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. H₀: σ₁/μ₁ = σ₂/μ₂) when the two samples are taken from normal distributions with positive means. The test statistic is

where V₁ and V₂ are the coefficients of variation for the two samples of size n₁ and n₂ 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).

Real Statistics Support

Real Statistics Functions: The Real Statistics Resource Pack provides the following array functions.

CVTEST(R1, lab, alpha): returns an array with the values from the one sample coefficient of variation (CV) test on the data in R1: sample CV, unbiased CV, standard error, p-value, lower and upper 1-alpha  confidence interval

CV2TEST(R1, R2, lab, alpha): returns an array with the values from the two sample coefficient of variation (CV) test on the data in R1 and R2: sample 1 CV, sample 2 CV, pooled CV, z-stat, p-value, lower and upper 1-alpha  confidence interval

alpha is the significance level of the test (default .05). If lab = TRUE (default FALSE) then a column of labels is appended to the output.

The output for Example 1 is shown on the left side of Figure 3, as calculated by the array formula =CVTEST(A4:A13,TRUE). The output for Example 2 is shown on the right side of the figure, as calculated by the array formula =CV2TEST(A4:A13,B4:B14,TRUE).

Figure 3 – Real Statistics output