5.6 Mean, Variance and Standard Deviation for Grouped Data

 5.6.1 5.6.2
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