1. Describing Data: Percentiles
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

2. LEARNING OBJECTIVES
LO1. Compute and understand quartiles, deciles, and percentiles. LO2. Construct and interpret box plots.

3. Quartiles, Deciles and Percentiles
Learning Objective 2 Compute and understand quartiles, deciles, and percentiles.
Quartiles, Deciles and Percentiles
The median splits the data into equal sized halves
Quartiles split the data into quarters
Deciles into tenths
And percentiles can be any split of our choosing
These measures include quartiles, deciles, a

4. - 50%
50% -
Lowest Data
Value
Median
50% value
Highest Data
Value
Quartiles
25%
25%
25%
25% Q1 Q2 Q3
Deciles 1/10
10%
10%
10%
10%
10%
10%
10%
10%
10%
10%

5. Percentile Computation
LO2
Percentile Computation
To formalize the computational procedure, let Lp refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L33 and if we wanted the median, the 50th percentile, then L50.
The number of observations is n, so if we want to locate the median, its position is at (n + 1)/2, or we could write this as
(n + 1)(P/100), where P is the desired percentile.

6. LO2
Percentiles - Example
Listed below are the commissions earned last month by a sample of 15 brokers at Salomon Smith Barney’s Oakland, California, office.
$2,038 $1,758 $1,721 $1,637
$2,097 $2,047 $2,205 $1,787
$2,287 $1,940 $2,311 $2,054
$2,406 $1,471 $1,460
Locate the median, the first quartile, and the third quartile for the commissions earned.

7. Percentiles – Example (cont.)
LO2
Percentiles – Example (cont.)
Step 1: Organize the data from lowest to largest value
$1,460 $1,471 $1,637 $1,721
$1,758 $1,787 $1,940 $2,038
$2,047 $2,054 $2,097 $2,205
$2,287 $2,311 $2,406

8. Percentiles – Example (cont.)
LO2
Percentiles – Example (cont.)
Step 2: Compute the first and third quartiles. Locate L25 and L75 using:
4-8

9. Learning Objective 3 Construct and interpret box plots.
A box plot is a graphical display, based on quartiles, that helps us picture a set of data.
To construct a box plot, we need only five statistics:
the minimum value,
Q1(the first quartile),
the median,
Q3 (the third quartile), and
the maximum value.

10. LO3
Boxplot - Example

11. LO3
Boxplot Example
Step1: Create an appropriate scale along the horizontal axis.
Step 2: Draw a box that starts at Q1 (15 minutes) and ends at Q3 (22
minutes). Inside the box we place a vertical line to represent the median (18 minutes).
Step 3: Extend horizontal lines from the box out to the minimum value (13
minutes) and the maximum value (30 minutes).

12. Example: Draw a Box & Whisker for
Sample Data in Ordered Array:
(n = 9)
Q1 (L25) is in the (9+1)*25/100 = 2.5 position of the ranked data
so use the value half way between the 2nd and 3rd values,
so Q1 = 12.5
Q1 and Q3 are measures of non-central location
Q2 = median, is a measure of central tendency

13. Quartile Measures Calculating The Quartiles: Example
Sample Data in Ordered Array:
(n = 9)
Q1 is in the (9+1)*25/100 = 2.5 position of the ranked data,
so Q1 = (12+13)/2 = 12.5
Q2 is in the (9+1)*50/100 = 5th position of the ranked data,
so Q2 = median = 16
Q3 is in the (9+1)*75/100 = 7.5 position of the ranked data,
so Q3 = (18+21)/2 = 19.5
Q1 and Q3 are measures of non-central location
Q2 = median, is a measure of central tendency

14. Quartile Measures- Calculation Rules
When calculating the ranked position use the following rules
If the result is a whole number then it is the ranked position to use
If the result is a fractional half (e.g. 2.5, 7.5, 8.5, etc.) then average the two corresponding data values.
If the result is not a whole number or a fractional half then interpolate between the data points.

15. Quartile Measures: The Interquartile Range (IQR)
The IQR is Q3 – Q1 and measures the spread in the middle 50% of the data
The IQR is a measure of variability that is not influenced by outliers or extreme values
Measures like Q1, Q3, and IQR that are not influenced by outliers are called resistant measures

16. The Interquartile Range
Example:
Median
(Q2) X X Q1 Q3
maximum
minimum
25% % % %
Interquartile range
= 19.5 – 12.5 = 7

17. Distribution Shape and The Boxplot
Negatively-Skewed
Symmetrical
Positively-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3
Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc..

18. Interpolation
If you found that the first quartile was the 13.75th value then you interpolate like this: Take the 13th and 14th data values Find the difference |14th-15th| Multiply the difference by 0.75 Add the calculated value to the 13th value

19. Exercises – To Do
Page 116 –
Q4-21
Q4-23
Q4-25

20. Stem and Leaf Diagrams 35 23 1 2 3 4 1 2 3 4 6 7 5 8 9 8 6 7 3 5 2 7 7 7 9 8 5 3 7 1 2 1
means 18

21. Stem and Leaf Diagrams

22. Raw data
The following data were collected on the ages of cyclists involved in road accidents

23. 66 6 62 19 20 15 21 8 63 44 10 26 35 61 13 28 7 52 22 64 11 39 9 17 32 36 37 18
138 16 67 45 55 14 49 23 12 88 46 59 25 42 29 60 50 31 34
Total 92

24. Ages of cyclists in road accidents
Always include a title
Always include a Key
Key 6|7 means 67 years