Download Measures of Dispersion

Chapter S4
Measures of dispersion

Learning Objectives
• Calculate common measures of dispersion from grouped
and ungrouped data (including the range, interquartile
range, mean deviation, and standard deviation)
• Calculate and interpret the coefficient of variation
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 1
1
The range




Simply the difference between the largest and
smallest values in a set of data
Useful for: daily temperature fluctuations or share
price movement
Is considered primitive as it considers only the
extreme values which may not be useful indicators of
the bulk of the population.
The formula is:
Range = largest observation - smallest observation
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 2
2
Interquartile range


Measures the range of the middle 50% of the values
only
Is defined as the difference between the upper and
lower quartiles
Interquartile range = upper quartile - lower quartile
= Q3 - Q1
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 3
3
The mean deviation

Measures the ‘average’ distance of each observation
away from the mean of the data

Gives an equal weight to each observation

Generally more sensitive than the range or
interquartile range, since a change in any value will
affect it
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 4
4
Actual and absolute deviations from
mean
A set of x values has a mean of

The residual of a particular x-value is:
Residual or deviation = x

x
- x
The absolute deviation is:
x-x
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 5
5
Mean deviation

The mean of the absolute deviations
Mean deviation 
 xx
n
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 6
6
To calculate mean deviation
1. Calculate mean of data
Find
x
2. Subtract mean from each For each x, find
observation
xx
Record the differences
3. Record absolute value of Find
each residual
xx
for each x
4. Calculate the mean of
the absolute values
Mean deviation 
 xx
n
Add up absolute values
and divide by n
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 7
7
The standard deviation





Measures the variation of observations from
the mean
The most common measure of dispersion
Takes into account every observation
Measures the ‘average deviation’ of
observations from mean
Works with squares of residuals not absolute
values—easier to use in further calculations
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 8
8
Standard deviation of a population
δ

Every observation in the population is
used.
2
Standard deviation  δ 

 x  x
n
The square of the population standard
deviation is called the variance.
Variance  δ 2
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 9
9
Standard deviation of a sample
s

In practice, most populations are very
large and it is more common to
calculate the sample standard deviation.
 x  x 
2
Sample standard deviation  s 

n 1
Where: (n-1) is the number of observations in the sample
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 10
10
To calculate standard deviation
1. Calculate the mean
2. Calculate the residual for each x
3. Square the residuals
x
xx
( x  x )2

2

2

2
4. Calculate the sum of the squares
 xx
5. Divide the sum in Step 4 by (n-1)
 xx
n 1
6. Take the square root of quantity
in Step 5
 xx
n 1
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 11
11
Standard deviations for frequency
distributions

If data is in a frequency distribution
No. Units
n
1
Frequency
f
85
2
192
3
123
Total
400
Total

Calculate standard deviation using:
s

 x  x
  1

2
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 12
12
Coefficient of variation

Is a measure of relative variability used
to:
– measure changes that have occurred in a
population over time
– compare variability of two populations that are
expressed in different units of measurement
– expressed as a percentage rather than in terms of
the units of the particular data
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 13
13
Formula for coefficient of variation

Denoted by V
s
V  100 %
x
where
x = the mean of the sample
s = the standard deviation of the sample
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 14
14
Summary

Measures of central tendency
– no ideal measure of dispersion exists
– standard deviation is the most important measure
of central tendency
• it is the most frequently used
• the value is affected by the value of every observation in
the data
• extreme values in the population may distort the data
© 2002 McGraw-Hill Australia, PPTs t/a Introductory
Mathematics & Statistics for Business 4e by John S. Croucher
Slide 15
15