Page 127 - Computational Statistics Handbook with MATLAB

114 Computational Statistics Handbook with MATLAB

These plots are shown in Figure 5.1. Notice that the shapes of the histograms
are the same in both types of histograms, but the vertical axis is different.
From the shape of the histograms, it seems reasonable to assume that the data
are normally distributed.

Frequency Histogram Relative Frequency Histogram
30 0.25

25
0.2
20
0.15
15
0.1
10

0.05
5

0 0
16 18 20 22 16 18 20 22
Length (inches) Length (inches)

U
F FI IG URE G 5. RE 5. 1 1
1
GU
F F II GU RE RE 5. 5. 1
On the left is a frequency histogram of the forearm data, and on the right is the relative
frequency histogram. These indicate that the distribution is unimodal and that the normal
distribution is a reasonable model.
One problem with using a frequency or relative frequency histogram is that
they do not represent meaningful probability densities, because they do not
integrate to one. This can be seen by superimposing a corresponding normal
distribution over the relative frequency histogram as shown in Figure 5.2.
A density histogram is a histogram that has been normalized so it will inte-
grate to one. That means that if we add up the areas represented by the bars,
then they should add up to one. A density histogram is given by the follow-
ing equation

ˆ ν k
f x() = ------ x in B k , (5.1)
nh

represents the number of data points that
where B k denotes the k-th bin, ν k
fall into the k-th bin and h represents the width of the bins. In the following

© 2002 by Chapman & Hall/CRC