Basic Business Statistics, 10/e
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Transcript Basic Business Statistics, 10/e
Basic Business Statistics
10th Edition
Chapter 3
Numerical Descriptive Measures
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc..
Chap 3-1
Learning Objectives
In this chapter, you learn:
To describe the properties of central tendency,
variation, and shape in numerical data
To calculate descriptive summary measures for a
population
To calculate the coefficient of variation and Zscores
To construct and interpret a box-and-whisker plot
To calculate the covariance and the coefficient of
correlation
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-2
Chapter Topics
Measures of central tendency, variation, and
shape
Mean, median, mode, geometric mean
Quartiles
Range, interquartile range, variance and standard
deviation, coefficient of variation, Z-scores
Symmetric and skewed distributions
Population summary measures
Mean, variance, and standard deviation
The empirical rule and Chebyshev rule
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-3
Chapter Topics
(continued)
Five number summary and box-and-whisker
plot
Covariance and coefficient of correlation
Pitfalls in numerical descriptive measures and
ethical issues
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-4
Summary Measures
Describing Data Numerically
Central Tendency
Quartiles
Variation
Arithmetic Mean
Range
Median
Interquartile Range
Mode
Variance
Geometric Mean
Standard Deviation
Shape
Skewness
Coefficient of Variation
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-5
Measures of Central Tendency
Overview
Central Tendency
Arithmetic Mean
Median
Mode
n
X
X
Geometric Mean
XG ( X1 X2 Xn )1/ n
i
i 1
n
Midpoint of
ranked
values
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Most
frequently
observed
value
Chap 3-6
Arithmetic Mean
The arithmetic mean (mean) is the most
common measure of central tendency
For a sample of size n:
n
X
X
i 1
n
Sample size
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
i
X1 X 2 Xn
n
Observed values
Chap 3-7
Arithmetic Mean
(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
Basic Business Statistics, 10e © 2006 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 3-8
Median
In an ordered array, the median is the “middle”
number (50% above, 50% below)
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
Not affected by extreme values
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-9
Finding the Median
The location of the median:
n 1
Median position
position in the ordered data
2
If the number of values is odd, the median is the middle number
If the number of values is even, the median is the average of
the two middle numbers
n 1
is not the value of the median, only the
2
position of the median in the ranked data
Note that
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-10
Mode
A measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical
(nominal) 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 = 9
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
0 1 2 3 4 5 6
No Mode
Chap 3-11
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-12
Review Example:
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
($3,000,000/5)
= $600,000
Chap 3-13
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-14
Quartiles
Quartiles split the ranked data into 4 segments with
an equal number of values per segment
25%
25%
Q1
25%
Q2
25%
Q3
The first quartile, Q1, is the value for which 25% of the
observations are smaller and 75% are larger
Q2 is the same as the median (50% are smaller, 50% are
larger)
Only 25% of the observations are greater than the third
quartile
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-15
Quartile Formulas
Find a quartile by determining the value in the
appropriate position in the ranked data, where
First quartile position:
Q1 = (n+1)/4
Second quartile position: Q2 = (n+1)/2 (the median position)
Third quartile position:
Q3 = 3(n+1)/4
where n is the number of observed values
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-16
Quartiles
Example: Find the first quartile
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)
Q1 is in the (9+1)/4 = 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 noncentral location
Q2 = median, a measure of central tendency
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-17
Quartiles
(continued)
Example:
Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22
(n = 9)
Q1 is in the (9+1)/4 = 2.5 position of the ranked data,
so Q1 = 12.5
Q2 is in the (9+1)/2 = 5th position of the ranked data,
so Q2 = median = 16
Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data,
so Q3 = 19.5
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-18
Geometric Mean
Geometric mean
Used to measure the rate of change of a variable
over time
XG ( X1 X2 Xn )
1/ n
Geometric mean rate of return
Measures the status of an investment over time
RG [(1 R1) (1 R2 ) (1 Rn )]
1/ n
1
Where Ri is the rate of return in time period i
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-19
Example
An investment of $100,000 declined to $50,000 at the
end of year one and rebounded to $100,000 at end
of year two:
X1 $100,000
X2 $50,000
50% decrease
X3 $100,000
100% increase
The overall two-year return is zero, since it started and
ended at the same level.
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-20
Example
(continued)
Use the 1-year returns to compute the arithmetic
mean and the geometric mean:
Arithmetic
mean rate
of return:
( 50%) (100%)
X
25%
2
Geometric
mean rate
of return:
R G [(1 R1 ) (1 R 2 ) (1 Rn )]1/ n 1
Misleading result
[(1 ( 50%)) (1 (100%))]1/ 2 1
[(.50) (2)]1/ 2 1 11/ 2 1 0%
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
More
accurate
result
Chap 3-21
Measures of Variation
Variation
Range
Interquartile
Range
Variance
Standard
Deviation
Coefficient
of Variation
Measures of variation give
information on the spread
or variability of the data
values.
Same center,
different variation
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-22
Range
Simplest measure of variation
Difference between the largest and the smallest
values in a set of data:
Range = Xlargest – Xsmallest
Example:
0 1 2 3 4 5 6 7 8 9 10 11 12
13 14
Range = 14 - 1 = 13
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-23
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-24
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
= Q3 – Q1
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-25
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-26
Variance
Average (approximately) of squared deviations
of values from the mean
n
Sample variance:
S
2
Where
(X X)
i1
2
i
n -1
X = mean
n = sample size
Xi = ith value of the variable X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-27
Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Is the square root of the variance
Has the same units as the original data
n
Sample standard deviation:
S
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
(X X)
2
i
i1
n -1
Chap 3-28
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
130
7
4.3095
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
A measure of the “average”
scatter around the mean
Chap 3-29
Measuring variation
Small standard deviation
Large standard deviation
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-30
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 = 0.926
20 21
Mean = 15.5
S = 4.567
Data B
11
12
13
14
15
16
17
18
19
Data C
11
12
13
14
15
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
16
17
18
19
Chap 3-31
Advantages of Variance and
Standard Deviation
Each value in the data set is used in the
calculation
Values far from the mean are given extra
weight
(because deviations from the mean are squared)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-32
Coefficient of Variation
Measures relative variation
Always in percentage (%)
Shows variation relative to mean
Can be used to compare two or more sets of
data measured in different units
S
100%
CV
X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-33
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Both stocks
have the same
standard
deviation, but
stock B is less
variable relative
to its price
Chap 3-34
Z Scores
A measure of distance from the mean (for example, a
Z-score of 2.0 means that a value is 2.0 standard
deviations from the mean)
The difference between a value and the mean, divided
by the standard deviation
A Z score above 3.0 or below -3.0 is considered an
outlier
XX
Z
S
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-35
Z Scores
(continued)
Example:
If the mean is 14.0 and the standard deviation is 3.0,
what is the Z score for the value 18.5?
X X 18.5 14.0
Z
1.5
S
3.0
The value 18.5 is 1.5 standard deviations above the
mean
(A negative Z-score would mean that a value is less
than the mean)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-36
Shape of a Distribution
Describes how data are distributed
Measures of shape
Symmetric or skewed
Left-Skewed
Symmetric
Right-Skewed
Mean < Median
Mean = Median
Median < Mean
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-37
Using Microsoft Excel
Descriptive Statistics can be obtained
from Microsoft® Excel
Use menu choice:
tools / data analysis / descriptive statistics
Enter details in dialog box
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Chap 3-38
Using Excel
Use menu choice:
tools / data analysis /
descriptive statistics
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-39
Using Excel
(continued)
Enter dialog box
details
Check box for
summary statistics
Click OK
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-40
Excel output
Microsoft Excel
descriptive statistics output,
using the house price data:
House Prices:
$2,000,000
500,000
300,000
100,000
100,000
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-41
Numerical Measures
for a Population
Population summary measures are called parameters
The population mean is the sum of the values in the
population divided by the population size, N
N
Where
X
i1
N
i
X1 X2 XN
N
μ = population mean
N = population size
Xi = ith value of the variable X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-42
Population Variance
Average of squared deviations of values from
the mean
N
Population variance:
σ2
Where
(X μ)
i1
2
i
N
μ = population mean
N = population size
Xi = ith value of the variable X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-43
Population Standard Deviation
Most commonly used measure of variation
Shows variation about the mean
Is the square root of the population variance
Has the same units as the original data
N
Population standard deviation:
σ
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
2
(X
μ)
i
i1
N
Chap 3-44
The Empirical Rule
If the data distribution is approximately
bell-shaped, then the interval:
μ 1σ contains about 68% of the values in
the population or the sample
68%
μ
μ 1σ
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Chap 3-45
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-46
Chebyshev Rule
Regardless of how the data are distributed,
at least (1 - 1/k2) x 100% of the values will
fall within k standard deviations of the mean
(for k > 1)
Examples:
At least
within
(1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ)
(1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ)
(1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ)
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-47
Approximating the Mean from a
Frequency Distribution
Sometimes only a frequency distribution is available, not
the raw data
Use the midpoint of a class interval to approximate the
values in that class
c
X
Where
m f
j j
j1
n
n = number of values or sample size
c = number of classes in the frequency distribution
mj = midpoint of the jth class
fj = number of values in the jth class
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-48
Approximating the Standard Deviation
from a Frequency Distribution
Assume that all values within each class interval
are located at the midpoint of the class
Approximation for the standard deviation from a
frequency distribution:
c
S
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
(m X)
j1
j
2
fj
n -1
Chap 3-49
Exploratory Data Analysis
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
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
25%
Median
Median
25%
3rd
3rd
Quartile
Maximum
Maximum
Quartile
Chap 3-50
Shape of Box-and-Whisker Plots
The Box and central line are centered between the
endpoints if data are symmetric around the median
Min
Q1
Median
Q3
Max
A Box-and-Whisker plot can be shown in either vertical
or horizontal format
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Chap 3-51
Distribution Shape and
Box-and-Whisker Plot
Left-Skewed
Q1
Q2 Q3
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Symmetric
Q1 Q2 Q3
Right-Skewed
Q1 Q2 Q3
Chap 3-52
Box-and-Whisker Plot Example
Below is a Box-and-Whisker plot for the following
data:
Min
0
Q1
2
2
Q2
2
00 22 33 55
3
3
Q3
4
5
5
Max
10
27
27
27
The data are right skewed, as the plot depicts
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-53
The Sample Covariance
The sample covariance measures the strength of the
linear relationship between two variables (called
bivariate data)
The sample covariance:
n
cov ( X , Y )
( X X)(Y Y )
i 1
i
i
n 1
Only concerned with the strength of the relationship
No causal effect is implied
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-54
Interpreting Covariance
Covariance between two random variables:
cov(X,Y) > 0
X and Y tend to move in the same direction
cov(X,Y) < 0
X and Y tend to move in opposite directions
cov(X,Y) = 0
X and Y are independent
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-55
Coefficient of Correlation
Measures the relative strength of the linear
relationship between two variables
Sample coefficient of correlation:
cov(X, Y)
r
SX SY
where
n
cov (X, Y)
(X X)(Y Y)
i1
i
n
i
n 1
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
SX
(X X)
i1
i
n 1
n
2
SY
(Y Y)
i1
2
i
n 1
Chap 3-56
Features of
Correlation Coefficient, r
Unit free
Ranges between –1 and 1
The closer to –1, the stronger the negative linear
relationship
The closer to 1, the stronger the positive linear
relationship
The closer to 0, the weaker the linear relationship
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-57
Scatter Plots of Data with Various
Correlation Coefficients
Y
Y
Y
X
X
r = -1
r = -.6
Y
r=0
Y
Y
r = +1
X
X
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
X
r = +.3
X
r=0
Chap 3-58
Using Excel to Find
the Correlation Coefficient
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Select
Tools/Data Analysis
Choose Correlation from
the selection menu
Click OK . . .
Chap 3-59
Using Excel to Find
the Correlation Coefficient
(continued)
Input data range and select
appropriate options
Click OK to get output
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-60
Interpreting the Result
r = .733
Scatter Plot of Test Scores
100
There is a relatively
strong positive linear
relationship between
test score #1
and test score #2
Test #2 Score
95
90
85
80
75
70
70
75
80
85
90
95
100
Test #1 Score
Students who scored high on the first test tended
to score high on second test, and students who
scored low on the first test tended to score low on
the second test
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-61
Pitfalls in Numerical
Descriptive Measures
Data analysis is objective
Should report the summary measures that best meet
the assumptions about the data set
Data interpretation is subjective
Should be done in fair, neutral and clear manner
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-62
Ethical Considerations
Numerical descriptive measures:
Should document both good and bad results
Should be presented in a fair, objective and
neutral manner
Should not use inappropriate summary
measures to distort facts
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-63
Chapter Summary
Described measures of central tendency
Mean, median, mode, geometric mean
Discussed quartiles
Described measures of variation
Range, interquartile range, variance and standard
deviation, coefficient of variation, Z-scores
Illustrated shape of distribution
Symmetric, skewed, box-and-whisker plots
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-64
Chapter Summary
(continued)
Discussed covariance and correlation
coefficient
Addressed pitfalls in numerical descriptive
measures and ethical considerations
Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.
Chap 3-65