vacuum switch controls the vacuum that is applied to a vacuum motor operating a valve in the intake snorkel of the air cleaner. As the engine warms up, the temperature‐sensing unit shuts off the vacuum applied to the motor, allowing the valve to close so that heated air shuts off and outside cooler air is drawn into the engine. The following data give the temperatures (coded) at which the sensing unit shuts off the vacuum:105101120116108112118119107100107120113113101102102100101100118106114100104101107113110100109108100104110113118100119120Prepare a complete frequency distribution table, that is, a table having frequency, relative frequency, percentage, and cumulative frequency columns.
2.4 Graphical Description of Qualitative and Quantitative Data
2.4.1 Dot Plot
A dot plot is one of the simplest graphs. To construct this graph, the value of each observation is plotted on a real line. It provides visual information about the distribution of a single variable. For illustration, we consider the following example.
Example 2.4.1 (Defective motors) The following data give the number of defective motors received in 20 different shipments:
| 8 | 12 | 10 | 16 | 10 | 25 | 21 | 15 | 17 | 5 |
| 26 | 21 | 29 | 8 | 6 | 21 | 10 | 17 | 15 | 13 |
Construct a dot plot for these data.
Solution: To construct a dot plot, draw a horizontal line with its scale beginning with a number less than the smallest observation and ending at a number greater than the largest observation. Then, construct the dot plot by plotting the given data points as shown in Figure 2.4.1.
Figure 2.4.1 Dot plot for the data on defective motors received in 20 different shipments.
Dot plots are more useful when the sample size is small. A dot plot gives us, for example, information about how the data are scattered and where most of the observations are concentrated. For instance, in this example, we see that the minimum number of defective motors and the maximum number of defective motors received in any shipment were 5 and 29, respectively. Also, we can see that 75% of the time, the number of defective motors was between 8 and 21 (inclusive) for these shipments, and so on.
2.4.2 Pie Chart
Pie charts are commonly used to describe qualitative data from one population. It is constructed by dividing a circle into various slices that represent different categories of a population. As examples: allocation of the federal budget by sector, revenues of a large manufacturing company by region or by plant, technicians in a large corporation who are classified according to their basic qualification: high‐school diploma, an associate degree, an undergraduate degree, a graduate degree, and so on. The pie chart helps us better understand at a glance the composition of the population with respect to the characteristic of interest.
To construct a pie chart, divide a circle into slices such that each slice representing a category is proportional to the size of that category. Since the total angle of a circle is 360°, the angle of a slice corresponding to a given category is determined as follows:
(2.4.1)
We illustrate the construction of a pie chart with the following example:
Example 2.4.2 (Manufacturing defect types) In a manufacturing operation, we are interested in understanding defect rates as a function of various process steps. The inspection points (categories) in the process are initial cutoff, turning, drilling, and assembly. The frequency distribution table for these data is shown in Table 2.4.1. Construct a pie chart for these data.
Table 2.4.1 Understanding defect rates as a function of various process steps.
| Process steps | Frequency | Relative frequency | Angle size |
| Initial cutoff | 86 | 86/361 = 23.8% | 85.76 |
| Turning | 182 | 182/361 = 50.4% | 181.50 |
| Drilling | 83 | 83/361 = 23.0% | 82.77 |
| Assembly | 10 | 10/361 = 2.8% | 9.97 |
| Total | 361 | 100% | 360.00 |
Solution: The pie chart for these data is constructed by dividing the circle into four slices. The angle of each slice is given in the last column of Table 2.4.1. Then, the pie chart for the data of Table 2.4.1 is as shown in the MINITAB printout in Figure 2.4.2. Clearly, the pie chart gives us a better understanding at a glance about the rate of defects occurring at different steps of the process.
Figure 2.4.2 Pie chart for the data in Table 2.4.1 using MINITAB.
MINITAB
Using MINITAB, the pie chart is constructed by taking the following steps:
1 Enter the category in column C1.
2 Enter frequencies of the categories in column C2.
3 From the Menu bar, select Graph Pie Chart. Then, check the circle next to Chart values from a table on the pie chart dialog box that appears on the screen.
4 Enter C1 under Categorical values and C2 under Summary variables.
