1.8. 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	4 9 6 4 2

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	4 9 6 4 2	5 15 25 35 45	20 135 150 140 90

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.

1.9. The Median for grouped data

Recall that the median is different for odd and for even numbers of observations when the data are not in the grouped form. However, if the n data are written in grouped form, then median is simply defined as the observation.

Thus, if we have the frequency distribution of 100 observations, then the observation in order of size would be the median; if we have 101 observations then the observation would be the median.

To find median, first, we need to find the class which contains the middle observation. Let M denotes the number of this class, where M is the some integers from 1 to k. If the median occurs in the fifth class then M=5; if it occurs in the seventh class, then M=7; and so on.

Let the frequency of the class be denoted by. Next, note how many observations are in classes preceding the median class; denote this cumulative frequency by.

The general formula for median is

where

lower boundary of the median class

number of observations

the number of observations in the median class

the number of observations in the classes

preceding the median class

width of the median class

Example: Find the median of the frequency distribution

Starting monthly salary(in dollars)	Frequency
900-1000 1000-1100 1100-1200 1200-1300 1300-1400 1400-1500 1500-1600	2 4 3 1 1 0 1
	n=12

Solution:

First of all, let us divide n (the number of all observations) to find the halfway point.

To find the class that contains observation it is necessary to form cumulative frequency distribution. This class is called the median class; it contains the median:

Starting monthly salary(in dollars)	Frequency	Cumulative frequency
900-1000 1000-1100 1100-1200 1200-1300 1300-1400 1400-1500 1500-1600	2 4 3 1 1 0 1	2 6 9 10 11 11 12

observation is in class. So, the median class is 1000-1100.

Now let us apply

In our case

; ; ;

After substituting we get

The median is 1100. In other words, median as a measure of center indicates that average value of monthly salaries of 12 employees is 1100$.