### Histograms

For the histograms used in digital image processing, see Image histogram and Color histogram.
Histogram
First described by Karl Pearson
Purpose To roughly assess the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values

In statistics, a histogram is a graphical representation of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson.[1] A histogram is a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. A histogram may also be normalized displaying relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size.[2] The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.[3]

Histograms are used to plot the density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x-axis are all 1, then a histogram is identical to a relative frequency plot.

An alternative to the histogram is kernel density estimation, which uses a kernel to smooth samples. This will construct a smooth probability density function, which will in general more accurately reflect the underlying variable.

The histogram is one of the seven basic tools of quality control.[4]

## Etymology

The etymology of the word histogram is uncertain. Sometimes it is said to be derived from the Greek histos 'anything set upright' (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); and gramma 'drawing, record, writing'. It is also said that Karl Pearson, who introduced the term in 1891, derived the name from "historical diagram".[5]

## Examples

The U.S. Census Bureau found that there were 124 million people who work outside of their homes.[6] Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times "at least 30 but less than 35 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time. The problem of reporting values as somewhat arbitrarily rounded numbers is a common phenomenon when collecting data from people.

Data by absolute numbers
Interval Width Quantity Quantity/width
0 5 4180 836
5 5 13687 2737
10 5 18618 3723
15 5 19634 3926
20 5 17981 3596
25 5 7190 1438
30 5 16369 3273
35 5 3212 642
40 5 4122 824
45 15 9200 613
60 30 6461 215
90 60 3435 57

This histogram shows the number of cases per unit interval as the height of each block, so that the area of each block is equal to the number of people in the survey who fall into its category. The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers, with Q in thousands.

Data by proportion
Interval Width Quantity (Q) Q/total/width
0 5 4180 0.0067
5 5 13687 0.0221
10 5 18618 0.0300
15 5 19634 0.0316
20 5 17981 0.0290
25 5 7190 0.0116
30 5 16369 0.0264
35 5 3212 0.0052
40 5 4122 0.0066
45 15 9200 0.0049
60 30 6461 0.0017
90 60 3435 0.0005

This histogram differs from the first only in the vertical scale. The area of each block is the fraction of the total that each category represents, and the total area of all the bars is equal to 1 (the fraction meaning "all"). The curve displayed is a simple density estimate. This version shows proportions, and is also known as a unit area histogram.

In other words, a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies: the height of each is the average frequency density for the interval. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also contiguous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)[7]

## Mathematical definition

In a more general mathematical sense, a histogram is a function mi that counts the number of observations that fall into each of the disjoint categories (known as bins), whereas the graph of a histogram is merely one way to represent a histogram. Thus, if we let n be the total number of observations and k be the total number of bins, the histogram mi meets the following conditions:

$n = \sum_\left\{i=1\right\}^k\left\{m_i\right\}.$

### Cumulative 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 Mi of a histogram mj is defined as:

$M_i = \sum_\left\{j=1\right\}^i\left\{m_j\right\}.$

### Number of bins and width

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[8] until Sturges's work in 1926.[9]

Using wider bins where the density 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. None the less, 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.[10]

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

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

The braces indicate the ceiling function.

Square-root choice
$k = \sqrt\left\{n\right\}, \,$

which takes the square root of the number of data points in the sample (used by Excel histograms and many others).[11]

Sturges' formula

Sturges' formula[9] is derived from a binomial distribution and implicitly assumes an approximately normal distribution.

$k = \lceil \log_2 n + 1 \rceil, \,$

It implicitly bases the bin sizes on the range of the data and can perform poorly if n < 30. It may also perform poorly if the data are not normally distributed.

Rice Rule
$k = \lceil 2 n^\left\{1/3\right\}\rceil,$

The Rice Rule [12] is presented as a simple alternative to Sturges's rule.

Doane's formula

Doane's formula[13] is a modification of Sturges' formula which attempts to improve its performance with non-normal data.

$k = 1 + \log_2\left( n \right) + \log_2 \left\left( 1 + \frac \left\{ |g_1| \right\}\left\{\sigma_\left\{g_1\right\}\right\} \right\right)$

where $g_1$ is the estimated 3rd-moment-skewness of the distribution and

$\sigma_\left\{g_1\right\} = \sqrt \left\{ \frac \left\{ 6\left(n-2\right) \right\}\left\{ \left(n+1\right)\left(n+3\right) \right\} \right\}$
Scott's normal reference rule
$h = \frac\left\{3.5 \hat \sigma\right\}\left\{n^\left\{1/3\right\}\right\},$

where $\hat \sigma$ is the sample standard deviation. Scott's normal reference rule[14] is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.[8]

Freedman–Diaconis' choice
$h = 2 \frac\left\{\operatorname\left\{IQR\right\}\left(x\right)\right\}\left\{n^\left\{1/3\right\}\right\},$

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.

Choice based on minimization of an estimated L2[16] risk function

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

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

Remark

A good reason why the number of bins should be proportional to $n^\left\{1/3\right\}$ is the following: suppose that the data are obtained as $n$ independent realizations of a bounded probability distribution with smooth density. Then the histogram remains equally »rugged« as $n$ tends to infinity. If $s$ 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 $n h/s$ and the relative standard error is of order $\sqrt\left\{s/\left(n h\right)\right\}$. Comparing to the next bin, the relative change of the frequency is of order $h/s$ provided that the derivative of the density is non-zero. These two are of the same order if $h$ is of order $s/n^\left\{1/3\right\}$, so that $k$ is of order $n^\left\{1/3\right\}$.

This simple cubic root choice can also be applied to bins with non-constant width.