Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta
histogram is a graphical tool consisting of bars placed side by side on a set of intervals (classes, bins, or cells) of equal width. The bars represent the frequency or relative frequency of classes. The height of each bar is proportional to the frequency or relative frequency of the corresponding class.
To construct a histogram, we take the following steps:
1 Step 1. Prepare a frequency distribution table for the given data.
2 Step 2. Use the frequency distribution table prepared in Step 1 to construct the histogram. From here, the steps involved in constructing a histogram are exactly the same as those to construct a bar chart, except that in a histogram, there is no gap between the intervals marked on the horizontal axis (the ‐axis).
A histogram is called a frequency histogram or a relative frequency histogram depending on whether the scale on the vertical axis (the
Example 2.4.5 (Survival times) The following data give the survival times (in hours) of 50 parts involved in a field test under extraneous operating conditions.
| 60 | 100 | 130 | 100 | 115 | 30 |
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145 | 75 | 80 | 89 | 57 | 64 | 92 | 87 | 110 | 180 |
| 195 | 175 | 179 | 159 | 155 | 146 | 157 | 167 | 174 | 87 | 67 | 73 | 109 | 123 | 135 | 129 | 141 |
| 154 | 166 | 179 | 37 |
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89 |
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39 | 49 | 190 |
Construct a frequency distribution table for this data. Then, construct frequency and relative frequency histograms for these data.
Solution:
1 Step 1. Find the range of the data:Then, determine the number of classes (see for example the Sturges' formula , in (2.3.2))Last, compute the class width:As we noted earlier, the class width number is always rounded up to another convenient number that is easy to work with. If the number calculated using (2.3.4) is rounded down, then some of the observations will be left out as they will not belong to any class. Consequently, the total frequency will be less than the total count of the data. The frequency distribution table for the data in this example is shown in Table 2.4.3.
2 Step 2. Having completed the frequency distribution table, construct the histograms. To construct the frequency histogram, first mark the classes on the ‐axis and the frequencies on the ‐axis. Remember that when marking the classes and identifying the bins on the ‐axis, there must be no gap between them. Then, on each class marked on the ‐axis, place a rectangle, where the height of each rectangle is proportional to the frequency of the corresponding class. The frequency histogram for the data with the frequency distribution given in Table 2.4.3 is shown in Figure 2.4.5. To construct the relative frequency histogram, the scale is changed on the ‐axis (see Figure 2.4.5) so that instead of plotting the frequencies, we plot relative frequencies. The resulting graph for this example, shown in Figure 2.4.6, is called the relative frequency histogram for the data with relative frequency distribution given in Table 2.4.3.
Table 2.4.3 Frequency distribution table for the survival time of parts.
| Frequency | Relative | Cumulative | ||
| Class | Tally | or count | frequency | frequency |
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///// | 5 | 5/50 | 5 |
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///// ///// | 10 | 10/50 | 15 |
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