| 5.6.1 |
Mean for grouped data |
|
To calculate the mean for grouped
data, first find the midpoint of each class and then multiply the midpoint
by the frequencies of the corresponding classes. The sum of these
products gives an approximation for the sum of all values. To find
the value of mean, divide this sum by the total number of observations
in the data. The formulas used to calculate the mean for grouped
data are as follows. |
|
Mean for population data:  |
|
Mean for sample data:  |
|
where m is the midpoint and
f
is the frequency of a class. |
|
|
| 5.6.2 |
Variance and standard deviation
for grouped data |
|
Following are the basic formulas
used to calculate the population and sample variances for grouped data. |
|
 and 
|
|
where
is the population variance, 
is the sample variance and m is the midpoint of a class.
In either cases, the standard deviation
is obtained by taking the positive square root of the variance. |
|
Again, the following computational
formulas are more efficient for calculating the variance and standard deviation. |
|
and 
|
|
where
is the population variance,
is the sample variance and m is the midpoint of a class. The standard
deviation is obtained by taking the positive square root of the variance. |
|
The population standard deviation: 
The sample standard deviation:  |
|
Example 5.6-1
The following gives the frequency
distribution of the daily commuting time (in minutes) from home to work
for all 25 employees of a company. |
|
|
Daily commuting time
|
Number of employees
|
|
0
to less than 10
|
4
|
|
10 to
less than 20
|
9
|
|
20 to
less than 30
|
6
|
|
30 to
less than 40
|
4
|
|
40
to less than 50
|
2
|
|
|
Calculate the mean, variance and
standard deviation of the daily commuting times. |
|
|
Daily commuting time
|
|
|
|
|
|
0 to less
than 10
|
4
|
5
|
20
|
100
|
|
10 to
less than 20
|
9
|
15
|
135
|
2025
|
|
20 to
less than 30
|
6
|
25
|
150
|
3750
|
|
30 to
less than 40
|
4
|
35
|
140
|
4900
|
|
40 to
less than 50
|
2
|
45
|
90
|
4050
|
| |
N = 25
|
|
|
|
|
|
Mean 
minutes |
|
Variance  |
|
Standard deviation 
minutes |
|
|
