Solved – Coefficient of variation (CV) of log-transformed data

descriptive statisticsgeometric-distributionlognormal distributionnonparametric

I understand that with log-transformed data, the coefficient of variation (CV) on the original scale is equal to sqrt(exp(sigma^2)-1), where sigma is the standard deviation of log-transformed data.

But is there anything inherently wrong with simply calculating CV on log scale as sigma/xbar, where xbar is the mean of the log-transformed data?

For instance, would this calculation of CV on log-scale not really represent what is thought of as a coefficient of variation?

EDIT to explain my intended use of CV

My intended use is to report descriptive statistics for two sets of data:

  1. price data for homes in different cities in Europe and;
  2. price level indexes of homes for different cities in Europe using London as
    a 'base', i.e. (price home in city x/price home in London) x 100.

Because city prices and indexes generally, but not always, follow a log-normal distribution I decided to perform a log transformation to better visualize the distance of each city price or each city price level index from the center of each respective distribution.

Best Answer

One nice property of the CV is that it does not change if you scale all the data by a constant factor. The SD of the log-transformed data shares this property. Your proposed measure (SD/mean of the log-transformed data) does not share this property. Lewontin (1966) may help elucidate some of these issues.

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