Chapter 3Numerically

Summarizing DataSection 3.5

Five Number Summary; Boxplots

The Five-Number Summary

MINIMUM Q1 M Q3

MAXIMUM

EXAMPLE Finding the Five Number Summary

Find the five number summary for the employment ratio data from Section 3.4.

Steps for Drawing a BoxplotStep 1: Determine the lower and upper fence:

Lower Fence = Q1 - 1.5(IQR)Upper Fence = Q3 + 1.5(IQR)

Steps for Drawing a BoxplotStep 1: Determine the lower and upper fence:

Lower Fence = Q1 - 1.5(IQR)Upper Fence = Q3 + 1.5(IQR)

Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box.

Steps for Drawing a BoxplotStep 1: Determine the lower and upper fence:

Lower Fence = Q1 - 1.5(IQR)Upper Fence = Q3 + 1.5(IQR)

Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence.

Steps for Drawing a BoxplotStep 1: Determine the lower and upper fence:

Lower Fence = Q1 - 1.5(IQR)Upper Fence = Q3 + 1.5(IQR)

Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence.Step 4: Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence.

Steps for Drawing a BoxplotStep 1: Determine the lower and upper fence:

Lower Fence = Q1 - 1.5(IQR)Upper Fence = Q3 + 1.5(IQR)

Step 2: Draw vertical lines at Q1, M, and Q3. Enclose these vertical lines in a box. Step 3: Label the lower and upper fence.Step 4: Draw a line from Q1 to the smallest data value that is larger than the lower fence. Draw a line from Q3 to the largest data value that is smaller than the upper fence.Step 5: Any data values less than the lower fence or greater than the upper fence are outliers and are marked with an asterisk (*).

EXAMPLE Drawing a Boxplot

Draw a boxplot for the employment ratio data.

1. If the median is near the center of the box and each of the horizontal lines are approximately equal length, then the distribution is roughly symmetric.2. If the median is left of the center of the box and/or the right line is substantially longer than the left line, the distribution is right skewed.3. If the median is right of the center of the box and/or the left line is substantially longer than the right line, the distribution is left skewed

Distribution Shape Based Upon Boxplot

Symmetric

Skewed Right

Skewed Left

EXAMPLE Identify the Shape of a Distribution from a Boxplot

Determine the shape of the employment ratio data based on the boxplot.

EXAMPLE Comparing Two Data Sets Using Boxplots

The following data represent the birth rate for women 15 - 44 years of age in 1990 and 1997 for each state. Draw boxplots for each year using the same scale.

49.5 68.4 60.3 61.2 64.2 55.9 61.852.4 60.0 58.9 62.1 70.4 60.7 69.049.6 58.1 61.7 66.3 78.1 62.9 72.457.8 61.4 58.0 67.3 88.5 64.656.6 60.4 53.1 65.7 75.4 65.760.3 62.2 64.3 67.8 61.8 64.963.9 61.0 60.4 75.3 62.7 59.964.1 63.6 66.1 59.1 72.3 70.3

60.5 63.9 68.0 66.1 64.8 74.2 85.364.1 65.9 66.1 64.7 67.3 67.4 86.360.5 66.5 73.8 56.1 72.9 65.9 79.262.4 73.0 69.1 65.5 70.1 77.763.6 69.4 70.3 69.7 72.1 82.364.2 64.7 68.9 69.7 67.5 92.069.1 66.3 67.2 71.5 77.2 77.667.0 64.8 71.7 62.8 65.9 68.7

1990:

1997: