Section 4.3: Summarizing a Data Set: Boxplots

Construction of a Skeletal Boxplot

1. Draw a horizontal (or vertical) measurement scale.

2. Construct a rectangular box whose left (or lower) edge is at the lower quartile and whose right (or upper) edge is at the upper quartile (so box width = iqr)

3. Draw a vertical (or horizontal) line segment inside the box at the location of the median

4. Extend horizontal (or vertical) line segments, called whiskers, from each end of the box to the smallest and largest observations in the data set

Example: Revisiting Hospital Cost-to-Charge Ratios

Lower Half 45 48 50 54 57 60 60 62 63 63 64 65 67 68 69

Median 71

Upper Half 71 72 72 74 74 75 75 76 80 83 84 88 100 100 100

• To construct a boxplot you need:– Smallest observation– The lowest quartile– The median– The upper quartile– Largest observation

(Called the five-number summary)

• Smallest = 45

• Lowest quartile (median of lower) = 62

• Median = 71

• Upper quartile (median of upper) = 76

• Largest = 100

• Use these values to construct a boxplot

Boxplot for Cost-to-Charge Ratio

Modified Boxplot

• Outlier – if it is more than 1.5(iqr) away from the nearest end of the box

• Extreme outlier – more than 3(iqr) from the nearest end of the box and it is mild otherwise

• Boxplot – represents mild outliers by shaded circles and extreme outliers by open circles

Example: Golden Rectangles

• The accompanying data came from an anthropological study of rectangular shapes. Observations were made on the variable x = width/length for a sample of n = 20 beaded rectangles used in Indian leather handicrafts. The following is the data set:

0.553 0.570 0.576 0.601 0.606 0.6060.609 0.611 0.615 0.628 0.654 0.6620.668 0.670 0.672 0.690 0.693 0.7490.844 0.933

Median = 0.641Lower quartile = 0.606Upper quartile = 0.681Iqr = 0.681 – 0.606 = .0751.5(iqr) = 1.5(.075) =0.11253(iqr) = 3(.075) = 0.225

• Thus, – Upper edge of box + 1.5(iqr)

• 0.681 + 0.1125 = 0.7935

– Lower edge of box – 1.5(iqr)• 0.606 – 0.1125 = 0.4935

So 0.844 and 0.933 are both outliers on the upper end (because they are larger than 0.7935), and there are no outliers on the lower end.

• Upper edge of box + 3(iqr)0.681 + 0.225 = 0.906

0.933 is an extreme outlier and 0.844 is only a mild outlier. The upper whisker extends to the largest observation that is not an outlier and the lower whisker extends to 0.553.

Modified Boxplot

Comparative BoxplotsExample: The Cost of Coffee

• An article compared various brands of brewed coffee. The study included 31 regular-roast caffeinated brands (Type 1), 9 decaffeinated brands (Type 2), and 13 dark-roast brands (Type 3). The following is the comparative boxplot for the cost of coffee.

Classwork: worksheetActivity: Boxplot Shapes

Try This One:

Construct a modified boxplot using the given information.

20 37 4 20 0 84 1436 5 331 19 0 0 22 313 14 36 4 0 18 8 026 4 0 5 23 19 7 12 8

13 16 21 7 13 12 842