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CALCULATING THE MEAN OF GROUPED DATA


Sometimes quantitative data are presented in the form of a frequency distribution table (FDT). A typical FDT is as follows.



Suppose that the table above presents the duration of 16 cell phone conversations between pairs of teens. There are 2 conversations with duration from 30 seconds to 44 seconds, 3 conversations with duration from 45 seconds to 59 seconds, etc. How do we calculate the mean of the data?

Step 1: Determine the midpoint of each class
If Mi denotes the midpoint of class i, $M_{i} = \frac{LB_{i}+UB_{i}}{2}$ where LBi = lower bound of class i and UBi = upper bound of class i. The lower bounds of class 1, 2, 3, 4, 5 are 30, 45, 60, 75, 90, respectively and the upper bounds are 44, 59, 74, 89, 104, respectively. Then, $M_{1} = \frac{30+44}{2} = 37$.  Similarly, $M_{2} = \frac{45+59}{2} = 52$. Continuing this way, we have the following table.


Step 2: Multiply each class frequency fi by the corresponding class midpoint Mi, resulting in fiMi
For example, for class no. 1, f1M1 = 2 x 37 =  74. Similarly, for class no. 2, f2M2 = 3 x 52 = 156. Continuing this way, we have the following table.


Step 3: Calculate the sum of frequencies and the sum of fiMi’s
The sum of frequencies is $\sum_{i=1}^{5} f_{i} = 2+3+3+5+3 = 16$.
The sum of fiMi’s is $\sum_{i=1}^{5} f_{i} M_{i} = 74 + 156 + 201 + 410 + 291 = 1132$.
Insert these two sums into the table, and we get:

Step 4: Get the mean by dividing the sum of fiMi’s by the sum of frequencies
$\bar{X} = \frac{\sum_{i=1}^{5} f_{i} M_{i}}{\sum_{i=1}^{5} f_{i}} = \frac{1132}{16} seconds \: = \: 70.75 \: seconds$.

In general, if the FDT contains k classes then the arithmetic mean is:




EXERCISE 1

The following are data on the number of years of experience of a private television news anchors.








Find the mean of the number of years of experience of the news anchors.


EXERCISE 2

Below is a frequency distribution table showing last year’s advertising cost of 40 freight forwarding companies, in thousands of dollars.

Calculate the mean advertising cost.

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