Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP. Bhisham C. Gupta
Solution: In this example, the variable of interest is the number of bypass surgeries performed at a hospital in a period of 24 hours. Now, following the discussion in Example 2.3.1, we can see that the frequency distribution table for the data in this example is as shown in Table 2.3.3. Frequency distribution table defined by using a single numerical value is usually called a single‐valued frequency distribution table.
Table 2.3.3 Frequency distribution table for the hospital data.
| Frequency | Cumulative | Cumulative | |||
| Categories | Tally | or count | frequency | Percentage | percentage |
| 1 | ///// /// | 8 | 8 | 16.00 | 16.00 |
| 2 | ///// ///// //// | 14 | 22 | 28.00 | 44.00 |
| 3 | ///// //// | 9 | 31 | 18.00 | 62.00 |
| 4 | ///// //// | 9 | 40 | 18.00 | 80.00 |
| 5 | ///// / | 6 | 46 | 12.00 | 92.00 |
| 6 | //// | 4 | 50 | 8.00 | 100.00 |
| Total | 50 | 100.00 |
2.3.2 Quantitative Data
So far, we have discussed frequency distribution tables for qualitative data and quantitative data that can be treated as qualitative data. In this section, we discuss frequency distribution tables for quantitative data.
Let
1 Step 1. Find the range of the data that is defined as(2.3.1)
2 Step 2. Divide the data set into an appropriate number of classes. The classes are also sometimes called categories, cells, or bins. There are no hard and fast rules to determine the number of classes. As a rule, the number of classes, say , should be somewhere between 5 and 20. However, Sturges's formula is often used, given by(2.3.2) or(2.3.3) where is the total number of data points in a given data set and log denotes the log to base 10. The result often gives a good estimate for an appropriate number of intervals. Note that since , the number of classes, should always be a whole number, the reader may have to round up or down the value of obtained when using either equation (2.3.2) or (2.3.3).
3 Step 3. Determine the width of classes as follows:(2.3.4) The class width should always be a number that is easy to work with, preferably a whole number. Furthermore, this number should be obtained only by rounding up (never by rounding down) the value obtained when using equation (2.3.4).
4 Step 4. Finally, preparing the frequency distribution table is achieved by assigning each data point to an appropriate class. While assigning these data points to a class, one must be particularly careful to ensure that each data point be assigned to one, and only one, class and that the whole set of data is included in the table. Another important point is that the class at the lowest end of the scale must begin at a number that is less than or equal to the smallest data point and that the class at the highest end of the scale must end with a number that is greater than or equal to the largest data point in the data set.
Example 2.3.4 (Rod manufacturing) The following data give the lengths (in millimeters) of 40 randomly selected rods manufactured by a company:
| 145 | 140 | 120 | 110 | 135 | 150 | 130 | 132 | 137 | 115 |
| 142 | 115 | 130 | 124 | 139 | 133 | 118 | 127 | 144 | 143 |
| 131 | 120 | 117 | 129 | 148 | 130 | 121 |
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