Download 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 A
Measures of Location
 Measures of Variability

Slide 2
Measures of Location





Mean
Median
Mode
Percentiles
Quartiles
x
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
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 (mu).
 xi

N
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
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 of a data set is the value in the middle
when the data items are arranged in ascending order.
If there is an odd number of items, the median is the
value of the middle item.
If there is an even number of items, the median is the
average of the values for the middle two items.
Slide 7
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
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
Mode

The mode of a data set is the value that occurs with
greatest frequency.
Slide 9
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
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 10
Using Excel to Compute
the Mean, Median, and Mode

Formula Worksheet
1
2
3
4
5
6
A
Apartment
1
2
3
4
5
B
C
D
E
Monthly
Rent ($)
525
Mean =AVERAGE(B2:B71)
440
Median =MEDIAN(B2:B71)
450
Mode =MODE(B2:B71)
615
480
Note: Rows 7-71 are not shown.
Slide 11
Using Excel to Compute
the Mean, Median, and Mode

Value Worksheet
1
2
3
4
5
6
A
Apartment
1
2
3
4
5
B
C
D
Monthly
Rent ($)
525
Mean
440
Median
450
Mode
615
480
E
490.80
475.00
450.00
Note: Rows 7-71 are not shown.
Slide 12
Percentiles

The pth percentile of a data set is a value such that at
least p percent of the items take on this value or less
and at least (100 - p) percent of the items take on this
value or more.
• Arrange the data in ascending order.
• Compute index i, the position of the pth percentile.
i = (p/100)n
• If i is not an integer, round up. The pth percentile is
the value in the ith position.
• If i is an integer, the pth percentile is the average of
the values in positions i and i+1.
Slide 13
Example: Apartment Rents

90th Percentile
i = (p/100)n = (90/100)70 = 63
Averaging the 63rd and 64th data values:
90th Percentile = (580 + 590)/2 = 585
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 14
Quartiles




Quartiles are specific percentiles
First Quartile = 25th Percentile
Second Quartile = 50th Percentile = Median
Third Quartile = 75th Percentile
Slide 15
Example: Apartment Rents

Third Quartile
Third quartile = 75th percentile
i = (p/100)n = (75/100)70 = 52.5 = 53
Third quartile = 525
425
440
450
465
480
510
575
430
440
450
470
485
515
575
430
440
450
470
490
525
580
435
445
450
472
490
525
590
435
445
450
475
490
525
600
435
445
460
475
500
535
600
435
445
460
475
500
549
600
435
445
460
480
500
550
600
440
450
465
480
500
570
615
440
450
465
480
510
570
615
Slide 16
Using Excel to Compute
Percentiles and Quartiles

1
2
3
4
5
6
Unsorted Monthly Rent ($)
A
B
C
D
Apart- Monthly
ment Rent ($)
1
525
2
440
3
450
4
615
5
480
Note: Rows 7-71 are not shown.
E
F
Slide 17
Using Excel to Compute
Percentiles and Quartiles

Sorting Data
Step 1 Select any cell containing data in column B
Step 2 Select the Data pull-down menu
Step 3 Choose the Sort option
Step 4 When the Sort dialog box appears:
In the Sort by box, make sure that
Monthly Rent ($) appears and that
Ascending is selected
In the My list has box, make sure that
Header row is selected
Click OK
Slide 18
Using Excel to Compute
Percentiles and Quartiles

1
2
3
4
5
6
Sorted Monthly Rent ($)
A
B
C
D
Apart- Monthly
ment Rent ($)
1
425
2
430
3
430
4
435
5
435
Note: Rows 7-71 are not shown.
E
F
Slide 19
Using Excel to Compute
Percentiles and Quartiles

1
2
3
4
5
6
Formula Worksheet for 90th Percentile’s Index
A
B
C
D
E
F
Apart- Monthly
Number of
ment Rent ($)
Observations Percentile Index i
1
425
70
90
=(E2/100)*D2
2
430
3
430
4
435
5
435
Note: Rows 7-71 are not shown.
Slide 20
Using Excel to Compute
Percentiles and Quartiles

1
2
3
4
5
6
Value Worksheet for 90th Percentile’s Index
A
B
C
D
E
Apart- Monthly
Number of
ment Rent ($)
Observations Percentile
1
425
70
90
2
430
3
430
4
435
5
435
Note: Rows 7-71 are not shown.
F
Index i
63.00
Slide 21
Using Excel to Compute
Percentiles and Quartiles

1
2
3
4
5
6
Value Worksheet for 3rd Quartile’s Index
A
B
C
D
E
Apart- Monthly
Number of
ment Rent ($)
Observations Percentile
1
425
70
75
2
430
3
430
4
435
5
435
Note: Rows 7-71 are not shown.
F
Index i
52.50
Slide 22
Measures of Variability





Range
Interquartile Range
Variance
Standard Deviation
Coefficient of Variation
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.
Slide 24
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
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 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.
Slide 26
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
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 27
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.
2
(
x

x
)

i
s2 
n 1
If the data set is a population, the variance is denoted
by  2.
2
(
x


)

i
2 
N
Slide 28
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).

2
Slide 29
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)

Slide 30
Example: Apartment Rents


Variance
s2  
( xi  x ) 2
n 1
 2 , 996.16
Standard Deviation
s  s2  2996. 47  54. 74

Coefficient of Variation
s
54. 74
 100 
 100  11.15
x
490.80
Slide 31
Using Excel to Compute the
Sample Variance and Standard Deviation

Formula Worksheet
1
2
3
4
5
6
7
A
B
C
D
E
Apart- Monthly
ment Rent ($)
1
525
Mean =AVERAGE(B2:B71)
2
440
Median =MEDIAN(B2:B71)
3
450
Mode =MODE(B2:B71)
4
615
Variance =VAR(B2:B71)
5
480
Std. Dev. =STDEV(B2:B71)
6
510
Note: Rows 8-71 are not shown.
Slide 32
Using Excel to Compute the
Sample Variance and Standard Deviation

Value Worksheet
1
2
3
4
5
6
7
A
B
C
D
Apart- Monthly
ment Rent ($)
1
525
Mean
2
440
Median
3
450
Mode
4
615
Variance
5
480
Std. Dev.
6
510
E
490.80
475.00
450.00
2996.16
54.74
Note: Rows 8-71 are not shown.
Slide 33
Using Excel’s
Descriptive Statistics Tool
Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose Descriptive Statistics from the list of
Analysis Tools
… continued
Slide 34
Using Excel’s
Descriptive Statistics Tool
Step 4 When the Descriptive Statistics dialog box
appears:
Enter B1:B71 in the Input Range box
Select Grouped By Columns
Select Labels in First Row
Select Output Range
Enter D1 in the Output Range box
Select Summary Statistics
Select OK
Slide 35
Using Excel’s
Descriptive Statistics Tool

1
2
3
4
5
6
7
8
Value Worksheet (Partial)
A
B
C
D
E
Apart- Monthly
ment Rent ($)
Monthly Rent ($)
1
525
2
440
Mean
490.8
3
450
Standard Error
6.542348114
4
615
Median
475
5
480
Mode
450
6
510
Standard Deviation 54.73721146
7
575
Sample Variance
2996.162319
Slide 36
Using Excel’s
Descriptive Statistics Tool

Value Worksheet (Partial)
9
10
11
12
13
14
15
16
A
8
9
10
11
12
13
14
15
B
430
440
450
470
485
515
575
430
C
D
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
E
-0.334093298
0.924330473
190
425
615
34356
70
Slide 37
End of Chapter 3, Part A
Slide 38