Coefficient of Variation as a Measure of Relative Dispersion

In a sample of small firms, mean yearly pre-tax profits is 50 million and the standard deviation is 5 million. On another sample of larger firms, the mean is 200 million and the standard deviation is 6.3 million. Can you say which of the two samples has more variability in profits?

Incorrect. This answer regards only the higher standard deviation recorded for the sample of large firms. But things are not quite that simple.

Now, suppose that your sample for smaller firms also contains data on their number of employees, with mean 8 and standard deviation 3. Can you compare variability in employment with variability in profits?

That isn't quite right. It is true that you can't directly compare units of profit money and units of workers. But the coefficient of variation will take care of this problem for you.

To summarize: [[summary]]

That's right.

Correct.

The standard deviation of 6.3 for the larger firms might suggest higher variability than the standard deviation of 5 for the smaller firms. But 6.3 relative to the large mean profit of 200 seems in fact smaller than 5 relative to the smaller profit of 50. Taking into account the companies' scale of operations, there might well be more variability of profits in the sample of smaller firms in spite of its lower standard deviation.

So how might you standardize this comparison?

The CV for profits, as you established earlier, is 0.100. The CV for employment is 3 workers divided by 8 workers, or 0.375. The important point is that you can compare these two numbers to conclude that employment is more variable than profits in the sample.

You can make the comparison because the CV is a unit-free measure. True, for profits you have mean and standard deviation in millions of monetary units, whereas for employment you have them in number-of-workers units. But when you calculate the CV, the division washes out the units of measurement. So the CVs for different variables are perfectly comparable.

That wouldn't really help. You'd want to think of risk on a per-unit basis here.

Absolutely. In order to interpret the data and make comparisons across different samples, the correct way of thinking about variability is in relative terms. The standard deviation is an absolute measure of dispersion; a relative measure would be in relation to the sample mean. The __coefficient of variation (CV)__ does exactly that for you. You can obtain it simply by dividing the sample standard deviation by the mean to deliver a measure of relative dispersion: $$\displaystyle CV = \frac{s}{{\bar X}}$$ In your two samples, the CV is $$\frac{6.3}{200} = 0.0315$$ for the larger firms, and $$\frac{5}{50} = 0.100$$ for the smaller firms. The CV confirms that there is more variability of profits among the smaller firms.

That would leave you with the same, original issue in terms of comparability.

Yes, the sample of larger firms clearly has more variability

No, the sample of smaller firms might have more variability

No, because that would be comparing variables in different units

Yes, because the CV can serve to compare different variables.

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