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. A histogram is a representation of tabulated frequencies, shown as adjacent
rectangles or squares, erected over discrete intervals, with an area proportional 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 must
be adjacent, and often are chosen to be of the same size. The rectangles of a histogram
are drawn so that they touch each other to indicate that the original variable is continuous.
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.
Etymology
The etymology of the word histogram is uncertain. Sometimes it is said to be derived from the
Greek histos 'anything set upright'; and gramma 'drawing, record, writing'. It is also said
that Karl Pearson, who introduced the term in 1891, derived the name from "historical
diagram". Examples
The U.S. Census Bureau found that there were 124 million people who work outside of their
homes. 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.
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. This
type of histogram shows absolute numbers, with Q in thousands.
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 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. 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, 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:
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:
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 until
Sturges's work in 1926. Using wider bins where the density is low
reduces noise due to sampling randomness; using narrower bins where the density is high
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.
The number of bins k can be assigned directly or can be calculated from a suggested bin
width h as:
The braces indicate the ceiling function. Square-root choice
which takes the square root of the number of data points in the sample.
Sturges' formula Sturges' formula is derived from a binomial
distribution and implicitly assumes an approximately normal distribution.
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
The Rice Rule is presented as a simple alternative to Sturges's rule.
Doane's formula Doane's formula is a modification of Sturges'
formula which attempts to improve its performance with non-normal data.
where is the estimated 3rd-moment-skewness of the distribution and
Scott's normal reference rule
where is the sample standard deviation. Scott's normal reference rule is optimal for random
samples of normally distributed data, in the sense that it minimizes the integrated mean
squared error of the density estimate. Freedman–Diaconis' choice
The Freedman–Diaconis rule is:
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 risk function
where and are mean and biased variance of a histogram with bin-width , and .
Remark A good reason why the number of bins should
be proportional to is the following: suppose that the data are obtained as independent
realizations of a bounded probability distribution with smooth density. Then the histogram remains
equally »rugged« as tends to infinity. If is the »width« of the distribution,
then the number of units in a bin is of order and the relative standard error is of order
. Comparing to the next bin, the relative change of the frequency is of order provided
that the derivative of the density is non-zero. These two are of the same order if is of order
, so that is of order . This simple cubic root choice can also be
applied to bins with non-constant width. See also
Data binning Density estimation
Kernel density estimation, a smoother but more complex method of density estimation
Freedman–Diaconis rule Image histogram
Pareto chart Seven Basic Tools of Quality
V-optimal histograms References
Further reading Lancaster, H.O. An Introduction to Medical
Statistics. John Wiley and Sons. 1974. ISBN 0-471-51250-8
External links Journey To Work and Place Of Work
Smooth histogram for signals and images from a few samples
Histograms: Construction, Analysis and Understanding with external links and an application to
particle Physics. A Method for Selecting the Bin Size of a Histogram
Interactive histogram generator Matlab function to plot nice histograms
Dynamic Histogram in MS Excel Histogram construction and manipulation using
Java applets, and charts on SOCR