plotting - Probability density histogram with unequal bin widths

I am confused by output of Histogram and HistogramList for probability density ("PDF") when bin widths are not equal.

According to this and this pages and other sources, density histograms are computed by dividing counts by bar widths and total number of observations. But Histogram obviously uses another algorithm as one can see from the following example:

SeedRandom[1];
data = RandomVariate[NormalDistribution[0, 1], 15];
HistogramList[data, {{-2, 0, 1}}, "PDF"][[2]]
BinCounts[data, {{-2, 0, 1}}]/((Length[data] - 3) Differences[{-2, 0, 1}])

{1/6, 2/3}

{1/6, 2/3}

The outputs are identical when bin counts are divided by total number of observations minus 3 in this case. Why is it? What algorithm Histogram uses for determining this difference (in other cases I got other numbers)?

An addition

Rod has answered the original question but there is another issue: if one gives upper bin boundary equal to the upper datapoint value then this value will be excluded from the histogram. It does not contradict the documentation where stated that {{b1,b2,...}} will use the bins [b1,b2),[b2,b3),.... Here is an illustration:

HistogramList[data, {{-2, 0, Max[data] - 10^-13}}, 
   "PDF"][[2]] - HistogramList[data, {{-2, 0, Max[data]}}, "PDF"][[2]]
HistogramList[data, {{-2, 0, Max[data] + 10^-13}}, "PDF"][[2]] - 
 HistogramList[data, {{-2, 0, Max[data]}}, "PDF"][[2]]

{0, 2.99205*10^-14}

{-(1/105), 0.0123342}

One can see that subtraction of 10^-13 does not alter the result significantly as expected but addition of 10^-13 changes it considerably because now the point Max[data] is included in the histogram. One can check this directly:

HistogramList[data, {{-2, 0, Max[data]}}, "PDF"][[2]]
BinCounts[data, {{-2, 0, Max[data]}}]/ 
  Differences[{-2, 0, Max[data]}]/(Length[data] - 1)

HistogramList[data, {{-2, 0, Max[data] + 10^-13}}, "PDF"][[2]]
BinCounts[data, {{-2, 0, Max[data] + 10^-13}}]/ 

  Differences[{-2, 0, Max[data]}]/Length[data]

{1/7, 0.462531}

{1/7, 0.462531}

{2/15, 0.474865}

{2/15, 0.474865}

Answer

Using SeedRandom[1] you get 3 observations higher than 1. When you use Histogram[data,{{-2,0,1}}] you're excluding those 3 observations...

If you exclude those 3 observations, now your probability (i.e., "PDF") should be based on 12 observations, and not 15...

Comments

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