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Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2005 Thomson/South-Western
Slide 1
Chapter 3
Descriptive Statistics: Numerical Measures
Part B

Measures of Distribution Shape, Relative Location,
and Detecting Outliers

Exploratory Data Analysis

Measures of Association Between Two Variables

The Weighted Mean and
Working with Grouped Data
© 2005 Thomson/South-Western
Slide 2
Measures of Distribution Shape,
Relative Location, and Detecting Outliers





Distribution Shape
z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers
© 2005 Thomson/South-Western
Slide 3
Distribution Shape: Skewness

An important measure of the shape of a distribution
is called skewness.

The formula for computing skewness for a data set is
somewhat complex.

Skewness can be easily computed using statistical
software.
© 2005 Thomson/South-Western
Slide 4
Distribution Shape: Skewness

Symmetric (not skewed)
• Skewness is zero.
• Mean and median are equal.
Relative Frequency
.35
Skewness = 0
.30
.25
.20
.15
.10
.05
0
© 2005 Thomson/South-Western
Slide 5
Distribution Shape: Skewness

Moderately Skewed Left
• Skewness is negative.
• Mean will usually be less than the median.
Relative Frequency
.35
Skewness = - .31
.30
.25
.20
.15
.10
.05
0
© 2005 Thomson/South-Western
Slide 6
Distribution Shape: Skewness

Moderately Skewed Right
• Skewness is positive.
• Mean will usually be more than the median.
Relative Frequency
.35
Skewness = .31
.30
.25
.20
.15
.10
.05
0
© 2005 Thomson/South-Western
Slide 7
Distribution Shape: Skewness

Highly Skewed Right
• Skewness is positive (often above 1.0).
• Mean will usually be more than the median.
Relative Frequency
.35
Skewness = 1.25
.30
.25
.20
.15
.10
.05
0
© 2005 Thomson/South-Western
Slide 8
Distribution Shape: Skewness

Example: Apartment Rents
Seventy efficiency apartments
were randomly sampled in
a small college town. The
monthly rent prices for
these apartments are listed
in ascending order on the next slide.
© 2005 Thomson/South-Western
Slide 9
Distribution Shape: Skewness
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
© 2005 Thomson/South-Western
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 10
Distribution Shape: Skewness
Relative Frequency
.35
Skewness = .92
.30
.25
.20
.15
.10
.05
0
© 2005 Thomson/South-Western
Slide 11
z-Scores
The z-score is often called the standardized value.
It denotes the number of standard deviations a data
value xi is from the mean.
xi - x
zi 
s
© 2005 Thomson/South-Western
Slide 12
z-Scores
 An observation’s z-score is a measure of the relative
location of the observation in a data set.
 A data value less than the sample mean will have a
z-score less than zero.
 A data value greater than the sample mean will have
a z-score greater than zero.
 A data value equal to the sample mean will have a
z-score of zero.
© 2005 Thomson/South-Western
Slide 13
z-Scores

z-Score of Smallest Value (425)
xi - x 425 - 490.80
z

 - 1.20
s
54.74
Standardized Values for Apartment Rents
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
© 2005 Thomson/South-Western
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 14
Chebyshev’s Theorem
At least (1 - 1/z2) of the items in any data set will be
within z standard deviations of the mean, where z is
any value greater than 1.
© 2005 Thomson/South-Western
Slide 15
Chebyshev’s Theorem
At least 75% of the data values must be
within z = 2 standard deviations of the mean.
At least 89% of the data values must be
within z = 3 standard deviations of the mean.
At least 94% of the data values must be
within z = 4 standard deviations of the mean.
© 2005 Thomson/South-Western
Slide 16
Chebyshev’s Theorem
For example:
Let z = 1.5 with
x = 490.80 and s = 54.74
At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56%
of the rent values must be between
x - z(s) = 490.80 - 1.5(54.74) = 409
and
x + z(s) = 490.80 + 1.5(54.74) = 573
(Actually, 86% of the rent values
are between 409 and 573.)
© 2005 Thomson/South-Western
Slide 17
Empirical Rule
For data having a bell-shaped distribution:
68.26% of the values of a normal random variable
are within +/- 1 standard deviation of its mean.
95.44% of the values of a normal random variable
are within +/- 2 standard deviations of its mean.
99.72% of the values of a normal random variable
are within +/- 3 standard deviations of its mean.
© 2005 Thomson/South-Western
Slide 18
Empirical Rule
99.72%
95.44%
68.26%
m – 3s
m – 1s
m – 2s
© 2005 Thomson/South-Western
m
m + 3s
m + 1s
m + 2s
x
Slide 19
Detecting Outliers
 An outlier is an unusually small or unusually large
value in a data set.
 A data value with a z-score less than -3 or greater
than +3 might be considered an outlier.
 It might be:
• an incorrectly recorded data value
• a data value that was incorrectly included in the
data set
• a correctly recorded data value that belongs in
the data set
© 2005 Thomson/South-Western
Slide 20
Detecting Outliers
 The most extreme z-scores are -1.20 and 2.27
 Using |z| > 3 as the criterion for an outlier, there are
no outliers in this data set.
Standardized Values for Apartment Rents
-1.20
-0.93
-0.75
-0.47
-0.20
0.35
1.54
-1.11
-0.93
-0.75
-0.38
-0.11
0.44
1.54
-1.11
-0.93
-0.75
-0.38
-0.01
0.62
1.63
-1.02
-0.84
-0.75
-0.34
-0.01
0.62
1.81
-1.02
-0.84
-0.75
-0.29
-0.01
0.62
1.99
© 2005 Thomson/South-Western
-1.02
-0.84
-0.56
-0.29
0.17
0.81
1.99
-1.02
-0.84
-0.56
-0.29
0.17
1.06
1.99
-1.02
-0.84
-0.56
-0.20
0.17
1.08
1.99
-0.93
-0.75
-0.47
-0.20
0.17
1.45
2.27
-0.93
-0.75
-0.47
-0.20
0.35
1.45
2.27
Slide 21
Exploratory Data Analysis


Five-Number Summary
Box Plot
© 2005 Thomson/South-Western
Slide 22
Five-Number Summary
1
Smallest Value
2
First Quartile
3
Median
4
Third Quartile
5
Largest Value
© 2005 Thomson/South-Western
Slide 23
Five-Number Summary
Lowest Value = 425
First Quartile = 445
Median = 475
Third Quartile = 525
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
© 2005 Thomson/South-Western
Largest Value = 615
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 24
Box Plot
 A box is drawn with its ends located at the first and
third quartiles.
 A vertical line is drawn in the box at the location of
the median (second quartile).
375 400 425 450 475 500 525 550 575 600 625
Q1 = 445
Q3 = 525
Q2 = 475
© 2005 Thomson/South-Western
Slide 25
Box Plot



Limits are located (not drawn) using the interquartile
range (IQR).
Data outside these limits are considered outliers.
The locations of each outlier is shown with the
symbol * .
… continued
© 2005 Thomson/South-Western
Slide 26
Box Plot
 The lower limit is located 1.5(IQR) below Q1.
Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(75) = 332.5
 The upper limit is located 1.5(IQR) above Q3.
Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
 There are no outliers (values less than 332.5 or
greater than 637.5) in the apartment rent data.
© 2005 Thomson/South-Western
Slide 27
Box Plot

Whiskers (dashed lines) are drawn from the ends of
the box to the smallest and largest data values inside
the limits.
375 400 425 450 475 500 525 550 575 600 625
Smallest value
inside limits = 425
© 2005 Thomson/South-Western
Largest value
inside limits = 615
Slide 28
Measures of Association
Between Two Variables


Covariance
Correlation Coefficient
© 2005 Thomson/South-Western
Slide 29
Covariance
The covariance is a measure of the linear association
between two variables.
Positive values indicate a positive relationship.
Negative values indicate a negative relationship.
© 2005 Thomson/South-Western
Slide 30
Covariance
The correlation coefficient is computed as follows:
sxy 
s xy
 ( xi - x )( yi - y )
n -1
 ( xi - m x )( yi - m y )

N
© 2005 Thomson/South-Western
for
samples
for
populations
Slide 31
Correlation Coefficient
The coefficient can take on values between -1 and +1.
Values near -1 indicate a strong negative linear
relationship.
Values near +1 indicate a strong positive linear
relationship.
© 2005 Thomson/South-Western
Slide 32
Correlation Coefficient
The correlation coefficient is computed as follows:
rxy 
sxy
sx s y
for
samples
© 2005 Thomson/South-Western
 xy
s xy

s xs y
for
populations
Slide 33
Correlation Coefficient
Correlation is a measure of linear association and not
necessarily causation.
Just because two variables are highly correlated, it
does not mean that one variable is the cause of the
other.
© 2005 Thomson/South-Western
Slide 34
Covariance and Correlation Coefficient
A golfer is interested in investigating
the relationship, if any, between driving
distance and 18-hole score.
Average Driving
Average
Distance (yds.) 18-Hole Score
277.6
69
259.5
71
269.1
70
267.0
70
255.6
71
272.9
69
© 2005 Thomson/South-Western
Slide 35
Covariance and Correlation Coefficient
x
y
277.6
259.5
269.1
267.0
255.6
272.9
69
71
70
70
71
69
( xi - x ) ( yi - y ) ( xi - x )( yi - y )
10.65
-7.45
2.15
0.05
-11.35
5.95
Average 266.95 70.0
Std. Dev. 8.2192 .8944
© 2005 Thomson/South-Western
-1.0
1.0
0
0
1.0
-1.0
-10.65
-7.45
0
0
-11.35
-5.95
Total -35.40
Slide 36
Covariance and Correlation Coefficient

Sample Covariance
sxy

( x - x )( y - y ) -35.40




i
i
n-1
6-1
- 7.08
Sample Correlation Coefficient
rxy 
sxy
sx s y

-7.08
 -.9631
(8.2192)(.8944)
© 2005 Thomson/South-Western
Slide 37
The Weighted Mean and
Working with Grouped Data




Weighted Mean
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data
© 2005 Thomson/South-Western
Slide 38
Weighted Mean
 When the mean is computed by giving each data
value a weight that reflects its importance, it is
referred to as a weighted mean.
 In the computation of a grade point average (GPA),
the weights are the number of credit hours earned for
each grade.
 When data values vary in importance, the analyst
must choose the weight that best reflects the
importance of each value.
© 2005 Thomson/South-Western
Slide 39
Weighted Mean
wx

x
w
i i
i
where:
xi = value of observation i
wi = weight for observation i
© 2005 Thomson/South-Western
Slide 40
Grouped Data
 The weighted mean computation can be used to
obtain approximations of the mean, variance, and
standard deviation for the grouped data.
 To compute the weighted mean, we treat the
midpoint of each class as though it were the mean
of all items in the class.
 We compute a weighted mean of the class midpoints
using the class frequencies as weights.
 Similarly, in computing the variance and standard
deviation, the class frequencies are used as weights.
© 2005 Thomson/South-Western
Slide 41
Mean for Grouped Data

Sample Data
fM

x
i
i
n

Population Data
fM

m
i
i
N
where:
fi = frequency of class i
Mi = midpoint of class i
© 2005 Thomson/South-Western
Slide 42
Sample Mean for Grouped Data
Given below is the previous sample of monthly rents
for 70 efficiency apartments, presented here as grouped
data in the form of a frequency distribution.
Rent ($)
420-439
440-459
460-479
480-499
500-519
520-539
540-559
560-579
580-599
600-619
© 2005 Thomson/South-Western
Frequency
8
17
12
8
7
4
2
4
2
6
Slide 43
Sample Mean for Grouped Data
Rent ($)
420-439
440-459
460-479
480-499
500-519
520-539
540-559
560-579
580-599
600-619
Total
fi
8
17
12
8
7
4
2
4
2
6
70
Mi
429.5
449.5
469.5
489.5
509.5
529.5
549.5
569.5
589.5
609.5
© 2005 Thomson/South-Western
f iMi
3436.0
7641.5
5634.0
3916.0
3566.5
2118.0
1099.0
2278.0
1179.0
3657.0
34525.0
34,525
x
 493.21
70
This approximation
differs by $2.41 from
the actual sample
mean of $490.80.
Slide 44
Variance for Grouped Data

For sample data
2
f
(
M
x
)

i
i
s2 
n -1

For population data
2
f
(
M
m
)

i
i
s2 
N
© 2005 Thomson/South-Western
Slide 45
Sample Variance for Grouped Data
Rent ($)
420-439
440-459
460-479
480-499
500-519
520-539
540-559
560-579
580-599
600-619
Total
fi
8
17
12
8
7
4
2
4
2
6
70
Mi
429.5
449.5
469.5
489.5
509.5
529.5
549.5
569.5
589.5
609.5
Mi - x
-63.7
-43.7
-23.7
-3.7
16.3
36.3
56.3
76.3
96.3
116.3
(M i - x )2 f i (M i - x )2
4058.96 32471.71
1910.56 32479.59
562.16
6745.97
13.76
110.11
265.36
1857.55
1316.96
5267.86
3168.56
6337.13
5820.16 23280.66
9271.76 18543.53
13523.36 81140.18
208234.29
continued
© 2005 Thomson/South-Western
Slide 46
Sample Variance for Grouped Data

Sample Variance
s2 = 208,234.29/(70 – 1) = 3,017.89

Sample Standard Deviation
s  3,017.89  54.94
This approximation differs by only $.20
from the actual standard deviation of $54.74.
© 2005 Thomson/South-Western
Slide 47
End of Chapter 3, Part B
© 2005 Thomson/South-Western
Slide 48