The arbitrary choice is a good case for not doing histograms in the 21st century. Try SmoothHistogram instead. (Not to mention that no roughly continuous distribution looks as blocky as a histogram even with large amounts of data and small bin widths.)

What I’m saying is that due to the goofy way in which the first bin edge and the last bin edge are calculated, there can be first and last bin widths that are too narrow and thus the bin counts for those two bins will be low. When there are nonuniform bin widths and the histogram is being used to estimate a density function, the width of the bin has to be incorporated into the calculation of the histogram in order to get a proper pdf estimate. Mathematica counts data for these “thin” bins but then plots them as if they had the same widths as the other bins, in the process making short heights which do not accurately portray the pdf..

Here’s a bit of code you can play with:

SeedRandom[2];
numBins = 8;
uniformSamples = RandomReal[{-Pi/2, Pi/2}, 1000];
Histogram[uniformSamples, numBins, "PDF"]
histogram = HistogramList[uniformSamples, numBins, "PDF"];
histogramDistribution = HistogramDistribution[uniformSamples, numBins];
binEdges = histogram[[1]]
binCounts = histogram[[2]]
binWidths = 
 binEdges[[2 ;; Length[binEdges]]] - 
  binEdges[[1 ;; Length[binEdges] - 1]]
{numBins, Length[histogram[[1]]] - 1}
Total[binCounts]
Total[binCounts*binWidths]
Plot[CDF[histogramDistribution, x], {x, -2, 2}]

This produces the attached screen shot. As you can see, the first and last bins have a very low count and are plotted with a “full” bin width, seeming to indicate an incorrect pdf. Instead, they should be plotted with a full height and a reduced width. As you can see, the areas do total to one but only if you correct for the (constant) bin width, which is certainly an odd feature since the bin width is not reported by either of the histogram functions. Yet, HistogramDistribution takes care of the (constant) bin width issue but not the nonuniform bin width issue, as seen in the last plot.

Attachments:

Thanks, Sander. That is exactly what I needed to know. Indeed, the divisions returned for my example are simple rational numbers beginning with -8/5 to sort of match the -Pi/2 request. This strikes me as silly but at least it should be mentioned in the docs. In addition to usually giving an unexpected number of bins it also frequently gives nonuniform bin widths at the ends which results in unexpected bin counts. This happens even when “ProbabilityDensity” is given as an option, in which case (at least this case) the nonuniform bin widths need to be taken into account when present the probability density estimate. The more I think about this, the more it looks like a bug, not just a documentation problem.

Thanks again. {xmin, xmas, dx} it is!

Jerry