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Introduction to Statistics
Chapter 3
Using Statistics to summarize
Data sets
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, 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

Compute and explain the coefficient of variation

Use the empirical to describe the shape of data
sets
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-2
Chapter Topics

Measures of Center and Location


Other measures of Location


Mean, median, mode
Weighted mean, percentiles, quartiles
Measures of Variation

Range, interquartile range, variance and
standard deviation, coefficient of variation
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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

i1
i
i
i
N
x
wx


w
wx


w
i
N
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
W
i
i
i
Chap 3-5
Mean (Arithmetic Average)

The Mean is the arithmetic average of data
values

Sample mean
x

n = Sample Size
n
x
i1
x1  x 2    x n

n
i
n
Population mean
N = Population Size
N
x
x1  x 2    xN


N
N
i1
i
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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 
w x
w
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
i
i

(4  5)  (12  6)  (8  7)  (2  8)
4  12  8  2

164
 6.31 days
26
i
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-16
Quartiles

Quartiles split the ranked data into 4 equal
groups
25% 25%
25%
25%
Q1

Q2
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-17
Measures of Variation
Variation
Range
Interquartile
Range
Variance
Standard Deviation
Population
Variance
Population
Standard
Deviation
Sample
Variance
Sample
Standard
Deviation
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Coefficient of
Variation
Chap 3-18
Variation

Measures of variation give information on
the spread or variability of the data
values.
Same center,
different variation
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-19
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-20
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-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-21
Interquartile Range

Can eliminate some outlier problems by using
the interquartile range

Eliminate some high-and low-valued
observations and calculate the range from the
remaining values.

Interquartile range = 3rd quartile – 1st quartile
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-22
Interquartile Range
Example:
X
minimum
Q1
25%
12
Median
(Q2)
25%
30
25%
45
X
Q3
maximum
25%
57
70
Interquartile range
= 57 – 30 = 27
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-23
Variance

Average of squared deviations of values from
the mean

Sample variance:
n
s2 

i1
i
σ2 
 x)
n -1
N
Population variance:
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
 (x
2
 (x
i1
i
 μ)
2
N
Chap 3-24
Standard Deviation



Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data

n
Sample standard deviation:
s

Population standard deviation:
i1
n -1
N
σ
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
2
(x

x
)
 i
2
(x

μ)
 i
i1
N
Chap 3-25
Calculation Example:
Sample Standard Deviation
Sample
Data (Xi) :
10
12
14
n=8
s 
15
17
18
18
24
Mean = x = 16
(10  x )2  (12  x )2  (14  x )2    (24  x )2
n 1

(10  16)2  (12  16)2  (14  16)2    (24  16)2
8 1

126
7

4.2426
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-26
Comparing Standard Deviations
Data A
11
12
13
14
15
16
17
18
19
20 21
Mean = 15.5
s = 3.338
20 21
Mean = 15.5
s = .9258
20 21
Mean = 15.5
s = 4.57
Data B
11
12
13
14
15
16
17
18
19
Data C
11
12
13
14
15
16
17
18
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
19
Chap 3-27
Coefficient of Variation

Measures relative variation

Always in percentage (%)

Shows variation relative to mean

Is used to compare two or more sets of data
measured in different units
Population
σ
CV  
μ
  100%
 
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Sample
 s 

CV  
 x   100%


Chap 3-28
Comparing Coefficient
of Variation

Stock A:
 Average price last year = $50
 Standard deviation = $5
s
$5


CVA     100% 
 100%  10%
$50
x

Stock B:


Average price last year = $100
Standard deviation = $5
s
$5
CVB     100% 
 100%  5%
$100
x
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
Chap 3-29
The Empirical Rule


If the data distribution is bell-shaped, then
the interval:
μ  1σ contains about 68% of the values in
the population or the sample
X
68%
μ
μ  1σ
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-30
The Empirical Rule


μ  2σ contains about 95% of the values in
the population or the sample
μ  3σ contains about 99.7% of the values
in the population or the sample
95%
99.7%
μ  2σ
μ  3σ
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-31
Chapter Summary

Described measures of center and location

Mean, median, mode

Discussed percentiles and quartiles

Described measure of variation

Range, interquartile range, variance,
standard deviation, coefficient of variation

Described the empirical rule
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 3-32