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I have a dataset where the mean is below the median. According to this, it implies that the distribution is obviously tailed but that the mass of the distribution is placed at higher values of y.

Looking at 2 strata of my data, stratum 1 has a bigger difference between the median and mean than stratum 2. This means that the distribution of stratum 1 deviates even more from a normal distribution than stratum 2 (i.e., it is more tailed). What else does it tell us?

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    $\begingroup$ Best to plot your data to see what is going on. Mean < median is consistent with various different distribution shapes. $\endgroup$ – Nick Cox Dec 14 '17 at 10:54
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    $\begingroup$ "Tailed" and "more tailed" are not in common usage in what I read. "Skewed" with qualifiers (left or negative, right or positive) is the more usual term for what I think you mean. In any case "tailed" might be taken to mean "tail weight" which is worth checking but not directly related to mean and median. $\endgroup$ – Nick Cox Dec 14 '17 at 10:57
  • $\begingroup$ It's hard for me to add to this without knowing more about your data, which you have not expanded upon. Your statement is that the mean is below the median. As said, that is consistent with various distribution shapes, including those quantified as right skewed and left skewed. The entertaining riff by @DJohnson raises several possibilities that could (most obviously) apply if by "below" you meant "above" or "much above". Outliers and long tails are however possible with skewness of any kind. $\endgroup$ – Nick Cox Dec 15 '17 at 12:25
  • $\begingroup$ I add here, FWIW, the information that I didn't downvote the answer mentioned just above. $\endgroup$ – Nick Cox Dec 15 '17 at 12:26
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    $\begingroup$ Seeing the one very elaborate answer, and this relatively short question, I'm left to wonder what exactly the question is. Does it refer to data being normally distributed just like that, or from some idea another statistical model or test requires data like that, or something else? FWIW I also did not downvote @DJohnson 's answer, but before posting an elaborate answer which starts by assuming what the OP would be expressing with the question, I'd suggest we need (way) more details from OP (like those asked in prior comments). Only afterwards, Djohnson's answer might prove the right one. $\endgroup$ – IWS Dec 15 '17 at 12:35
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You wrote:

Looking at 2 strata of my data. Stratum 1 has a bigger difference between the median and mean than stratum 2. This means that the distribution of stratum 1 deviates even more from a normal distribution than stratum 2 (i.e. it is more tailed). What else does it tell us?

I think the answer is "nothing". That is, by itself, the fact that the difference between the mean and median is larger in one stratum doesn't tell you anything other than that the difference is larger. In fact, depending on what you mean by "deviates" it might not even tell you that it is farther from normal, because it could be that the strata have very different means and medians.

Suppose that the distributions had the same shape, but that one stratum had much larger values, then that stratum would have a larger difference.

As people said in the comments, the best way to examine this is to graph the data. You could start with overlaid density plots of each distribution, a parallel box plot, and a qq plot.

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