**One Sample Testing**

In Measures of Variability, we describe the unitless measure of dispersion called the coefficient of variation. It turns out that *s*/*x̄* is a biased estimator for the population coefficient of variation *σ/μ*. A nearly unbiased estimator is

where *n* is the sample size.

When the coefficient of variation is calculated from a sample drawn from a normal population, then the standard error can be calculated by

Using the unbiased sample coefficient of variation, we get

For normally distributed data, we can use the following test statistic

**Example 1**: Determine whether the population coefficient of variation for the data in range A4:A13 of Figure 1 (representing the length of certain biological organisms) is significantly different from 0. Also find the 95% confidence interval for the population coefficient of variation.

**Figure 1 – Test of Coefficient of Variation**

We see from the figure that p-value < alpha, and so the coefficient of variation is significantly different from zero. The 95% confidence interval is (.1079, .3403).

**Two Sample Testing**

For two samples you can test whether their populations have the same coefficient of variation (i.e. H_{0}: *σ _{1}/μ_{1} = σ_{2}/μ_{2}*) when the two samples are taken from normal distributions with positive means. The test statistic is

where *V*_{1} and *V*_{2} are the coefficients of variation for the two samples of size *n*_{1} and *n*_{2} and the pooled coefficient of variation is

The 1 – *α* confidence interval for the difference between the population coefficients of variation is

The test works best when the sample sizes are at least 10 and the population coefficients are at most .33.

**Example 2**: Determine whether there is a significant difference between the population coefficient of variation for weight and height based on the two independent samples in range of A3:B14 of Figure 2. Also find the 95% confidence interval for the difference between the population coefficients of variation.

**Figure 2 – Two sample test for coefficient of variation**

As you can see from Figure 2, there is no significant difference between the two coefficients of variation (p-value =.18) and the 95% confidence interval for the difference between the coefficients is (-.1614, .2306).

Would you please tell me what test you have used for two sample testing?

Ignacio,

The test is as described on the referenced webpage. I don’t know what the name of this test is (except maybe “two sample testing of the coefficient of variation”).

Charles

Hi,

In your previous response you mentioned av”reference webpage”. What is the webpage you reference? And, what reference are you using for the tests of significant differences between CV? I’d appreciate more background information.

Thanks!

Elizabeth,

I was referring to the webpage from which the person made his/her comment.

Charles

Can we use the ‘two sample testing’ for more than 2 CVs, for example 6 CVs?

Faizal,

Just like ANOVA is the extension of the t test to more than two samples, you will need an extension of the test given to more than two samples. I am not familiar with such a test, but I have found such a test on the Internet. Here is the link.

https://cran.r-project.org/web/packages/cvequality/vignettes/how_to_test_CVs.html

I have not vetted this approach, and am merely passing it on to you.

Charles