Naively, one might compare two normal distributions based on their mean or median to determine which one is more appropriate. However, if I know that a metric follows a log-normal distribution, such as the time taken for an event or process, is there a more accurate way to compare the two distributions?
The mean is overly sensitive to outliers but also captures the effect of extreme values, which can strongly influence the distribution, especially for very bad outcomes.
One approach could be comparing the mean of the log-transformed values. This would reduce the skewness but would not give as much weight to outliers.
A more robust approach would be to compare quantiles.
However, I was wondering if there is a method that strikes a balance—something that might not be as effective as quantile comparison but still an improvement over comparing the naive median or mean of the distribution as if it were normal.
It feels counterintuitive that knowing the distribution is log-normal doesn't significantly change how we compare the distributions.
