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 |
|
|
///// | 5 | 5/50 | 5 |
|
|
///// ///// | 10 | 10/50 | 15 |
|
|
///// //// | 9 | 9/50 | 24 |
|
|
///// // | 7 | 7/50 | 31 |
|
|
///// / | 6 | 6/50 | 37 |
|
|
///// / | 6 | 6/50 | 43 |
|
|
///// // | 7 | 7/50 | 50 |
| Total | 50 | 1 |
Figure 2.4.5 Frequency histogram for survival time of parts under extraneous operating conditions.
Figure 2.4.6 Relative frequency histogram for survival time of parts under extraneous operating conditions.
Another graph that becomes the basis of probability distributions, which we will study in later chapters, is called the frequency polygon or relative frequency polygon depending on which histogram is used to construct this graph. To construct the frequency or relative frequency polygon, first mark the midpoints on the top ends of the rectangles of the corresponding histogram and then simply join these midpoints. Note that classes with zero frequencies at the lower as well as at the upper end of the histogram are included so that we can connect the polygon with the
Quite often a data set consists of a large number of observations that result in a large number of classes of very small widths. In such cases, frequency polygons or relative frequency polygons become smooth curves. Figure 2.4.8
