# Histogram

A method for representing experimental data. A histogram is constructed as follows. The entire range of the observed values $ X_1, \dots, X_n $ of some random variable $ X $ is subdivided into $ k $ grouping intervals (which are usually all of equal length) by points $ x_1, \dots, x_{k+1} $; the number of observations $ m_i $ per interval $ [x_i, x_{i+1}] $ and the frequency $ h_i=m_i/n $ are computed. The points $ x_1, \dots, x_{k+1} $ are marked on the abscissa, and the segments $ x_ix_{i+1} \quad (i = 1,\dots, k) $ are taken as the bases of rectangles with heights $ h_i/(x_{i+1}-x_i) $. If the intervals $ [x_i, x_{i+1}) $ have equal lengths, the altitudes of the rectangles are taken as $ h_i $ or as $ m_i $. Thus, let the measurements of trunks of 1000 firs give the following results:

diameter in cm. | 22–27 | 27–32 | 32–37 | 37–42 | 42–52 |

number of trunks | 100 | 130 | 500 | 170 | 100 |

The histogram for this example is shown in the figure. diameter in cm. number of trunks

Figure: h047450a

#### Comments

The histogram can be considered as a technique of density estimation (cf. also Density of a probability distribution), and there is much literature on its properties as a statistical estimator of an unknown probability density as $ n\to\infty $ and the grouping intervals are made smaller (grouping intervals of lengths $ \approx n^{-1/3} $ seem optimal).

#### References

[a1] | D. Freedman, P. Diaconis, "On the histogram as a density estimator: $ L_2 $ theory" Z. Wahrsch. Verw. Geb. , 57 (1981) pp. 453–476 |

**How to Cite This Entry:**

Histogram.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Histogram&oldid=29844