tool called the box‐whisker plot or simply box plot, and invented by J. Tukey, helps us answer this question. Figure 2.8.1 illustrates the construction of a box plot for any data set.
Figure 2.8.1 Box‐whisker plot.
2.8.1 Construction of a Box Plot
1 Step 1. Find the quartiles , , and for the given data set.
2 Step 2. Draw a box with its outer lines of the box standing at the first quartile () and the third quartile (), and then draw a line at the second quartile (). The line at divides the box into two boxes, which may or may not be of equal size.
3 Step 3. From the outer lines, draw straight lines extending outwardly up to three times the IQR and mark them as shown in Figure 2.8.1. Note that each distance between the points A and B, B and C, D and E, and E and F is equal to one and a one‐half times distance between the points C and D, or one and one‐half times IQR. The points S and L are, respectively, the smallest and largest data points that fall within the inner fences. The lines from S to C and D to L are called the whiskers.
2.8.2 How to Use the Box Plot
About the Outliers
1 Any data points that fall beyond the lower and upper outer fences are the extreme outliers. These points are usually excluded from the analysis.
2 Any data points between the inner and outer fences are the mild outliers. These points are excluded from the analysis only if we were convinced that these points are somehow recorded or measured in error.
About the Shape of the Distribution
1 If the second quartile (median) is close to the center of the box and each of the whiskers is approximately of equal length, then the distribution is symmetric.
2 If the right box is substantially larger than the left box and/or the right whisker is much longer than the left whisker, then the distribution is right‐skewed.
3 If the left box is substantially larger than the right box and/or the left whisker is much longer than the right whisker, then the distribution is left‐skewed.
Example 2.8.1 (Noise levels) The following data gives the noise level measured in decibels (a usual conversation by humans produces a noise level of about 75 dB) produced by 15 different machines in a very large manufacturing plant:
85 80 88 95 115 110 105 104 89 87 96 140 75 79 99
Construct a box plot and examine whether the data set contains any outliers.
Figure 2.8.2 Box plot for the data in Example 2.8.1.
Solution: First we arrange the data in the ascending order and rank them.
| Data values | 75 | 79 | 80 | 85 | 88 | 89 | 95 | 96 | 97 | 99 | 104 | 105 | 110 | 115 | 140 |
| Ranks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
We then find the ranks of the quartiles
Rank of
Rank of
Rank of
Therefore, the values of
Figure 2.8.2 shows the box plot for these data. Figure 2.8.2 shows that the data include one outlier. In this case, action should be taken to modify the machine that produced a noise level of 140 dB. Use MINITAB to create a box plot:
MINITAB
1 Enter the data in column C1.
2 From the Menu bar, select Graph Boxplot. This prompts a dialog box to appear. In this dialog box, select simple and click OK. This prompts another dialog box to appear.
3 In this dialog box, enter C1 in the box under the Graph variables and click
