Download Section 3.4: Five%Number Summary and more Slides Statistics in PDF only on Docsity!
Section 3.4: Five-Number Summary
1 The Five Number Summary
DeÖnition 1 For any set of data the Öve-number summary is, in order, the Öve summary statistics:
minimum; Q 1 ; median; Q 3 ; maximum:
Example 2 Find the Öve-number summary for the Chicago Bullís salaries.
Obs Player Salary 1 Michael Jordan $33,140, 2 Ron Harper^ $4,560, 3 Toni Kukoc $4,560, 4 Dennis Rodman $4,500, 5 Luc Longley^ $3,184, 6 Scottie Pippen^ $2,775, 7 Bill Wennington^ $1,800, 8 Scott Burrell^ $1,430, 9 Randy Brown^ $1,260, 10 Robert Parish^ $1,150, 11 Jason^ Caffey^ $850, 12 Steve Kerr $750, 13 Keith Booth^ $597, 14 Jud Buechler^ $500, 15 Joe Kleine $272,
Chicago Bulls Salaries 1997-1998 Season
The Öve number summary for the player salaries for the 1997-1998 Chicago Bulls is
Excel TI-83/ minimum $272,250 $272, Q 1 $800,460 $750, median $1,430,000 $1,430, Q 3 $3,842,450 $4,500, maximum $33,140,000 $33,140,
Analysis Variable : Salary Salary
Minimum
Lower Quartile Median
Upper Quartile Maximum Mean Std Dev 10th Pctl 80th Pctl 272250.00 750000.00 1430000.00 4500000.00 33140000.00 4088711.33 8182474.38 500000.00 4530000.
SAS output
2 IQR
The range is very susceptible to unusually large or small values in a date set. A single extreme value skews the range of a set of data. The interquartile range (IQR) is much more resistant to skew. The IQR measures the range of the central 50% of the data.
DeÖnition 3 For a given set of data, the IQR is the (positive or occasionally
- di§erence between Q 3 and Q 1 in a quantitative data set.
Example 4 For the player salaries of the 1997-1998 Chicago Bulls the range is 33140000 272250 = 32 867 750 and the IQR = 3842450 800460 = 3041 990.
3 Outliers and IQR
We consider values that are unusually large or small compared to the rest of a data set to be outliers. We formally deÖne unusually large or small via the IQR.
lower outlier boundary = Q 1 1 : 5 IQR
upper outlier boundary = Q 3 + 1: 5 IQR Values that fall outside these boundaries are outliers.
Problem 5 Determine the salary outliers for the í97-í98 Bulls roster.
Weíve already determined that Q 1 = $800; 460 , Q 3 = $3; 842 ; 450 and IQR = 3842450 800460 = 3041 990. So, the lower outlier boundary is 800460 1 : 5 3041 990 = 3762500. Clearly, nothing falls below this lower outlier boundary. The upper outlier boundary is 3842450 + 1: 5 3041 990 =
- Only Jordanís salary is larger than $8,405,400. The only outlier is this data set is the salary for Michael Jordan.
Cancer
Disease
Vascular
Cerebral
Disease
Heart
Coronary Other Unknown
100
200
300
400
500
Cholesterol
Distribution of Cholesterol by Death Cause
The modiÖed box and whisker plot illustrates outliers. Another type of box plot is the standard box plot. The whiskers in the standard box plot represent the min and max values in the data set. Thus, no outliers are represented. My preference is for the modiÖed box and whisker plot (which are the examples given above).
5 Exercises
- Kokoska 3rd edition Section 3.4: 3.110a, 3.114a, b,
