Editing Histogram (section)

==== Variable bin widths ====

Rather than choosing evenly spaced bins, for some applications it is preferable to vary the bin width. This avoids bins with low counts. A common case is to choose ''equiprobable bins'', where the number of samples in each bin is expected to be approximately equal. The bins may be chosen according to some known distribution or may be chosen based on the data so that each bin has <math>\approx n/k</math> samples. When plotting the histogram, the ''frequency density'' is used for the dependent axis. While all bins have approximately equal area, the heights of the histogram approximate the density distribution.

For equiprobable bins, the following rule for the number of bins is suggested:<ref>{{cite web |author1=Jack Prins |author2=Don McCormack |author3=Di Michelson |author4=Karen Horrell |title=Chi-square goodness-of-fit test |url=https://itl.nist.gov/div898/handbook/prc/section2/prc211.htm |website=NIST/SEMATECH e-Handbook of Statistical Methods |publisher=NIST/SEMATECH |access-date=29 March 2019 |page=7.2.1.1}}</ref>

:<math>k = 2 n^{2/5}</math>

This choice of bins is motivated by maximizing the power of a [[Pearson chi-squared test]] testing whether the bins do contain equal numbers of samples. More specifically, for a given confidence interval <math>\alpha</math> it is recommended to choose between 1/2 and 1 times the following equation:<ref>{{cite book |last1=Moore |first1=David |editor1-last=D'Agostino |editor1-first=Ralph |editor2-last=Stephens |editor2-first=Michael |title=Goodness-of-Fit Techniques |date=1986 |publisher=Marcel Dekker Inc. |location=New York, NY, US |isbn=0-8247-7487-6 |page=70 |chapter=3}}</ref>

:<math>k = 4 \left( \frac{2 n^2}{\Phi^{-1}(\alpha)} \right)^\frac{1}{5}</math>

Where <math>\Phi^{-1}</math> is the [[probit]] function. Following this rule for <math>\alpha = 0.05</math> would give between <math>1.88n^{2/5}</math> and <math>3.77n^{2/5}</math>; the coefficient of 2 is chosen as an easy-to-remember value from this broad optimum.