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I was playing around with bootstrapping today. In particular, I bootstrapped the mean, median, and midrange statistics. Below are the bootstrap distributions.

enter image description here

Interestingly BOTH the midrange and the mean have good kurtosis (peakedness) - however the median seems to have extremely poor kurtosis. I cannot think of a reason why this would be so. Could someone think of a reason why the median would have poor kurtosis HOWEVER the mean and midrange have EXTREMELY good kurtosis?

I can't seem to think of a good reason for the plots above.

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    $\begingroup$ What makes "peakedness" good? It seems an odd way to think about what you're seeing. $\endgroup$
    – Glen_b
    Mar 20, 2016 at 2:57
  • $\begingroup$ It means that there is one well defined value for this distribution. It would be great if there was one value that is likely to appear. $\endgroup$
    – user46925
    Mar 20, 2016 at 11:19
  • $\begingroup$ How small is this dataset? $\endgroup$
    – whuber
    Mar 20, 2016 at 15:09
  • $\begingroup$ Only about 4,000 observations - it's just a toy one that I found online $\endgroup$
    – user46925
    Mar 20, 2016 at 15:13

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You should provide the link to the data set so others can look at it. Then it will be easy to discuss. BTW, kurtosis does not measure "peakedness", it measures outliers. The "peakedness" definition is a used because of inertia, all started by Pearson's mis-characterization of the kurtosis statistic in 1905, and then promoted for over a century because apparently no one wanted to disagree with Pearson.

See http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/

I suspect your concern is related to discreteness, not "peakedness", but without the actual data set it will be hard to tell.

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  • $\begingroup$ Hmm, idk about that. Kurtosis seems to be synonymous with peakedness from my online scavenging $\endgroup$
    – user46925
    Mar 20, 2016 at 17:28
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    $\begingroup$ Right, inertia. No one has a clue, so they just parrot others. Please read my article. If you find a compelling counter-argument, I'd love to hear it. $\endgroup$ Mar 20, 2016 at 17:53
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    $\begingroup$ "no one wanted to disagree with Pearson" isn't a good summary of the history at all, even though you are very well informed about it. Your own paper and many others show people disagreeing about what kurtosis measures. $\endgroup$
    – Nick Cox
    Mar 21, 2016 at 0:32
  • $\begingroup$ Well, why do you think they got it wrong all those years, Nick? $\endgroup$ Mar 21, 2016 at 13:11

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