Download weighted mean

Anderson
u
Sweeney
u
Williams
CONTEMPORARY
BUSINESS
STATISTICS
WITH MICROSOFT EXCEL
u Slides Prepared by JOHN LOUCKS u
© 2001 South-Western /Thomson Learning
Slide 1
Chapter 3
Descriptive Statistics II:
Numerical Methods - Part B




Measures of Relative
Location and Detecting
Outliers
Exploratory Data Analysis
Measures of Association
between Two Variables
The Weighted Mean and
Working with Grouped
Data
Slide 2
Measures of Relative Location
and Detecting Outliers




z-Scores
Chebyshev’s Theorem
The Empirical Rule
Detecting Outliers
Slide 3
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



A data value less than the sample mean will have a zscore 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 zscore of zero.
Slide 4
Example: Apartment Rents

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
-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 5
Chebyshev’s Theorem
At least (1 - 1/k2) of the items in any data set will be
within k standard deviations of the mean, where k is
any value greater than 1.
• At least 75% of the items must be within k = 2
standard deviations of the mean.
• At least 89% of the items must be within k = 3
standard deviations of the mean.
• At least 94% of the items must be within k = 4
standard deviations of the mean.
Slide 6
Example: Apartment Rents

Chebyshev’s Theorem
Let k = 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 - k(s) = 490.80 - 1.5(54.74) = 409
and
x + k(s) = 490.80 + 1.5(54.74) = 573
Slide 7
Example: Apartment Rents

Chebyshev’s Theorem (continued)
Actually, 86% of the rent values
are between 409 and 573.
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
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 8
The Empirical Rule
For data having a bell-shaped distribution:
• Approximately 68% of the data values will be
within one standard deviation of the mean.
• Approximately 95% of the data values will be
within two standard deviations of the mean.
• Almost all of the items (99.7%) will be
within three standard deviations of the mean.
Slide 9
Example: Apartment Rents

The Empirical Rule
Within +/- 1s
Within +/- 2s
Within +/- 3s
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
Interval
436.06 to 545.54
381.32 to 600.28
326.58 to 655.02
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
% in Interval
48/70 = 69%
68/70 = 97%
70/70 = 100%
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 10
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.
It might be a data value that was incorrectly included
in the data set.
It might be a correctly recorded data value that
belongs in the data set !
Slide 11
Example: Apartment Rents

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
-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 12
Exploratory Data Analysis


Five-Number Summary
Box Plot
Slide 13
Five-Number Summary





Smallest Value
First Quartile
Median
Third Quartile
Largest Value
Slide 14
Example: Apartment Rents

Five-Number Summary
Lowest Value = 425
First Quartile = 450
Median = 475
Third Quartile = 525
Largest Value = 615
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
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 15
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.
Fences are located (not drawn) using the interquartile
range (IQR).
• The inner fences are located 1.5(IQR) below Q1
and 1.5(IQR) above Q3.
• The outer fences are located 3(IQR) below Q1 and
3(IQR) above Q3.
… continued
Slide 16
Box Plot (Continued)



Whiskers (dashed lines) are drawn from the ends of
the box to the smallest and largest data values inside
the inner fences.
The locations of mild outliers are shown with the
symbol * .
The locations of extreme outliers are shown with the
symbol o .
Slide 17
Example: Apartment Rents

Box Plot
Inner Fences: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5
Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
Outer Fences: Q1 - 3(IQR) = 450 - 3(75) = 225
Q3 + 3(IQR) = 525 + 3(75) = 750
There are no mild or extreme outliers.
37
5
40
0
42
5
45
0
47
5
50
0
52
5
550
575 600
625
Slide 18
Measures of Association
between Two Variables


Covariance
Correlation Coefficient
Slide 19
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.
Slide 20
Covariance

If the data sets are samples, the covariance is denoted
by sxy.
 ( xi  x )( yi  y )
sxy 
n 1

If the data sets are populations, the covariance is
denoted by  xy .
 xy
 ( xi   x )( yi   y )

N
Slide 21
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.
If the data sets are samples, the coefficient is rxy.
rxy 

sxy
sx s y
If the data sets are populations, the coefficient is  xy .
 xy
 xy

 x y
Slide 22
Using Excel to Compute the
Covariance and Correlation Coefficient

1
2
3
4
5
6
7
8
Formula Worksheet
A
Average
Drive
277.6
259.5
269.1
267.0
255.6
272.9
B
18-Hole
Score
69
71
70
70
71
69
C
D
E
Pop. Covariance =COVAR(A2:A7,B2:B7)
Samp. Correlation =CORREL(A2:A7,B2:B7)
Slide 23
Using Excel to Compute the
Covariance and Correlation Coefficient

1
2
3
4
5
6
7
8
Value Worksheet
A
Average
Drive
277.6
259.5
269.1
267.0
255.6
272.9
B
18-Hole
Score
69
71
70
70
71
69
C
D
Pop. Covariance
Samp. Correlation
E
-5.9
-0.963073682
Slide 24
The Weighted Mean and
Working with Grouped Data




The Weighted Mean
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data
Slide 25
The 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.

Slide 26
The Weighted Mean
xwt =  wi xi
 wi
where:
xi = value of observation i
wi = weight for observation i
Slide 27
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.
Slide 28
Mean for Grouped Data

Sample Data
fM

x
f
i
fM


i
i
i

Population Data
i
where:
N
fi = frequency of class i
Mi = midpoint of class i
Slide 29
Example: Apartment Rents
Given below is the previous sample of monthly rents
for one-bedroom apartments presented here as grouped
data in the form of a frequency distribution.
Rent ($) Frequency
420-439
8
440-459
17
460-479
12
480-499
8
500-519
7
520-539
4
540-559
2
560-579
4
580-599
2
600-619
6
Slide 30
Example: Apartment Rents

Mean for Grouped Data
Rent ($)
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
f iM i
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 31
Variance for Grouped Data

Sample Data

Population Data
2
f
(
M

x
)

i
i
s2 
n 1
2
f
(
M


)
 i
i
2 
N
Slide 32
Example: Apartment Rents

Variance for Grouped Data
s2  3, 017.89

Standard Deviation for Grouped Data
s  3, 017.89  54. 94
This approximation differs by only $.20
from the actual standard deviation of $54.74.
Slide 33
End of Chapter 3, Part B
Slide 34