McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

LEARNING OBJECTIVES
LO1. Compute and understand quartiles, deciles , and percentiles .
LO2. Construct and interpret box plots .
4-2

Learning Objective 2
Compute and understand 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
4-3

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

 To formalize the computational procedure, let L p refer to the location of a desired percentile. So if we wanted to find the 33rd percentile we would use L
33
50th percentile, then L
50
. and if we wanted the median, the
LO2
 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.
4-5

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.
LO2
4-6

Step 1: Organize the data from lowest to largest value
LO2
$1,460
$1,758
$2,047
$2,287
$1,471
$1,787
$2,054
$2,311
$1,637
$1,940
$2,097
$2,406
$1,721
$2,038
$2,205
4-7

LO2
Step 2: Compute the first and third quartiles. Locate L
25 and L
75 using:
L
25

( 15

1 )
25
100

4 L
75

( 15

1 )
75
100

12
Therefore, the first and third quartiles are located at the 4th and 12th positions, respective ly
L
25
L
75

$ 1 , 721

$ 2 , 205
4-8

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:
1. the minimum value,
2. Q1(the first quartile),
3. the median,
4. Q3 (the third quartile), and
5. the maximum value.
4-9

LO3
4-10

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).
LO3
4-11

Example: Draw a Box & Whisker for
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)
Q
1
(L
25
) is in the (9+1)*25/100 = 2.5 position of the ranked data so use the value half way between the 2 nd and 3 rd values, so Q
1
= 12.5
Q
1
Q
2 and Q
3 are measures of non-central location
= median, is a measure of central tendency
4-12

Quartile Measures
Calculating The Quartiles: Example
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)
Q
1 is in the (9+1)*25/100 = 2.5 position of the ranked data, so Q
1
= (12+13)/2 = 12.5
Q
2 is in the (9+1)*50/100 = 5 th position of the ranked data, so Q
2
= median = 16
Q
3 is in the (9+1)*75/100 = 7.5 position of the ranked data, so Q
3
= (18+21)/2 = 19.5
Q
1
Q
2 and Q
3 are measures of non-central location
= median, is a measure of central tendency
4-13

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.
4-14

Quartile Measures:
The Interquartile Range (IQR)
― The IQR is Q
3
– Q
1 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 Q
1
, Q
3
, and IQR that are not influenced by outliers are called resistant measures
4-15

The Interquartile Range
Example:
X minimum
Q
1
Median
(Q
2
)
Q
3
25% 25% 25% 25%
X maximum
11 12.5 16 19.5 22
Interquartile range
= 19.5 – 12.5 = 7
4-16

Negatively-Skewed Symmetrical Positively-Skewed
Q
1
Q
2
Q
3
Q
1
Q
2
Q
3
Q
1
Q
2
Q
3
Basic Business Statistics, 11e © 2009
Prentice-Hall, Inc..
4-17

If you found that the first quartile was the
13.75
th value then you interpolate like this:
Take the 13 th and 14 th data values
Find the difference |14 th -15 th |
Multiply the difference by 0.75
Add the calculated value to the 13 th value
4-18

 Page 116 –
Q4-21
Q4-23
Q4-25
4-19

35 23 18 25 20
16 22 27 33 41
27 37 17 25 27
29 28 31 32 40
1
2
3
4
8
3
5
1
6 7
3
0
5 0 2 7
7 1 2
7 7 9 8
1 8 means 18
1
2
3
4
6
0
1
0
7 8
2
1
2 3 5 7
3 5 7
7 7 8 9

5.2
6.6
4.3
8.3
5.1
7.5
8.6
7.1
7.8
2.2
6.6
5.8
3.5
7.5
6.1
3.8
2.5
2.7
8.8
4.8

The following data were collected on the ages of cyclists involved in road accidents
4-22

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

Ages of cyclists in road accidents
Always include a
1 0 0 0 1 1 2 2 3 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 8 9 9
2 0 0 0 1 1 1 2 2 2 3 5 6 6 8 8 9
3 1 2 4 4 5 6 6 7 7 9
4 2 4 4 5 6 9
5 0 2 2 5 9
8 8
Key 6|7 means 67 years
4-24