Mean for grouped data

The statistical measures we have presented for the central location and dispersion of data sets are computed using the individual data values. The computational procedures we have discussed provide the most common methods for computing measures of central location and dispersion. However, in some situations the data available only in grouped or frequency distribution form. In these cases special procedures are used in order to obtain approximations to the common measures of central location and dispersion.

The formulas used to calculate the mean for grouped data are as follows:

Mean for the population data:

Mean for the sample data:

Where is the midpoint of class, is the frequency of class, is the total number of classes.

To calculate the mean for grouped data, first find the midpoint of each class and then multiply by the frequencies of the corresponding classes. The sum of these products, denoted by , gives an approximation for the sum of all values. To find the value of the mean, divide this sum by the total number of observations in the data.

Example:

The following table gives the frequency distribution of daily commuting time (in minutes) from home to work for all 25 employees of a company

Daily commuting time (minutes)	Number of employees
0 to less than 10 10 to less than 20 20 to less than 30 30 to less than 40 40 to less than 50

Calculate the mean of daily commuting time.

Solution:

Note that because the data set includes all 25 employees of the company, it represents the population. Table 1.12 shows the calculation of .

In table 1.12- denotes the midpoint of the classes.

Table 1.12

Daily commuting time (minutes)
0 to less than 10 10 to less than 20 20 to less than 30 30 to less than 40 40 to less than 50

To calculate the mean, we first find the midpoint of each class. The class midpoints are recorded in the third column of Table 1.12. The products of the midpoints and the corresponding frequencies are listed in the fourth column of that table. The sum of column, denoted by , gives the approximate total daily commuting time(in minutes) for all 25 employees. The mean is obtained by dividing this sum by the total frequency. Therefore minutes

Thus, the employees of this company spend an average of 21.40 minutes a day commuting from home to work.

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