Download weighted mean

Chapter 3
Descriptive Statistics: 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
x
Measures of Relative Location
and Detecting Outliers




z-Scores
Chebyshev’s Theorem
Empirical Rule
Detecting Outliers
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.
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
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.
• At least 75% of the items must be within
z = 2 standard deviations of the mean.
• At least 89% of the items must be within
z = 3 standard deviations of the mean.
• At least 94% of the items must be within
z = 4 standard deviations of the mean.
Example: Apartment Rents

Chebyshev’s Theorem
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
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
Empirical Rule
For data having a bell-shaped distribution:
• Approximately 68% of the data values will be
within one standard deviation of the mean.
Empirical Rule
For data having a bell-shaped distribution:
• Approximately 95% of the data values will be
within two standard deviations of the mean.
Empirical Rule
For data having a bell-shaped distribution:
• Almost all (99.7%) of the items will be
within three standard deviations of the mean.
Example: Apartment Rents

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
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
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
Exploratory Data Analysis


Five-Number Summary
Box Plot
Five-Number Summary





Smallest Value
First Quartile
Median
Third Quartile
Largest Value
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
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.
Limits are located (not drawn) using the interquartile
range (IQR).
• The lower limit is located 1.5(IQR) below Q1.
• The upper limit is located 1.5(IQR) above Q3.
• Data outside these limits are considered outliers.
… continued
Box Plot (Continued)


Whiskers (dashed lines) are drawn from the ends of
the box to the smallest and largest data values inside
the limits.
The locations of each outlier is shown with the
symbol * .
Example: Apartment Rents

Box Plot
Lower Limit: Q1 - 1.5(IQR) = 450 - 1.5(75) = 337.5
Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(75) = 637.5
There are no outliers.
37
5
40
0
42
5
45
0
47
5
50
0
52
5
550
575 600
625
Measures of Association
Between Two Variables


Covariance
Correlation Coefficient
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.
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
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

 x y
 xy
.
The Weighted Mean and
Working with Grouped Data




Weighted Mean
Mean for Grouped Data
Variance for Grouped Data
Standard Deviation for Grouped Data
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.
Weighted Mean
x =  wi xi
 wi
where:
xi = value of observation i
wi = weight for observation i
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.
Mean for Grouped Data

Sample Data
fM

x
f
i
fM


i
i
i

Population Data
i
N
where:
fi = frequency of class i
Mi = midpoint of class i
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 ($) Frequency
420-439
8
440-459
17
460-479
12
480-499
8
500-519
7
520-539
4
540-559
2
560-579
4
580-599
2
600-619
6
Example: Apartment Rents

Mean for Grouped Data
Rent ($)
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 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
 493. 21
70
This approximation
differs by $2.41 from
the actual sample
mean of $490.80.
x
Variance for Grouped Data

Sample Data
2
f
(
M

x
)

i
i
s2 
n 1

Population Data
2
f
(
M


)

i
i
2 
N
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.