### Chapter 3, Part A - Salisbury University

```Slides by
John
Loucks
St. Edward’s
University
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 3, Part A
Descriptive Statistics: Numerical Measures


Measures of Location
Measures of Variability
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Measures of Location

Mean

Median
Mode



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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Sample Mean x
x
x
Sum of the values
of the n observations
i
n
Number of
observations
in the sample
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Population Mean m
m
x
Sum of the values
of the N observations
i
N
Number of
observations in
the population
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Sample Mean
 Example: Apartment Rents
Seventy efficiency apartments were randomly
sampled in a small college town. The monthly rent
prices for these apartments are listed below.
445
440
465
450
600
570
510
615
440
450
470
485
515
575
430
440
525
490
580
450
490
590
525
450
472
470
445
435
435
425
450
475
490
525
600
600
445
460
475
500
535
435
460
575
435
500
549
475
445
600
445
460
480
500
550
435
440
450
465
570
500
480
430
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
615
450
480
465
480
510
440
Slide 7
Sample Mean
 Example: Apartment Rents
x

x
34, 356

 490.80
n
70
445
440
465
450
600
570
510
615
440
450
470
485
515
575
430
440
525
490
580
450
490
590
525
450
472
470
445
435
i
435
425
450
475
490
525
600
600
445
460
475
500
535
435
460
575
435
500
549
475
445
600
445
460
480
500
550
435
440
450
465
570
500
480
430
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
615
450
480
465
480
510
440
Slide 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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Median
 For an odd number of observations:
26 18 27 12 14 27 19
12
14
18
19
26
27 27
7 observations
in ascending order
the median is the middle value.
Median = 19
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Median
 For an even number of observations:
26 18 27 12 14 27 30 19
12
14
18
19
26
27 27
30
8 observations
in ascending order
the median is the average of the middle two values.
Median = (19 + 26)/2 = 22.5
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 11
Median
 Example: Apartment Rents
Averaging the 35th and 36th data values:
Median = (475 + 475)/2 = 475
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 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.
 Caution: If the data are bimodal or multimodal,
Excel’s MODE function will incorrectly identify a
single mode.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 14
Mode
 Example: Apartment Rents
450 occurred most frequently (7 times)
Mode = 450
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Percentiles
Arrange the data in ascending order.
Compute index i, the position of the pth percentile.
i = (p/100)n
If i is not an integer, round up. The pth percentile
is the value in the ith position.
If i is an integer, the pth percentile is the average
of the values in positions i and i+1.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 17
80th Percentile
 Example: Apartment Rents
i = (p/100)n = (80/100)70 = 56
Averaging the 56th and 57th data values:
80th Percentile = (535 + 549)/2 = 542
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
80th Percentile
 Example: Apartment Rents
“At least 80% of the
items take on a
value of 542 or less.”
“At least 20% of the
items take on a
value of 542 or more.”
56/70 = .8 or 80%
14/70 = .2 or 20%
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
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
440
450
465
480
510
570
615
Slide 19
Quartiles
 Quartiles are specific percentiles.
 First Quartile = 25th Percentile
 Second Quartile = 50th Percentile = Median
 Third Quartile = 75th Percentile
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Third Quartile
 Example: Apartment Rents
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
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
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Measures of Variability
 Range
 Interquartile Range
 Variance
 Standard Deviation
 Coefficient of Variation
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Range
 The range of a data set is the difference between the
largest and smallest data values.
 It is the simplest measure of variability.
 It is very sensitive to the smallest and largest data
values.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Range
 Example: Apartment Rents
Range = largest value - smallest value
Range = 615 - 425 = 190
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Interquartile Range
 Example: Apartment Rents
3rd Quartile (Q3) = 525
1st Quartile (Q1) = 445
Interquartile Range = Q3 - Q1 = 525 - 445 = 80
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
Note: Data is in ascending order.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Variance
The variance is the average of the squared
differences between each data value and the mean.
The variance is computed as follows:
2  ( xi  x )
s 
n 1
for a
sample
2
 ( xi  m )
 
N
2
2
for a
population
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
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.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Standard Deviation
The standard deviation is computed as follows:
s  s2
  2
for a
sample
for a
population
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
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

 100  %
x



 100  %
m

for a
sample
for a
population
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Sample Variance, Standard Deviation,
And Coefficient of Variation
 Example: Apartment Rents
• Variance
s2 
2
(
x

x
)
 i
n1

2, 996.16
• Standard Deviation
s  s 2  2996.16 
• Coefficient of Variation
54.74
the standard
deviation is
about 11%
of the mean
s

 54.74

  100  %  
 100  %  11.15%
x

 490.80

© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
End of Chapter 3, Part A
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
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