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I've collected some datasets for which I want to report a summary statistic.

I produced normal probability plots for the datasets, and the data does not conform to a Gaussian distribution - there are extremely long vertical tails for the lowest and highest values, and very few values correspond to the mean. Therefore, it appears the data follows an inverse bell curve (which I have come to learn is a speacial case of a bimodal distribution).

I need to select a single summary statistic to report the results of the datasets. Which summary statistic would be best to report results from an inverted bell curve?

EDIT: Here is an image of the distribution via the Normal Probability Plot:

enter image description here

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    $\begingroup$ "Inverse bell curve" is not a term whose meaning is clear. Do you mean to refer to the inverse of a cdf of a normal? Please clarify (can you show the plot? -- or upload it to say imgur and give the link). $\endgroup$ – Glen_b Jan 21 '17 at 4:56
  • $\begingroup$ I've attached the normal probability plot - as you can see, the vast majority of the points fall within the lowest and highest possible values, with the smallest number of returns at the mean - a true 'Inverse Bell Curve'. $\endgroup$ – AggroCrag Jan 21 '17 at 9:47
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    $\begingroup$ Thanks for the image. The description "inverse bell curve" isn't standard and can't be recommended. More importantly, the data show a discrete variable with values 0, ..., 11: the normal not only doesn't fit in practice, it's in principle irrelevant to such data. The mean of bimodal distributions is still well defined; it just doesn't fall in a zone of high frequency. There can't be a single summary statistic that tells you everything about distributions in general, and this kind of distribution is no exception. At some point, show a histogram. Are values >11 possible in principle? $\endgroup$ – Nick Cox Jan 21 '17 at 10:06
  • $\begingroup$ @rgf95 1. could you clarify what the two axes are in your question? (I believe I can infer them but it should be explicit for every reader -- we shouldn't be guessing). 2. What does your variable represent? How is it measured? 3. I fully agree with Nick Cox's comments. I think he touches on pretty much everything I'd have wanted to say. $\endgroup$ – Glen_b Jan 22 '17 at 3:32
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There is no good single summary statistic for the type of distribution you have plotted, or, really, for any multimodal distribution.

That is, you can calculate anything you'd like: Mean, median, mode, interquartile range .... whatever. But none of these are good representations of data that has multiple modes.

Even for data that is perfectly normally distributed, you need two numbers: Mean and standard deviation. But let's assume you want a single measure of central tendency or location. For the normal, that's the mean. For highly skewed distributions, the usual choice is the median (although sometimes the mean is best, or even the mode), for distributions with a few extreme outliers, you might consider the trimmed or Winsorized mean.

But for multimodal distributions, none of these really work. The math is fine, but the intuition fails.

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Inverse bell curve is essentially bimodal distribution. Without going into much details here, and in response to the "intuition fails" comment, I would suggest to look at the gender orientation measures including all parameters together. Extreme males and extreme females are there, and the ideal 50-50 behaviour is the rarest - AND the aggregate (integral) between the two extremes and the rarest trough covers the maximum number of people. This is also known as Well curve. Mathematically, this is a combination of two bell curves with a translation along the x-axis.

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