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Home Content 5.1 Introduction 5.2 Table Representation 5.3 Graphical Representation 5.4 Measures of Central Tendency 5.5 Measure of Variability 5.6 Mean, Variance and Standard Deviation for Grouped Data

5.5 Measures of Variability

 
 
5.5.1
5.5.2

 
The measure of central tendency, such as mean, median, and mode, do not reveal the whole picture of the distribution of a data set.  Two data sets with the same mean may have completely different spreads.  The variation among values of observations for one data set may be much larger or smaller than for the other data set.
Note that the words dispersion, spread, and variation have the same meaning.

 
5.5.1 Range
The range is the simplest measure of dispersion to calculate.  It is obtained by taking the difference between the largest and the smallest values in a data set.  For a set of n values : , the
    Range =
Example 5.5-1
Consider the following two data sets on the ages of all workers for each of the two small company
 
Company 1:
47
38
35
40
36
45
39
Comapny 2:
70
33
18
52
27

Find the mean and range for these data sets.
 
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5.5.2 Variance and standard deviation
The standard deviation is the most frequently used measure of dispersion.  The value of the standard deviation tells how closely the values of a data are spread around the mean. In general, a lower value of the standard deviation of a data set indicates that the values of that data set are spread over a relatively smaller range around the mean.  On the other hand, a large value of the standard deviation for a data set indicates that the values of that data set are spread over a relatively larger range around the mean.
The standard deviation is obtained by taking the positive square root of the variance.  The following formulas are used to calculate the variance.
and
where  is the population variance and is the sample variance.  The quantity  or  in the above formulas is called the deviation of x value from the mean.  The sum of the deviations of x values from the mean is always zero.
However, for large number of data, it would be easier and more efficient to calculate the variance and hence the standard deviation by using the following computational formulas.
and 
Example 5.5-2
A building subcontractor pays his eight employees the daily wages (in dollar) is given in the following table.  Find the variance and standard deviation for these data.
Employee
1
2
3
4
5
6
7
8
Wage (W) $
1000
600
700
1000
600
1000
1300
800
1000
1000000
600
360000
700
490000
1000
1000000
600
360000
1000
1000000
1300
1690000
800
6400000
Variance 

Standard deviation 

By using the mean and standard deviation, we can find the proportion or percentage of the total observations that fall within a given interval about the mean.
Empirical Rule
For a bell-shaped distribution, approximately
1. 68% of the observations lie within one standard deviation of the mean
2. 95% of the observations lie within two standard deviation of the mean
3. 99.7% of the observations lie within three standard deviation of the mean
The following figure illustrates the empirical rule
The empirical rule applies to both population and sample data.
 
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Home Content 5.1 Introduction 5.2 Table Representation 5.3 Graphical Representation 5.4 Measures of Central Tendency 5.5 Measure of Variability 5.6 Mean, Variance and Standard Deviation for Grouped Data