolute measure of dispersion | Relative measure of dispersion |
| Definition | Mean of the squared deviations from the mean | SD as a percentage of the mean |
| Unit | Square of the data's unit (₹², kg²) | Pure number, a percentage |
| Comparison | Only between series in the same unit and of similar size | Between series in different units or with different (positive) means |
| Change of origin | Unaffected (adding a constant leaves it unchanged) | Changes, because the mean changes |
| Change of scale | Multiplied by the square of the scale factor | Unaffected by multiplying every value by a positive constant |
| Typical use | Theory: combined variance, moments, regression | Practice: consistency and risk comparisons |
The properties in the change-of-origin and change-of-scale rows follow from the SD being unaffected by a change of origin but changing with the scale: the variance is the SD squared, while in the CV a change of scale alters the SD and the mean in the same ratio, which cancels.
Where the 3 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Purpose of the CV | 1 |
| 2 | Distinction between variance and CV | 2 |
| Total | 3 |
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