Homogeneity of Coefficients of Variation

Basic Concepts

We now extend the test to determine whether two coefficients of variation are equal (described in Coefficient of Variation Testing) to more than two samples.

For k samples you can test whether their populations have the same coefficient of variation (i.e. H0: σ1/μ1 = σ2/μ2 = … = σk/μk) when the k samples are taken from normal distributions with positive means. The test statistic is

Statistic homogeneity coefficients variation

where the Vj are the coefficients of variation for the k samples of size nj with n = \sum_{j=1}^k n_j and the pooled coefficient of variation is

Pooled coefficient of variation

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

Example

Example 1: Determine whether there is a significant difference between the population coefficient of variation for the three independent samples in range of A3:C14 of Figure 1.

Homogeneity Coefficients of Variation

Figure 1 – Testing for homogeneity of coefficients of variation

As you can see from Figure 1, there is a significant difference between the two coefficients of variation (p-value =.02567 < .05 = alpha).

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Sokal, R. R. and Baumann, C. A. (1980) Significance tests of coefficients of variation and variability profiles
https://academic.oup.com/sysbio/article/29/1/50/1655230

Gilpin, A. R. (1993) A program to test equality of two or more coefficients of variation
https://link.springer.com/article/10.3758/BF03204452

6 thoughts on “Homogeneity of Coefficients of Variation”

  1. Hi Dr Zaiontz. I am an archaeologist researching the embodiment of cognitive innovation exhibited by archaeological assemblages. I am not a statistics expert by any means (pardon the pun). Looking at, e.g., blue glass from Egypt and Mesopotamia made around the 15th to 14th century BC I have calculated by a simple method of comparing the average STDEV divided by the Average MEAN of data sets, then multiplied by 100 to come up with an overall coefficient of Variance. I then subtract this coefficient of variance from 100 to obtain what I describe as a homogeneity value for the sample group. I use, as attributes, elements in parts per million and/or oxides in weight percentage. My results may exhibit differing homogeneity values of, e.g., 70% to 80% for two sets of glass samples. These data sets may or may not comply with an ANOVA significance of p = o.05 and generally will have a varying n value between sample groups. Do you believe my approach is empirically sound given the limited access to data?

    Reply

    • Hello George,
      Whether or not this approach is empirically sound, depends on what you plan to do with the results and what you expect it to show. In particular, what hypothesis are you trying to test?
      Charles

      Reply

      • Hi Charles. Generally, what I am trying to achieve is a comparison of the homogeneity embodied in ancient production of things compared with things made today. It is my belief that homogeneity is an empirically sound measure of either orthodoxy or agency in the production process. For example a high level of orthodoxy is apparent in the assembly-line process to build the Model-T Ford…this I can establish from numerical allocation to specific attributes. If, e.g., an attribute gives me a hiccup in the C of V homogeneity I would suggest either a mistake in the production process or an intentional deviation from the standard process (I.e., agency, innovative intention). My analysis is always within the 0.33 (33%).Therefore, my question is. Is it empirically sound to use my approach for this purpose. What I am able to present from my methodology is both a Levene’s-test value and a percentage of relative ‘sameness’. For example my large data sets (sometimes several thousand attribute values). One of my calculations recently gave me a 86% sameness, however, by ANOVA, I had a Levene’s-test value of 1.00 (homogeneous). I have, as a control, used this method comparing mint-produced coins to support it’s reliability. Your thoughts please

        Reply

        • George,
          I don’t have enough knowledge about your field to say whether or not this approach is applicable. It certainly sounds like you have thought through the details though.
          Charles

          Reply

  2. Dear Mr. Zaiontz, good morning.
    In my country (Brazil), teachers’ day is celebrated on October 15th.
    Because I have learned a lot of statistics on your site, I wish you a “happy teacher’s day”. Big hug and success!
    Igor

    Reply

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