# Can coefficients of variation be compared if the samples differ in size? [closed]

I have to do a study for trying a sanitary product that has an acceptance criterion for the %CV repeatability and %CV reproducibility of the measurands. The acceptance criterion of the performance of this product is based on %CV and the cut-off was obtained by doing a similar study (actually the same but with bigger sample size than the study that I am doing).

So, could I use the same value of the acceptance criterion for my study to accept if the measurements fit?

The coefficient of variation (CV) is just the standard deviation (SD) divided by the mean,. Often, although not necessarily, it is expressed in percent terms. (A referee of a paper of mine commented that a CV of 2 was extraordinarily low. He was presuming that I meant 2%. It really was 200%.)

As you are aware, the mean and SD both adjust for sample size. That is, the totals of 1 2 3 and 1 1 2 2 3 3 are quite different, but dividing the totals by the sample sizes of 3 and 6 is precisely what ensures getting the same mean. A similar but not quite identical argument underlies the calculation of the SD.

Hence the CV carries over an automatic adjustment for sample size.

What remains are general and specific warnings.

1. Generally, all measures are subject to sampling variation, so keeping an eye on the sample size being used is always prudent.

2. Specifically, the coefficient of variation is highly labile at the best of times. That follows mostly from its being a ratio and hence especially sensitive to variations in the mean as well as the SD.

In your circumstances it sounds as if you are obliged to use the CV, but several threads here underline its limitations. See for example this thread and others highly voted under the tag.

The bottom line for you is that you have a high risk of making the wrong decision, especially if your sample size is much smaller than that used in the previous study. One way to get a handle on the risk might be to bootstrap the CV calculation for your data.

• Thank you Nick Cox, for your answer but I have a doubt of it. Do you mean that I should do the computations of the correspondence CV for my new data? And another thing is that when I say different sample size is not only the size but also the values. For instance, the first study is made up by 8 observations 1, 4, 6.3, 4.1, 4.4, 7.9, 9, 8.2 and the new study is made by 4 observations 3, 2, 5.5, 6. Do you know if there is any standard measure that let us compare the spread of 2 different samples? – Laura Santulario Verdú Dec 15 '18 at 17:04
• I don't know what the "correspondence CV" is and I don't understand what you're asking. If you want to compare coefficients of variation, you need to calculate them first. I am missing your point. – Nick Cox Dec 16 '18 at 7:28