EXAMPLE 2 Constructing a Frequency Distribution and Histogram
Instead of using the individual data values from Table 5.3 for the classes, we set up class intervals for the COINS data and construct a frequency distribution based on the class intervals. We then display the frequency distribution graphically in a histogram.
Constructing a Frequency Distribution
Step 1: Choose the classes. Determine an interval that is wide enough to contain all the data. Subdivide this interval into a reasonable number of class intervals of equal width. Be sure to specify the classes precisely so that each individual falls into exactly one class.
The data in Table 5.3 range from 0 to 225. So here’s one way to choose the class intervals. All the data are between 0 and 250. We subdivide this interval into five class intervals of equal width:
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Step 2: Setting up the table. Set up a table with three columns for the following: class interval, tally, and frequency. (Remove the tally column in the final table.)
Step 3: To complete the table, determine the frequency with which data values fall into each class interval.
Step 4: If desired, add a fourth column for relative frequency. The entries in this column should be the frequencies divided by the number of data values.
Table 5.4 shows the construction of a frequency and relative frequency table for the COINS data (Table 5.3). Since there were 33 data values, the relative frequencies were determined by dividing the frequencies by 33.
| Class Interval | Tally | Frequency | Relative Frequency |
|---|---|---|---|
|
11 | ||
|
11 | ||
|
6 | ||
|
3 | ||
|
2 |
Making a Histogram
The best way to represent a frequency distribution graphically is with a histogram. Here are the steps for making a histogram.
- Step 1: Draw a set of axes. On the horizontal axis, mark the boundaries of the class intervals. On the vertical axis, set up a scale appropriate for the frequencies (or relative frequencies).
- Step 2: Label the horizontal axis with the name of the variable being measured and the units. Label the vertical axis as "Frequency" (or "Relative Frequency").
- Step 3: Over each class interval, draw a rectangle with the interval as its base. The height of the rectangle should match the frequency (or relative frequency) of data contained in that interval.
In the case of the coin data, we draw a horizontal axis with tick marks every 50 units from 0 to 250 to mark the class intervals. On the vertical axis, we place tick marks every two units from 0 to 12 for the frequencies. We then add the rectangular bars to make the histogram shown in Figure 5.2.
