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© Peter Broadfoot 2008
Histograms
Modal Class
The mode, along with the mean and the median, are three of the important statistics used to
summarise a data set. They were explained using examples of ungrouped data. For
example, as a very simple reminder, if you have a sample of just three values, 1, 2, and 3,
the median (the middle value) is 2. The mean is also 2. There is no mode. You will recall
that the mode (also called the modal value) is the value that occurs most often. For the set
1, 2, 3, 3, 5, 6, 7, the mode is 3 (there are two 3s). For a large set of data, once the data are
grouped, and without the raw data, there is no way of knowing the modal value. In fact, in
that situation, individual values are not important. Knowing the class that contains most
data is more useful. That class is called the modal class.
For the Employees example (see pages 18-19), 150 employees live between 3 and 5miles
from work (3≤d<5). That is the largest group and hence the modal class. Take care with
grouped data when the groups are different widths. The modal class is the class with the
largest frequency density, not necessarily the largest frequency. In the Employees example,
the classes are all the same width and so the class interval 3≤d<5 has the largest frequency
and the largest frequency density (see table, page 18).
Summary
The main points about frequency density and histograms are:
use frequency density, not frequency on the y-axis.
the area of a bar is the frequency.
the combined area of all the bars equals the total number of data in the sample.
the height of a bar is the frequency density.
frequency density is the frequency per unit class width, which means that you divide the
frequency by the width of the bar (fd=f/w).
the modal class is the class with the largest frequency density.
because frequency equals area, the ratio of two frequencies equals the ratio of the areas
of the corresponding bars. If one bar’s area is two 1cm squares, and a 2nd bar’s area is
six 1cm squares, the frequency for the 2nd bar is 6:2 or 3 times bigger. If the frequency
of the 1st bar is 10, the frequency of the bigger bar is 3×10 = 30. The example, next
page, shows a typical calculation.
If you are not convinced that plotting frequency density, instead of frequency, is so
important, see Appendix B – Variable Class Width.
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