Histogram

A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram M_i of a histogram m_j is defined as:

${\displaystyle M_{i}=\sum _{j=1}^{i}{m_{j}}.}$

There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is at least as old as Graunt's work in the 17th century, but no systematic guidelines were given[11] until Sturges' work in 1926.[12]

Using wider bins where the density of the underlying data points is low reduces noise due to sampling randomness; using narrower bins where the density is high (so the signal drowns the noise) gives greater precision to the density estimation. Thus varying the bin-width within a histogram can be beneficial. Nonetheless, equal-width bins are widely used.

Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb.[13]

The number of bins k can be assigned directly or can be calculated from a suggested bin width h as:

${\displaystyle k=\left\lceil {\frac {\max x-\min x}{h}}\right\rceil .}$

The braces indicate the ceiling function.

${\displaystyle k=\lceil {\sqrt {n}}\rceil \,}$

which takes the square root of the number of data points in the sample (used by Excel's Analysis Toolpak histograms and many other) and rounds to the next integer.[14]

Sturges' formula[12] is derived from a binomial distribution and implicitly assumes an approximately normal distribution.

${\displaystyle k=\lceil \log _{2}n\rceil +1,\,}$

Sturges' formula implicitly bases bin sizes on the range of the data, and can perform poorly if n < 30, because the number of bins will be small—less than seven—and unlikely to show trends in the data well. On the other extreme, Sturges' formula may overestimate bin width for very large datasets, resulting in oversmoothed histograms.[15] It may also perform poorly if the data are not normally distributed.

When compared to Scott's rule and the Terrell-Scott rule, two other widely accepted formulas for histogram bins, the output of Sturges' formula is closest when n ≈ 100.[15]

${\displaystyle k=\lceil 2{\sqrt[{3}]{n}}\rceil ,}$

The Rice Rule [16] is presented as a simple alternative to Sturges' rule.

Doane's formula[17] is a modification of Sturges' formula which attempts to improve its performance with non-normal data.

${\displaystyle k=1+\log _{2}(n)+\log _{2}\left(1+{\frac {|g_{1}|}{\sigma _{g_{1}}}}\right)}$

where ${\displaystyle g_{1}}$ is the estimated 3rd-moment-skewness of the distribution and

${\displaystyle \sigma _{g_{1}}={\sqrt {\frac {6(n-2)}{(n+1)(n+3)}}}}$

Bin width ${\displaystyle h}$ is given by

${\displaystyle h={\frac {3.49{\hat {\sigma }}}{\sqrt[{3}]{n}}},}$

where ${\displaystyle {\hat {\sigma }}}$ is the sample standard deviation. Scott's normal reference rule[18] is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.[11]

The Freedman–Diaconis rule gives bin width ${\displaystyle h}$ as:[19][11]

${\displaystyle h=2{\frac {\operatorname {IQR} (x)}{\sqrt[{3}]{n}}},}$

which is based on the interquartile range, denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.

This approach of minimizing integrated mean squared error from Scott's rule can be generalized beyond normal distributions, by using leave-one out cross validation:[20][21]

${\displaystyle {\underset {h}{\operatorname {arg\,min} }}{\hat {J}}(h)={\underset {h}{\operatorname {arg\,min} }}\left({\frac {2}{(n-1)h}}-{\frac {n+1}{n^{2}(n-1)h}}\sum _{k}N_{k}^{2}\right)}$

Here, ${\displaystyle N_{k}}$ is the number of datapoints in the kth bin, and choosing the value of h that minimizes J will minimize integrated mean squared error.

The choice is based on minimization of an estimated L² risk function[22]

${\displaystyle {\underset {h}{\operatorname {arg\,min} }}{\frac {2{\bar {m}}-v}{h^{2}}}}$

where ${\displaystyle \textstyle {\bar {m}}}$ and ${\displaystyle \textstyle v}$ are mean and biased variance of a histogram with bin-width ${\displaystyle \textstyle h}$, ${\displaystyle \textstyle {\bar {m}}={\frac {1}{k}}\sum _{i=1}^{k}m_{i}}$ and ${\displaystyle \textstyle v={\frac {1}{k}}\sum _{i=1}^{k}(m_{i}-{\bar {m}})^{2}}$.

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 ${\displaystyle \approx n/k}$ 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:[23]

${\displaystyle k=2n^{2/5}}$

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 ${\displaystyle \alpha }$ it is recommended to choose between 1/2 and 1 times the following equation:[24]

${\displaystyle k=4\left({\frac {2n^{2}}{\Phi ^{-1}(\alpha )}}\right)^{\frac {1}{5}}}$

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

A good reason why the number of bins should be proportional to ${\displaystyle {\sqrt[{3}]{n}}}$ is the following: suppose that the data are obtained as ${\displaystyle n}$ independent realizations of a bounded probability distribution with smooth density. Then the histogram remains equally "rugged" as ${\displaystyle n}$ tends to infinity. If ${\displaystyle s}$ is the "width" of the distribution (e. g., the standard deviation or the inter-quartile range), then the number of units in a bin (the frequency) is of order ${\displaystyle nh/s}$ and the relative standard error is of order ${\displaystyle {\sqrt {s/(nh)}}}$. Compared to the next bin, the relative change of the frequency is of order ${\displaystyle h/s}$ provided that the derivative of the density is non-zero. These two are of the same order if ${\displaystyle h}$ is of order ${\displaystyle s/{\sqrt[{3}]{n}}}$, so that ${\displaystyle k}$ is of order ${\displaystyle {\sqrt[{3}]{n}}}$. This simple cubic root choice can also be applied to bins with non-constant widths.

Histogram and density function for a Gumbel distribution[25]