Questions the calculation of Z score and confidence intervals in biology In biology, especially in some screening assays, people love to calculate a Z score between positive and negative controls. Essentially a distance calculation using the mean and sd to calculate how separated the controls are, hence how strong the signal could be by inferring the range

My question/concern is, shouldn’t we adjust for the actual distribution of the data ?
Most the data measured in biology would be from Zero to a positive number, in this case lets say we are measuring light intensity, it ranges from 0 to positive infinity in theory (detection limit, lets say) ,so if the distribution is log normal, then what would happen to the calculated Z score? will we get a smaller/bigger/the same distance? (sorry for my bad drawing)

And the next step is usually report the confidence intervals using the measured controls (usually a couple replicates). Given the log normal distribution, should we adjust that too?
And if so, how to?

Mean and sd have an interpretation that isn't exclusive to the normal distribution, so I don't think such a Z score should be seen as requiring normality (normality may be required for certain distributional statements for example used for computing confidence intervals and the like, but those can hold approximately also for other distributions due to the Central Limit Theorem).
However it is true that mean and sd are problematic for summarising a distribution's location and variation for clearly skew distributions or in presence of outliers, so in such cases such scores may be inappropriate. Ultimately it depends on the exact situation and the use made of and interpretation given to these scores.
For example, certain data distributions may be more symmetric and show fewer problems with outliers after a log transformation, but even then in certain applications interpretation is relevant in terms of the original values rather than in terms of logs.
