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A method for representing experimental data. A histogram is constructed as follows. The entire range of the observed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474501.png" /> of some random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474502.png" /> is subdivided into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474503.png" /> grouping intervals (which are usually all of equal length) by points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474504.png" />; the number of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474505.png" /> per interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474506.png" /> and the frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474507.png" /> are computed. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474508.png" /> are marked on the abscissa, and the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474509.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745010.png" />) are taken as the bases of rectangles with heights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745011.png" />. If the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745012.png" /> have equal lengths, the altitudes of the rectangles are taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745013.png" /> or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745014.png" />. Thus, let the measurements of trunks of 1000 firs give the following results:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">diameter in cm.</td> <td colname="2" style="background-color:white;" colspan="1">22–27</td> <td colname="3" style="background-color:white;" colspan="1">27–32</td> <td colname="4" style="background-color:white;" colspan="1">32–37</td> <td colname="5" style="background-color:white;" colspan="1">37–42</td> <td colname="6" style="background-color:white;" colspan="1">42–52</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">number of trunks</td> <td colname="2" style="background-color:white;" colspan="1">100</td> <td colname="3" style="background-color:white;" colspan="1">130</td> <td colname="4" style="background-color:white;" colspan="1">500</td> <td colname="5" style="background-color:white;" colspan="1">170</td> <td colname="6" style="background-color:white;" colspan="1">100</td> </tr> </tbody> </table>
 
A method for representing experimental data. A histogram is constructed as follows. The entire range of the observed values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474501.png" /> of some random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474502.png" /> is subdivided into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474503.png" /> grouping intervals (which are usually all of equal length) by points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474504.png" />; the number of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474505.png" /> per interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474506.png" /> and the frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474507.png" /> are computed. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474508.png" /> are marked on the abscissa, and the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h0474509.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745010.png" />) are taken as the bases of rectangles with heights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745011.png" />. If the intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745012.png" /> have equal lengths, the altitudes of the rectangles are taken as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745013.png" /> or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047450/h04745014.png" />. Thus, let the measurements of trunks of 1000 firs give the following results:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">diameter in cm.</td> <td colname="2" style="background-color:white;" colspan="1">22–27</td> <td colname="3" style="background-color:white;" colspan="1">27–32</td> <td colname="4" style="background-color:white;" colspan="1">32–37</td> <td colname="5" style="background-color:white;" colspan="1">37–42</td> <td colname="6" style="background-color:white;" colspan="1">42–52</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">number of trunks</td> <td colname="2" style="background-color:white;" colspan="1">100</td> <td colname="3" style="background-color:white;" colspan="1">130</td> <td colname="4" style="background-color:white;" colspan="1">500</td> <td colname="5" style="background-color:white;" colspan="1">170</td> <td colname="6" style="background-color:white;" colspan="1">100</td> </tr> </tbody> </table>
  

Revision as of 19:52, 15 February 2013


A method for representing experimental data. A histogram is constructed as follows. The entire range of the observed values of some random variable is subdivided into grouping intervals (which are usually all of equal length) by points ; the number of observations per interval and the frequency are computed. The points are marked on the abscissa, and the segments () are taken as the bases of rectangles with heights . If the intervals have equal lengths, the altitudes of the rectangles are taken as or as . Thus, let the measurements of trunks of 1000 firs give the following results:'
<tbody> </tbody>
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 and the grouping intervals are made smaller (grouping intervals of lengths seem optimal).

References

[a1] D. Freedman, P. Diaconis, "On the histogram as a density estimator: 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=29440
This article was adapted from an original article by V.N. Chugueva (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article