Business statistics, 6e, 2005, groebner CH03

Business Statistics:
A Decision-Making Approach
6th Edition

Chapter 3
Describing Data Using
Numerical Measures

Business Statistics: A Decision-Making Approach, 6e © 2005 PrenticeHall, Inc.

Chap 3-1


Chapter Goals
After completing this chapter, you should be able to:


Compute and interpret the mean, median, and mode for a
set of data



Compute the range, variance, and standard deviation and
know what these values mean



Construct and interpret a box and whiskers plot



Compute and explain the coefficient of variation and
z scores



Use numerical measures along with graphs, charts, and
tables to describe data
Business Statistics: A Decision-

Chap 3-2


Chapter Topics


Measures of Center and Location




Other measures of Location




Mean, median, mode, geometric mean,
midrange
Weighted mean, percentiles, quartiles

Measures of Variation



Range, interquartile range, variance and
standard deviation, coefficient of variation

Business Statistics: A Decision-

Chap 3-3


Summary Measures
Describing Data Numerically
Center and Location

Other Measures
of Location

Mean
Median
Mode

Variation
Range

Percentiles
Interquartile Range
Quartiles

Weighted Mean

Variance
Standard Deviation
Coefficient of
Variation

Business Statistics: A Decision-

Chap 3-4


Measures of Center and
Location
Overview
Center and Location

Mean

Median

Mode

Weighted Mean

n

x=

∑x
i=1

i

XW

n

µ=

i=1

i

i

i

N

∑x

wx
∑
=
∑w
wx
∑
=
∑w

i

N
Business Statistics: A Decision-

µW

i

i

i

Chap 3-5


Mean (Arithmetic Average)


The Mean is the arithmetic average of data values


Sample mean
n = Sample
Size

n



x=

Population mean

∑x
n

x1 + x 2 +  + x n
=
n

N

N = Population
Size

i=1

i

∑x

x1 + x 2 +  + x N
µ=
=
N
N
i=1

i

Business Statistics: A Decision-

Chap 3-6


Mean (Arithmetic Average)
(continued
)




The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)

0 1 2 3 4 5 6 7 8 9 10

Mean = 3
1 + 2 + 3 + 4 + 5 15
=
=3
5
5

Business Statistics: A Decision-

0 1 2 3 4 5 6 7 8 9 10

Mean = 4

1 + 2 + 3 + 4 + 10 20
=
=4
5
5

Chap 3-7


Median


Not affected by extreme values

0 1 2 3 4 5 6 7 8 9 10



0 1 2 3 4 5 6 7 8 9 10

Median = 3
Median = 3
In an ordered array, the median is the “middle” number


If n or N is odd, the median is the middle number



If n or N is even, the median is the average of the two middle

numbers

Business Statistics: A Decision-

Chap 3-8


Mode







A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Mode = 5

Business Statistics: A Decision-

0 1 2 3 4 5 6

No Mode

Chap 3-9


Weighted Mean


Used when values are grouped by frequency or relative
importance

Example: Sample of
26 Repair Projects
Days to
Complete

Frequency

5

4

6

12

7

8

8

2

Weighted Mean Days
to Complete:
XW

wx
∑
=
∑w

Business Statistics: A Decision-

i

i

i

(4 × 5) + (12 × 6) + (8 × 7) + (2 × 8)
=
4 + 12 + 8 + 2
=

164
= 6.31 days
26

Chap 3-10


Review Example


Five houses on a hill by the beach
$2,000 K

House Prices:
$2,000,000
500,000
300,000
100,000
100,000

$500 K
$300 K

$100 K
$100 K

Business Statistics: A Decision-

Chap 3-11


Summary Statistics
House Prices:
$2,000,000

500,000
300,000
100,000
100,000



Mean:



Median: middle value of ranked data
= $300,000



Mode: most frequent value
= $100,000

Sum 3,000,000

($3,000,000/5)
= $600,000

Business Statistics: A Decision-

Chap 3-12


Which measure of location

is the “best”?


Mean is generally used, unless extreme
values (outliers) exist



Then median is often used, since the
median is not sensitive to extreme values.


Example: Median home prices may be reported
for a region – less sensitive to outliers

Business Statistics: A Decision-

Chap 3-13


Shape of a Distribution


Describes how data is distributed



Symmetric or skewed

Left-Skewed

Symmetric

Mean < Median < Mode Mean = Median =
Mode
(Longer tail extends to left)

Business Statistics: A Decision-

Right-Skewed

Mode < Median < Mean
(Longer tail extends to right)

Chap 3-14


Other Location Measures
Other Measures
of Location
Percentiles
The pth percentile in a data array:




p% are less than or equal to this
value
(100 – p)% are greater than or
equal to this value

(where 0 ≤ p ≤ 100)

Business Statistics: A Decision-

Quartiles



1st quartile = 25th percentile



2nd quartile = 50th percentile
= median



3rd quartile = 75th percentile

Chap 3-15


Percentiles


The pth percentile in an ordered array of n values is the
value in ith position, where

p
i=

(n + 1)
100


Example: The 60th percentile in an ordered array of 19
values is the value in 12th position:

p
60
i=
(n + 1) =
(19 + 1) = 12
100
100
Business Statistics: A Decision-

Chap 3-16


Quartiles


Quartiles split the ranked data into 4 equal groups

25%

25%

Q1


25%

Q2

25%
Q3

Example: Find the first quartile

Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)

25 (9+1) = 2.5 position
100

Q1 = 25th percentile, so find the

so use the value half way between the 2nd and 3rd values,
so

Q1 = 12.5

Business Statistics: A Decision-

Chap 3-17


Box and Whisker Plot


A Graphical display of data using 5-number summary:
Minimum -- Q1 -- Median -- Q3 -- Maximum

Example:
25%

Minimum

Minimum

25%

1st
1st
Quartile

Quartile

25%

Median

Median

Business Statistics: A Decision-

25%

3rd

3rd
Quartile

Quartile

Maximum

Maximum

Chap 3-18


Shape of Box and Whisker Plots


The Box and central line are centered between the
endpoints if data is symmetric around the median



A Box and Whisker plot can be shown in either vertical
or horizontal format

Business Statistics: A Decision-

Chap 3-19


Distribution Shape and
Box and Whisker Plot

Left-Skewed

Q1

Q2 Q3

Symmetric

Q1 Q2 Q3

Business Statistics: A Decision-

Right-Skewed

Q1 Q2 Q3

Chap 3-20


Box-and-Whisker Plot
Example


Below is a Box-and-Whisker plot for the following data:
0 2
Min



2Q1 2

3

3 Q2
4

5

5

10Q3 27

Max

This data is very right skewed, as the plot depicts
00 22 33 55

Business Statistics: A Decision-

27
27

Chap 3-21


Measures of Variation
Variation
Range
Interquartile
Range

Variance

Standard Deviation

Population
Variance

Population
Standard
Deviation

Sample
Variance

Sample
Standard
Deviation

Business Statistics: A Decision-

Coefficient of
Variation

Chap 3-22


Variation


Measures of variation give information on the
spread or variability of the data values.

Same center,
different variation

Business Statistics: A Decision-

Chap 3-23


Range



Simplest measure of variation
Difference between the largest and the smallest
observations:

Range = xmaximum – xminimum
Example:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Range = 14 - 1 = 13
Business Statistics: A Decision-

Chap 3-24


Disadvantages of the Range



Ignores the way in which data are distributed
7

8

9

10

11

12

Range = 12 - 7 = 5


7

8

9

10

11

12

Range = 12 - 7 = 5

Sensitive to outliers
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = 5 - 1 = 4

1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = 120 - 1 = 119

Business Statistics: A Decision-

Chap 3-25