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### Chapter 5 Understanding and Comparing Distributions

Another Useful Graphical Method: Boxplots

Pulse Rates n = 138

Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70

Q1: median of lower half (lower half = 69 smallest pulses); Q1 = pulse in ordered position 35;

Q1 = 63

Q3 median of upper half (upper half = 69 largest pulses); Q3= pulse in position 35 from the high end; Q3=78

Recall the 5-number summary of data from Chapter 4

- Minimum Q1 median Q3 maximum
- Pulse data 5-number summary

45 63 70 78 111

- A boxplot is a graphical display of the 5-number summary

Example

- Consider the data shown at the left.
- The data values 6.1, 5.6, …, are in the right column
- They are arranged in decreasing order from 6.1 (data rank of 25 shown in far left column) to 0.6 (data rank of 1 in far left column)
- The center column shows the ranks of the quartiles (in blue) from each end of the data and from the overall median (in yellow)

Boxplot: display of 5-number summary

Largest = max = 6.1

BOXPLOT

Q3= third quartile

= 4.2

m = median = 3.4

Q1= first quartile

= 2.3

Five-number summary:

min Q1 m Q3 max

Smallest = min = 0.6

Boxplot: display of 5-number summary

- Example: age of 66 “crush” victims at rock concerts 1999-2000.

5-number summary:

13 17 19 22 47

Boxplot construction

1) construct box with ends located at Q1 and Q3; in the box mark the location of median (usually with a line or a “+”)

2) fences are determined by moving a distance 1.5(IQR) from each end of the box;

2a) upper fence is 1.5*IQR above the upper quartile

2b) lower fence is 1.5*IQR below the lower quartile

Note: the fences only help with constructing the boxplot; they do not appear in the final boxplot display

Box plot construction (cont.)

3) whiskers: draw lines from the ends of the box left and right to the most extreme data values found within the fences;

4) outliers: special symbols represent each data value beyond the fences;

4a) sometimes a different symbol is used for “far outliers” that are more than 3 IQRs from the quartiles

Boxplot: display of 5-number summary

Largest = max = 7.9

BOXPLOT

Distance to Q3

7.9 − 4.2 = 3.7

Q3= third quartile

= 4.2

Interquartile range

Q3 – Q1=

4.2 − 2.3 = 1.9

Q1= first quartile

= 2.3

1.5 * IQR = 1.5*1.9=2.85. Individual #25 has a value of 7.9 years, which is 3.7 years above the third quartile. This is more than 2.85 = 1.5*IQR above Q3. Thus, individual #25 is a suspected outlier.

Beg. of class pulses (n=138)

- Q1 = 63, Q3 = 78
- IQR=78 63=15
- 1.5(IQR)=1.5(15)=22.5
- Q1 - 1.5(IQR): 63 – 22.5=40.5
- Q3 + 1.5(IQR): 78 + 22.5=100.5

40.5

70

78

100.5

63

45

Below is a box plot of the yards gained in a recent season by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q3 ?

410

958

136

684

1232

0

273

1369

821

547

1095

Pass Catching Yards by Receivers

- 450
- 750
- 215
- 545

10

Countdown

Automating Boxplot Construction

- Excel “out of the box” does not draw boxplots.
- Many add-ins are available on the internet that give Excel the capability to draw box plots.
- Statcrunch (http://statcrunch.stat.ncsu.edu) draws box plots.

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