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What Is the Mean of Grouped Data?

Rule

Mean of Grouped Data

Multiply the midpoint xm of each interval with its frequency fn, then add all these products together and divide the sum by the total number N of observations:

x¯ = xm1 f1 + xm2 f2 + + xmN fN N

Rule

Midpoint of an Interval

To find the midpoint xm of each interval, just add together the lowest and greatest value in the interval and divide the sum by 2:

xm = 1 2 (lowest value in interval + greatest value in interval)

xm = lowest value in interval + greatest value in interval 2

Example 1

A group of skaters are distributed into different weight classes. Here’s a table showing the classes and the frequency of each class.



Weight Class Frequency


[55kg,60kg) 7


[60kg,65kg) 8


[65kg,70kg) 12


[70kg,75kg) 9


[75kg,80kg) 6


Find the mean weight of this group.

The first thing to do is to find the midpoint of each interval. You can use the formula above to do this.

Now you multiply the midpoint xm of each interval with the frequency f of that interval. The results are shown in this table:





Weight (kg) xm f xm f




[55, 60) 57.5 7 402.5




[60, 65) 62.5 8 500




[65, 70) 67.5 12 810




[70, 75) 72.5 9 652.5




[75, 80) 77.5 6 465




Finally, you find the mean weight of the group of skaters by adding together all the values for xm f and dividing that sum by the total number of observations (the sum of all the frequencies). In this case, that’s the number of skaters in the group. x¯ = the sum of xm f n = 402.5 + 500 + 810 + 652.5 + 465 42 = 2830 42 67kg

You’ve found the mean weight of the skaters to be 67 kg.

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