© 2003 Thomson/South-Western
Slide 1
Chapter 3
Descriptive Statistics: Numerical Methods
Part A


Measures of Location
Measures of Variability
x
© 2003 Thomson/South-Western
Slide 2
Measures of Location





Mean
Median
Mode
Percentiles
Quartiles
© 2003 Thomson/South-Western
Slide 3
Example: Apartment Rents
Given below is a sample of monthly rent values ($)
for one-bedroom apartments. The data is a sample of 70
apartments in a particular city. The data are presented
in ascending order.
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
© 2003 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 4
Mean


The mean of a data set is the average of all the data
values.
If the data are from a sample, the mean is denoted by
x.
 xi
x
n

If the data are from a population, the mean is
denoted by m (mu).
 xi

N
© 2003 Thomson/South-Western
Slide 5
Example: Apartment Rents

Mean
 xi 34, 356
x

 490.80
n
70
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
© 2003 Thomson/South-Western
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 6
Median


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.
© 2003 Thomson/South-Western
Slide 7
Median



The median of a data set is the value in the middle
when the data items are arranged in ascending order.
For an odd number of observations, the median is the
middle value.
For an even number of observations, the median is
the average of the two middle values.
© 2003 Thomson/South-Western
Slide 8
Example: Apartment Rents

Median
Median = 50th percentile
i = (p/100)n = (50/100)70 = 35.5
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
© 2003 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 9
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.
© 2003 Thomson/South-Western
Slide 10
Example: Apartment Rents

Mode
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
© 2003 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 11
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.
© 2003 Thomson/South-Western
Slide 12
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.
• 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.
© 2003 Thomson/South-Western
Slide 13
Example: Apartment Rents

90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 585
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
© 2003 Thomson/South-Western
435
445
460
475
500
535
600
435
445
460
475
500
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600
435
445
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500
550
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450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 14
Quartiles




Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile
© 2003 Thomson/South-Western
Slide 15
Example: Apartment Rents

Third Quartile
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
© 2003 Thomson/South-Western
435
445
460
475
500
535
600
435
445
460
475
500
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600
435
445
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500
550
600
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450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 16
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.
© 2003 Thomson/South-Western
Slide 17
Measures of Variability





Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation
© 2003 Thomson/South-Western
Slide 18
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.
© 2003 Thomson/South-Western
Slide 19
Example: Apartment Rents

Range
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
© 2003 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 20
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.
© 2003 Thomson/South-Western
Slide 21
Example: Apartment Rents

Interquartile Range
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
© 2003 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 22
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, 
for a population).
© 2003 Thomson/South-Western
Slide 23
Variance


The variance is the average of the squared differences
between each data value and the mean.
If the data set is a sample, the variance is denoted by
s2 .
s2 

2
(
x

x
)
 i
n 1
If the data set is a population, the variance is denoted
by  2.
2
(
x


)

i
2 
N
© 2003 Thomson/South-Western
Slide 24
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 comparable, than the variance, to the
mean.
If the data set is a sample, the standard deviation is
denoted s.
s  s2

If the data set is a population, the standard deviation
is denoted  (sigma).

© 2003 Thomson/South-Western
2
Slide 25
Coefficient of Variation


The coefficient of variation indicates how large the
standard deviation is in relation to the mean.
If the data set is a sample, the coefficient of variation
is computed as follows:
s
(100)
x

If the data set is a population, the coefficient of
variation is computed as follows:

(100)

© 2003 Thomson/South-Western
Slide 26
Example: Apartment Rents

Variance

s 
n 1
2

( xi  x ) 2
 2 , 996.16
Standard Deviation
s  s2  2996. 47  54. 74

Coefficient of Variation
s
54. 74
 100 
 100  11.15
x
490.80
© 2003 Thomson/South-Western
Slide 27
End of Chapter 3, Part A
© 2003 Thomson/South-Western
Slide 28