Download Ch 3 - faculty.fairfield.edu

Slides by
John
Loucks
St. Edward’s
University
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
1
Chapter 3, Part A
Descriptive Statistics: Numerical Measures


Measures of Location
Measures of Variability
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
2
Measures of Location

Mean
Median
Mode

Weighted Mean

Geometric Mean

Percentiles
Quartiles



If the measures are computed
for data from a sample, they
are called sample statistics.
If the measures are computed for
data from a population, they are
called population parameters.
A sample statistic is referred to as
the point estimator of the
corresponding population parameter.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
3
Mean




Perhaps the most important measure of location is
the mean.
The mean provides a measure of central location.
The mean of a data set is the average of all the data
values.
The sample mean 𝑥 is the point estimator of the
population mean m.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
4
Sample Mean 𝑥
Sum of the values
of the n observations
𝑥𝑖
𝑥=
𝑛
Number of
observations
in the sample
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
5
Population Mean m
Sum of the values
of the N observations
𝑥𝑖
𝜇=
𝑁
Number of
observations in
the population
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
6
Sample Mean 𝑥
 Example: Apartment Rents
Seventy efficiency apartments were randomly
sampled in a college town. The monthly rents for
these apartments are listed below.
545
540
565
550
700
670
610
715
540
550
570
585
615
675
530
540
625
590
680
550
590
690
625
550
572
570
545
535
535
525
550
575
590
625
700
700
545
560
575
600
635
535
560
675
535
600
649
575
545
700
545
560
580
600
650
535
540
550
565
670
600
580
530
715
550
580
565
580
610
540
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
7
Sample Mean 𝑥
 Example: Apartment Rents
𝑥=
545
540
565
550
700
670
610
715
540
550
570
585
615
675
530
540
625
590
680
550
590
𝑥𝑖
𝑛
690
625
550
572
570
545
535
=
41,356
70
535
525
550
575
590
625
700
= 590.80
700
545
560
575
600
635
535
560
675
535
600
649
575
545
700
545
560
580
600
650
535
540
550
565
670
600
580
530
715
550
580
565
580
610
540
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
8
Median
 The median of a data set is the value in the middle
when the data items are arranged in ascending order.
 Whenever a data set has extreme values, the median
is the preferred measure of central location.
 The median is the measure of location most often
reported for annual income and property value data.
 A few extremely large incomes or property values
can inflate the mean.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
9
Median
 For an odd number of observations:
26 18 27 12 14 27 19
7 observations
12 14 18 19 26 27 27
in ascending order
the median is the middle value.
Median = 19
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
10
Median
 For an even number of observations:
26 18 27 12 14 27 30 19
8 observations
12 14 18 19 26 27 27 30
in ascending order
the median is the average of the middle two values.
Median = (19 + 26)/2 = 22.5
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
11
Median
 Example: Apartment Rents
Averaging the 35th and 36th data values:
Median = (575 + 575)/2 = 575
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
12
Trimmed Mean
 Another measure, sometimes used when extreme
values are present, is the trimmed mean.
 It is obtained by deleting a percentage of the
smallest and largest values from a data set and then
computing the mean of the remaining values.
 For example, the 5% trimmed mean is obtained by
removing the smallest 5% and the largest 5% of the
data values and then computing the mean of the
remaining values.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
13
Mode
 The mode of a data set is the value that occurs with
greatest frequency.
 The greatest frequency can occur at two or more
different values.
 If the data have exactly two modes, the data are
bimodal.
 If the data have more than two modes, the data are
multimodal.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
14
Mode
 Example: Apartment Rents
550 occurred most frequently (7 times)
Mode = 550
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
15
Excel’s Mean, Median, and Mode Functions
 Excel’s Mean function
=AVERAGE(data cell range)
 Excel’s Median function
=MEDIAN(data cell range)
 Excel’s Mode function
=MODE.SNGL(data cell range)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
16
Using Excel to Compute
the Mean, Median, and Mode
 Excel Formula Worksheet
1
2
3
4
5
6
A
Apartment
1
2
3
4
5
B
Monthly
Rent ($)
545
715
530
690
535
C
D
E
Mean =AVERAGE(B2:B71)
Median =MEDIAN(B2:B71)
Mode =MODE.SNGL(B2:B71)
Note: Rows 7-71 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
17
Using Excel to Compute
the Mean, Median, and Mode
 Excel Value Worksheet
1
2
3
4
5
6
A
Apartment
1
2
3
4
5
B
Monthly
Rent ($)
545
715
530
690
535
C
D
E
Mean
Median
Mode
590.80
575.00
550.00
Note: Rows 7-71 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
18
Weighted Mean

In some instances the mean is computed by giving
each observation a weight that reflects its relative
importance.

The choice of weights depends on the application.

The weights might be the number of credit hours
earned for each grade, as in GPA.
In other weighted mean computations, quantities
such as pounds, dollars, or volume are frequently
used.

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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
19
Weighted Mean
If data is from
a population,
m replaces 𝑥.
𝑥=
𝑤𝑖 𝑥𝑖
𝑤𝑖
Numerator:
sum of the weighted
data values
Denominator:
sum of the
weights
where:
xi = value of observation i
wi = weight for observation i
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
20
Weighted Mean
 Example: Construction Wages
Ron Butler, a home builder, is looking over the
expenses he incurred for a house he just built. For the
purpose of pricing future projects, he would like to
know the average wage ($/hour) he paid the workers
he employed. Listed below are the categories of
worker he employed, along with their respective wage
and total hours worked.
Worker
Carpenter
Electrician
Laborer
Painter
Plumber
Wage ($/hr) Total Hours
21.60
520
28.72
230
11.80
410
19.75
270
24.16
160
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
21
Weighted Mean
 Example: Construction Wages
xi
21.60
28.72
11.80
19.75
24.16
Worker
Carpenter
Electrician
Laborer
Painter
Plumber
𝑥=
𝑤𝑖 𝑥𝑖
𝑤𝑖
=
31,873.7
1,590
wi
520
230
410
270
160
1590
wi x i
11232.0
6605.6
4838.0
5332.5
3865.6
31873.7
= 20.0464 = $20.05
FYI, equally-weighted (simple) mean = $21.21
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
22
Trimmed Mean
 Another measure, sometimes used when extreme
values are present, is the trimmed mean.
 It is obtained by deleting a percentage of the
smallest and largest values from a data set and then
computing the mean of the remaining values.
 For example, the 5% trimmed mean is obtained by
removing the smallest 5% and the largest 5% of the
data values and then computing the mean of the
remaining values.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
23
Geometric Mean
 The geometric mean is calculated by finding the nth
root of the product of n values.
 It is often used in analyzing growth rates in
financial data (where using the arithmetic mean
will provide misleading results).
 It should be applied anytime you want to determine
the mean rate of change over several successive
periods (be it years, quarters, weeks, . . .).
 Other common applications include: changes in
populations of species, crop yields, pollution levels,
and birth and death rates.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
24
Geometric Mean
𝑥𝑔 =
𝑛
𝑥1 𝑥2 … (𝑥𝑛 )
= [(x1)(x2)…(xn)]1/n
 Excel’s geometric mean function is:
=GEOMEAN(data cell range)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
25
Geometric Mean
 Example: Rate of Return
Period
1
2
3
4
5
𝑥𝑔 =
Return (%)
-6.0
-8.0
-4.0
2.0
5.4
5
Growth Factor
0.940
0.920
0.960
1.020
1.054
.94 . 92)(.96)(1.02)(1.054)
= [.89254]1/5 = .97752
Average growth rate per period
is (.97752 - 1) (100) = -2.248%
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
26
Percentiles
 A percentile provides information about how the
data are spread over the interval from the smallest
value to the largest value.
 Admission test scores for colleges and universities
are frequently reported in terms of percentiles.

The pth percentile of a data set is a value such that at
least p percent of the items take on this value or less
and at least (100 - p) percent of the items take on this
value or more.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
27
Percentiles
Arrange the data in ascending order.
Compute Lp, the location of the pth percentile.
Lp = (p/100)(n + 1)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
28
80th Percentile
 Example: Apartment Rents
Lp = (p/100)(n + 1) = (80/100)(70 + 1) = 56.8
(the 56th value plus .8 times the
difference between the 57th and 56th values)
80th Percentile = 635 + .8(649 – 635) = 646.2
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
29
80th Percentile
 Example: Apartment Rents
“At least 80% of the
“At least 20% of the
items take on a
items take on a
value of 646.2 or less.”
value of 646.2 or more.”
56/70 = .8 or 80%
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
14/70 = .2 or 20%
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
30
Using Excel to Compute Percentiles
Excel’s percentile function is:
PERCENTILE.EXC(data range, p/100)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
31
Using Excel to Compute the 80th Percentile
 Excel Formula Worksheet
1
2
3
4
5
6
80th percentile
A
B
C
D
Apart- Monthly
ment Rent ($)
80th Percentile
1
545
=PERCENTILE.EXC(B2:B71,.8)
2
715
3
530
It is not necessary
4
690
5
535
to put the data
Note: Rows 7-71 are not shown.
in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
32
Using Excel to Compute the 80th Percentile
 Excel Value Worksheet
1
2
3
4
5
6
A
B
C
Apart- Monthly
ment Rent ($)
1
545
2
715
3
530
4
690
5
535
D
80th Percentile
646.2
Note: Rows 7-71 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
33
Quartiles
 Quartiles are specific percentiles.
 First Quartile = 25th Percentile
 Second Quartile = 50th Percentile = Median
 Third Quartile = 75th Percentile
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
34
Third Quartile (75th Percentile)
 Example: Apartment Rents
Lp = (p/100)(n + 1) = (75/100)(70 + 1) = 53.25
(the 53rd value plus .25 times the
difference between the 54th and 53rd values)
Third quartile = 625 + .25(625 – 625) = 625
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
35
Using Excel to Compute Quartiles
 Excel’s quartile function is:
QUARTILE.EXC (data range, quartile number)
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
36
Using Excel to Compute the Third Quartile
 Excel Formula Worksheet
1
2
3
4
5
6
A
B
Apart- Monthly
ment Rent ($)
1
545
2
715
3
530
4
690
5
535
C
3rd quartile
D
Third Quartile
=QUARTILE.EXC(B2:B71,3)
It is not necessary
to put the data
in ascending order.
Note: Rows 7-71 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
37
Using Excel to Compute the Third Quartile
 Excel Value Worksheet
1
2
3
4
5
6
A
B
Apart- Monthly
ment Rent ($)
1
545
2
715
3
530
4
690
5
535
C
D
Third Quartile
625
Note: Rows 7-71 are not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
38
Measures of Variability
 It is often desirable to consider measures of variability
(dispersion), as well as measures of location.
 For example, in choosing supplier A or supplier B we
might consider not only the average delivery time for
each, but also the variability in delivery time for each.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
39
Measures of Variability
 Range
 Interquartile Range
 Variance
 Standard Deviation
 Coefficient of Variation
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
40
Range
 The range of a data set is the difference between the
largest and smallest data values.
Range = Largest value – Smallest value
 It is the simplest measure of variability.
 It is very sensitive to the smallest and largest data
values.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
41
Range
 Example: Apartment Rents
Range = largest value - smallest value
Range = 715 - 525 = 190
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
42
Interquartile Range
 The interquartile range of a data set is the difference
between the third quartile and the first quartile.
 It is the range for the middle 50% of the data.
 It overcomes the sensitivity to extreme data values.
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43
Interquartile Range (IQR)
 Example: Apartment Rents
3rd Quartile (Q3) = 625
1st Quartile (Q1) = 545
IQR = Q3 - Q1 = 625 - 545 = 80
525
540
550
565
580
610
675
530
540
550
570
585
615
675
530
540
550
570
590
625
680
535
545
550
572
590
625
690
535
545
550
575
590
625
700
535
545
560
575
600
635
700
535
545
560
575
600
649
700
535
545
560
580
600
650
700
540
550
565
580
600
670
715
540
550
565
580
610
670
715
Note: Data is in ascending order.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
44
Variance
The variance is a measure of variability that utilizes
all the data.
It is based on the difference between the value of
each observation (xi) and the mean (𝑥 for a sample,
m for a population).
The variance is useful in comparing the variability
of two or more variables.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
45
Variance
The variance is the average of the squared
differences between each data value and the mean.
The variance is computed as follows:
𝑠2 =
𝑥𝑖 − 𝑥
𝑛−1
for a
sample
2
𝜎2 =
𝑥𝑖 − 𝜇
𝑁
2
for a
population
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
46
Standard Deviation
The standard deviation of a data set is the positive
square root of the variance.
It is measured in the same units as the data, making
it more easily interpreted than the variance.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
47
Standard Deviation
The standard deviation is computed as follows:
s = 𝑠2
s= s2
for a
sample
for a
population
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
48
Excel’s Variance and
Standard Deviation Functions
 Excel’s Sample Variance function
=VAR.S(data cell range)
 Excel’s Sample Standard Deviation function
=STDEV.S(data cell range)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
49
Coefficient of Variation
The coefficient of variation indicates how large the
standard deviation is in relation to the mean.
The coefficient of variation is computed as follows:
𝑠
𝑥
x 100 %
for a
sample
𝜎
𝜇
x 100 %
for a
population
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
50
Sample Variance, Standard Deviation,
And Coefficient of Variation
 Example: Apartment Rents
•
Variance
𝑥𝑖 −𝑥
𝑛−1
s2 =
•
2
= 2,996.16
Standard Deviation
s = 𝑠2 =
2,996.16 = 54.74
Standard
deviation is
about 9%
of the mean
• Coefficient of Variation
𝑠
𝑥
x 100 % =
54.74
590.80
x 100 % = 9.27%
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
51
Using Excel to Compute the Sample Variance,
Standard Deviation, and Coefficient of Variation
 Formula Worksheet
1
2
3
4
5
6
7
A
B
C
D
E
Apart- Monthly
ment Rent ($)
1
545
Mean =AVERAGE(B2:B71)
2
715
Median =MEDIAN(B2:B71)
3
530
Mode =MODE.SNGL(B2:B71)
4
690
Variance =VAR.S(B2:B71)
5
535
Std. Dev. =STDEV.S(B2:B71)
6
700
C.V. =E6/E2*100
Note: Rows 8-71 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
52
Using Excel to Compute the Sample Variance,
Standard Deviation, and Coefficient of Variation
 Value Worksheet
1
2
3
4
5
6
7
A
B
C
D
Apart- Monthly
ment Rent ($)
1
545
Mean
2
715
Median
3
530
Mode
4
690
Variance
5
535
Std. Dev.
6
700
C.V.
E
590.80
575.00
550.00
2996.16
54.74
9.27
Note: Rows 8-71 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
53
End of Chapter 3, Part A
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
54