Example 34

EXAMPLE 34 Constructing a boxplot by hand

Construct a boxplot by hand for the export data.

Solution

From Example 31, the fve-number summary for the state export data is $Min = 1.6$ , $Q 1 = 3.4$ , $Med = 4.65$ , $Q 3 = 5.4$ , $Max = 7.7$ . The interquartile range for the state export data is $IQR = Q 3 - Q 1 = 5.4 - 3.4 = 2.0$ .

Step 1 Determine the lower and upper fences:
1. Lower $fence = Q 1 - 1.5 (IQR) = 3.4 - 1.5 (2) = 0.4$
2. Upper $fence = Q 3 + 1.5 (IQR) = 5.4 + 1.5 (2) = 8.4$
Step 2 Draw a horizontal number line that encompasses the range of your data, including the fences. Above the number line, draw vertical lines at $Q 1 = 3.4$ , $median = 4.65$ , and $Q 3 = 5.4$ . Connect the lines for Q1 and Q3 to each other so as to form a box, as shown in Figure 35a.

FIGURE 35A Constructing a boxplot by hand: Steps 1 and 2.
Step 3 Temporarily indicate the fences ( $lower fence = 0.4$ and $upper fence = 8.4$ ) as brackets above the number line. (See Figure 35b.)

FIGURE 35B Constructing a boxplot by hand: Step 3.
Step 4 Draw a horizontal line from $Q 1 = 3.4$ to the smallest data value greater than the lower fence. The lowest data value is $min = 1.6$ . This is greater than the lower $fence = 0.4$ , so draw the line from 3.4 to 1.6. Draw a horizontal line from $Q 3 = 5.4$ to the largest data value smaller than the upper fence. The largest data value is $Max = 7.7$ , which is smaller than the upper fence, so draw the line from 5.4 to 7.7. (See Figure 35c.)
Page 175

FIGURE 35C Constructing a boxplot by hand: Step 4.
Step 5 No data values are lower than the lower fence or greater than the upper fence. Thus, no outliers exist in this data set. Therefore, simply remove the temporary brackets, and the boxplot is complete, as shown in Figure 35d.

FIGURE 35D The completed boxplot.

NOW YOU CAN DO

Exercises 9–10, 15–16, and 21–22.