M08-Numerical Summaries 2 1 Department of ISM, University of Alabama, 1995-2003

Lesson Objectives

Learn what percentiles are andhow to calculate quartiles.

Learn to find the five number summary.

Learn how to construct and use Boxplots.

M08-Numerical Summaries 2 2 Department of ISM, University of Alabama, 1995-2003

If x = the 100pth percentile, then at least 100p% of data is x at least 100(1-p)% of data is x.

Sample 100pth percentile:

Example: You are told you scored 47;then you hear “47” is at the 82nd percentile.

82% of the sample have scores 47, AND 18% have scores 47.

M08-Numerical Summaries 2 3 Department of ISM, University of Alabama, 1995-2003

1. Order the data.

2. Calculate np.

3a. If np is NOT an integer,

Finding 100pth percentile:

n = 25, p = 1/3

np = 8.333,

round up; find the obs. inthis position.

9th positionwill be the33.333 %tile.

M08-Numerical Summaries 2 4 Department of ISM, University of Alabama, 1995-2003

1. Order the data.

2. Calculate np.

3b. If np IS an integer, say k,

Finding 100pth percentile:

n = 25, p = .40np = _____ ,

then avg thekth and (k+1)th

ordered values.

average of

______ & ____positions will bethe 40th %tile.

M08-Numerical Summaries 2 5 Department of ISM, University of Alabama, 1995-2003

1. Maximum 2. 3rd Quartile, Q3 = 75th p’tile

3. Median4. 1st Quartile, Q1 = 25th p’tile

5. Minimum

Five Number Summary

M08-Numerical Summaries 2 6 Department of ISM, University of Alabama, 1995-2003

1st Quartile (25th percentile) : at least 25% of the data values

lie at or below it.

3rd Quartile (75th percentile) : at least 75% of the data values

lie at or below it.

Quartiles:

M08-Numerical Summaries 2 7 Department of ISM, University of Alabama, 1995-2003

Method 1: Percentile method

Q1 located at position (n+1)*1/4

Q2 located at position (n+1)*2/4

Q3 located at position (n+1)*3/4n Q1 Q2 Q3

5

8

11

M08-Numerical Summaries 2 8 Department of ISM, University of Alabama, 1995-2003

Step 1: Order the data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Max = Q3 =Median = Q1 =

Min =

Example 6

M08-Numerical Summaries 2 9 Department of ISM, University of Alabama, 1995-2003

median of observations below the median’s position.

Q3 = median of observations above the median’s position.

Q1 =

Method 2: Median method

M08-Numerical Summaries 2 10 Department of ISM, University of Alabama, 1995-2003

Ordered data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Max = Q3 =Median = Q1 =

Min =

Example 6

M08-Numerical Summaries 2 11 Department of ISM, University of Alabama, 1995-2003

IQR = Q - Q 13

IQR is the range of the middle 50% of the data.

Observations more than 1.5 IQR’s beyond quartiles are considered outliers.

4. Interquartile Range (IQR)

M08-Numerical Summaries 2 12 Department of ISM, University of Alabama, 1995-2003

Which summary statistics should I use?

Shape?

Location?

Variation?

M08-Numerical Summaries 2 13 Department of ISM, University of Alabama, 1995-2003

Boxplot

A graphically display ofthe five number summary

(also called a box-and-whiskers plot)

M08-Numerical Summaries 2 14 Department of ISM, University of Alabama, 1995-2003

Ordered data:12, 14, 16, 18, 19, 21, 22, 25, 27

Max = Q3 =Median = Q1 =

Min =

27.0

19.0

12.0

Q1 = 15.0 Q3 = 23.5

23.5

15.0IQR = 8.5

Example 6

M08-Numerical Summaries 2 15 Department of ISM, University of Alabama, 1995-2003

Ordered data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Example 6A What if . . . .

19, 19, 19,

Ordered data:

12, 14, 16, 18, 19, 21, 22, 25, 27

Example 6B What if . . . .

X

M08-Numerical Summaries 2 16 Department of ISM, University of Alabama, 1995-2003

28

22

24

12

14

16

18

20

26

Note:Middle 50% of data are within the range of the box

Note:Middle 50% of data are within the range of the box

Max = Q3 =Median = Q1 =

Min =

27.0

19.0

12.0

23.5

15.0IQR = 8.5

M08-Numerical Summaries 2 17 Department of ISM, University of Alabama, 1995-2003

Use side-by-side boxplots to display two variables when

one is quantitative, and one is categorical.

Useful tool for comparing distributions.

M08-Numerical Summaries 2 18 Department of ISM, University of Alabama, 1995-2003

A B C

Part Suppliers;who is best?

Part Suppliers;who is best?

15.000

14.980

15.020

15.040

14.960

M08-Numerical Summaries 2 19 Department of ISM, University of Alabama, 1995-2003

Modified Boxplot

Observations more than 1.5 IQR’s beyond quartiles are considered outliers.

Useful in detecting outliers:

More accurate picture of data.

Available in Minitab (boxplot); not in Excel.

M08-Numerical Summaries 2 20 Department of ISM, University of Alabama, 1995-2003

13, 24, 26, 26, 27, 28, 36, 46

Maximum =3rd Quartile =Median =1st Quartile =Minimum =

26.5

46.0

26.5

13.0

25.0 32.0

32.0

25.0IQR = 7.0

Example 7

1.5 IQR = 1.5 • 7.0 = 10.5

M08-Numerical Summaries 2 21 Department of ISM, University of Alabama, 1995-2003

Q3 = 32.0

Q1 = 25.0

Q - 1.5 • IQR = 14.51

Q + 1.5 • IQR = 42.53

*

Note:Whiskers go to themost extreme valuewithin the limits,not to the limits.

Note:Whiskers go to themost extreme valuewithin the limits,not to the limits.

48

44

40

36

32

28

24

20

16

12 *

1.5•IQR

1.5•IQR

Data: 13, 24, 26, 26, 27, 28, 36, 46

M08-Numerical Summaries 2 22 Department of ISM, University of Alabama, 1995-2003

*48

44

40

36

32

28

24

20

16

12 *

Data: 13, 24, 26, 26, 27, 28, 36, 46

FinishedBox PlotFinishedBox Plot

Formula Sheet Example

Q1 Q3M MaxMin

Q3 +1.5 IQRQ1 -1.5 IQRLines extend to thesmallest & largest

obs. inside of limits.

ModifiedModifiedBox Plot:Box Plot:

Box Plot:Box Plot:

Plot each obs. that

is beyond the“outlier limits”on each end.

Note: For this problem, no data are below thelower “outlier limit”.

1.5 IQR1.5 IQR

M08-Numerical Summaries 2 24 Department of ISM, University of Alabama, 1995-2003

Match each of the following descriptions to one of the following histograms.

1. Scores on an EASY Math exam.2. Heights of a group of students.3. Number of medals won by medal winning countries in the 1996 Winter Olympics.4. SAT scores for some college students.5. Last digit in SSN for 100 people.

M08-Numerical Summaries 2 25 Department of ISM, University of Alabama, 1995-2003

AA

BB CC

DD EE

Match descriptions to a Histograms.

1. Scores on an EASY Math exam.2. Heights of a group of students.3. Number of medals won by medal winning countries in the 1996 Winter Olympics.4. SAT scores for some college students.5. Last digit in SSN for 100 people.

M08-Numerical Summaries 2 26 Department of ISM, University of Alabama, 1995-2003

Match each of the following Boxplots (1,2,3,4,5) to one

of the Histograms (A-E) above.

M08-Numerical Summaries 2 27 Department of ISM, University of Alabama, 1995-2003

11

22

33

44

55

M08-Numerical Summaries 2 28 Department of ISM, University of Alabama, 1995-2003

Descriptive Statistics

Variable N Mean Median Range A 100 50.6 51.0 20.0 B 100 49.9 50.1 42.6 C 100 49.9 50.6 12.9 D 100 54.1 32.9 415.4 E 100 50.4 49.8 32.9

M08-Numerical Summaries 2 29 Department of ISM, University of Alabama, 1995-2003

Descriptive Statistics

Variable N Mean Median Range A 100 50.6 51.0 20.0 B 100 49.9 50.1 42.6 C 100 49.9 50.6 12.9 D 100 54.1 32.9 415.4 E 100 50.4 49.8 32.9

11

33

22

44

55

M08-Numerical Summaries 2 30 Department of ISM, University of Alabama, 1995-2003