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# Chapter 5 Understanding and Comparing Distributions - PowerPoint PPT Presentation

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. Q 1 : median of lower half (lower half = 69 smallest pulses); Q 1 = pulse in ordered position 35; Q 1 = 63.

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### Chapter 5 Understanding and Comparing Distributions

Another Useful Graphical Method: Boxplots

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

• 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

• 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)

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

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

5-number summary:

13 17 19 22 47

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

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.

• 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

• 450

• 750

• 215

• 545

10

Countdown

Rock concert deaths: histogram and boxplot by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

Automating Boxplot Construction by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

• 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.

Statcrunch Boxplot by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

Largest = max = 7.9

Q3= third quartile

= 4.2

Q1= first quartile

= 2.3

Tuition 4-yr Colleges by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

Statcrunch: 2012-13 NFL Salaries by Position by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

College Football Head Coach Salaries by Conference by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

2013 Major League Baseball Salaries by Team by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q

End of Chapter 5 by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q