When comparing two independent groups with normal and non-normal data distribution, it is common to report the central tendency and spread of the data using different measures depending on the distribution. For normally distributed data, the mean and standard deviation are appropriate measures, while for non-normally distributed data, the median and interquartile range are often used.

If the statistical test reveals a significant difference between the two groups, it is important to interpret the results in light of the differences in data distribution and the different measures of central tendency and spread used to present the data. It is not appropriate to directly compare the mean and standard deviation of one group with the median and interquartile range of the other group, as these measures reflect different aspects of the data distribution.

Instead, it is recommended to focus on the direction and magnitude of the difference observed between the two groups, rather than attempting to directly compare the means or medians. For example, you might report that Group A had a significantly higher mean value than Group B for normally distributed data, and that Group A had a significantly higher median value than Group B for non-normally distributed data. Additionally, you might report the effect size of the difference between the groups, such as Cohen's d or a non-parametric effect size measure like the Hodges-Lehmann estimator, which can help to contextualize the magnitude of the difference observed.

Overall, it is important to report the appropriate measures of central tendency and spread for each group based on the data distribution, and to interpret the results in terms of the direction and magnitude of the difference observed rather than attempting to directly compare different measures of central tendency or spread.

What test are you using ?

For descriptive statistics, it might be best to include all the statistics for both groups. That is, include mean, median, sd, IQR, and so on.

I agree that in your case you would compare apples with oranges. But not just because meadiean and mean don't have the same meaning. If the distributions of the variable in groups you like to compare are considerably different, then a simple difference in means (or medians) may not capture the relevant difference at all. So the most important question should be why you find different distributions. Understanding this might hint you towards an entirely different way of analysis, or you may identify a faulty experimental design or problems with the data.

Most likely, the reason for your finding is that you wrongly assumed that the data should be normal distributed, and that in one group the distribution is not sufficiently different from a normal distribution to recognize this. An example: the variable is actually Poisson distributed. In one group, the expected values is close to zero, say it's about 2, and in the other group the expected value is 10. The distribution of the first group is very strongly skewed and it is easy to see that this is not a normal distribution. The distribution of the second group is much less skewed (more symmetric) and it is not that easy to recognize that it is nor normal distributed. But both distributes are Poisson, and a Poisson model would in this case be the correct choice to analyze the data (typically investigating relative changes). In this example (Poisson) the data would be whole numbers - something that would alone already strongly speak against using a normal distribution, but similar examples are possible with lesser known continuous distributions like the beta or gamma distributions.

What is the dependent variable? And what is your research question?

I think you're confusing several things.

First, usage about summarizing « normal » data with mean ± standard deviation and other kind of distributions, especially asymetric, by median and IQR is just a matter of convention. As descriptive statistics, both mean and median, SD and IQR exist in any sample. So you should well report both of them for both groups. That would be the most informative. Intuitive interpretation of mean, and even more SD, is not correct when data are not Gaussian, but that's another problem.

Second, non-normal data are NOT abnormal, there are many many cases where non-normal data are just what is expected. in fact, practically, NO real data are really normally distributed (because NO real data can range from -infinity to +infinity with non-null probability, even if tiny).

Third, as Jochen Wilhelm explained in details, apparently normal data are often just lack of detection of non-normality for various reasons. I would add to his examples the log-normal distribution, which is very frequent in some areas and has theoretical justifications for several kinds of biological data.

Last, do not forget that statistical test do test precise hypothesis, that concern either means (T-test), medians (Median-test) or the whole distribution (Wilcoxon-Mann-Whitney). There is *no* (useful, usual) test that compares a mean in group 1 to a median in group 2, whatever you present in your results tables.

As a bonus : remember that for a Gaussian distribution, median = mean (for the theoretical values), so…

although, everything is right in one way or another,

I would like to recap a few things.

So from here, the question. what are you looking for?. Comparison can be done from different perspectives, (and this is always the "blessed problem that statisticians receive", they do the data collection process, and then, since it doesn't work out what they want to do, they consult us, what can we do? , with this data?)

Although some still associate the mean and standard deviation to the normal or symmetric distribution (t-student, triangular, cauchy, etc), it should be remembered that in terms of Distribution they are parameters of position, shape and scale. and that using the mean or another is due more to the type of variable and the quality of the data.

what do you want to compare?

You may not be comparing Apples with Oranges, but FUJI and GoldenDelicius Apples, or you could be comparing from the point of view that both are Fruits.

If the positions of the population are similar, then you can't compare mean with median (obvious), but why can't you compare the mean? So if it makes you uncomfortable, then you're not sure what you want to find? fruits better than another?, so what measure do I need and from where can I compare them, should be the question,

When you have two independent groups, for the normally distributed data use parametric test to identify the difference between the groups and present the p value with mean and standard deviation and interpret the result. For the non-normal data do a non-parametric test to identify the difference between the groups and interpret the result based on the p value, provide median and inter quartile range. It is possible in a study to have both normal and non-normal data and the proper analysis will be to apply parametric and non-parametric test appropriately. Thus it is not like comparing apple with orange. Also, as said by Emmanuel, there is nothing like abnormally distributed data. Either it is normally distributed or skewed (non-normally distributed) data.

Kandasamy Ravichandran, what did you mean when you wrote that final sentence above?

"Either it is normally distributed or skewed (non-normally distributed) data."

That makes no sense to me. Many distributions that are perfectly symmetrical are not normal. E.g.,

• A uniform distribution
• A binomial distribution with p = q = 0.5 (and n < infinity)
• A t-distribution with df < infinity

Thanks for clarifying.

... for more examples of symmetric (non-skewed) distributions that are not normal, see https://en.wikipedia.org/wiki/Symmetric_probability_distribution.

And additionally: Beta with α = β (https://en.wikipedia.org/wiki/Beta_distribution), which has the continuous uniform distribution as a special case (α = β = 1).

Hi Bruce Weaver, the last sentence (Either it is normally distributed or …data.) to be understood along with the previous sentence. The word ‘skewed’ is introduced to replace the word ‘abnormal’ used by Kurniawan and don’t have any further interpretation. Hope it's clear now. It is well known that all normal distributions are symmetrical but not all symmetrical distributions are normal. Thanks to Jochen Wilhelm for sharing more examples.