MathsDirect

Means

MathsDirect

The mean of a set of values, is defined as the sum of the values, divided by the number of terms. So if we take the following values for a variable x

x=6;4;6;8;7;3;9

then the mean is defined as		where the bar over the x indicates mean.
So in this case

Grouped Data

When finding the mean of grouped data, the difficulty is in determining the sum of all the terms. This is relatively straightforward when dealing with discrete data, but requires a little more work if your data is continuous.

Discrete Variables

Consider the tabulated data below: (f = frequency)

x	1	2	3	4	5	6
f	2	4	5	4	3	1

To find the sum of all the results, we need to take into account the fact that different values occurred a different number of times. 1 appears twice, 2 appears 4 times, and so on.

So the formula for the sum of grouped discrete data is

The subscript i just means that you multiply each value of x by the frequency with which that value of x occurs. So for the example above:

To find the size of the population (n), you just need to add together all of the frequencies.


which in the example gives

We can therefore give a formula for the mean, using grouped discrete data:

Which for our first example gives

A Second Example

Find the mean of the following data

x	3	4	5	6	7	8	9
f	2	0	3	7	3	2	3

You can make the calculation easier by adding an extra row

x	3	4	5	6	7	8	9
f	2	0	3	7	3	2	3
xf	6	0	15	42	21	16	27

Now

Add up the bottom two rows.

Continuous Variables

The same method is used for determining the mean value of a continuous variable, but there is an extra step involved.

Length (cm)
Frequency	4	12	10	5

For discrete data, we multiplied each x by the frequency with which it occurred, but this table only gives a range for x, so we must assign a value of x to each group. The value we choose for this is the midpoint of the group. The mean is now given by:

Again, add a couple of rows to the table, to make the calculation easier.

Length (cm)
Frequency	4	12	10	5
Midpoint	5	15	25	35
mf	20	180	250	175

So the mean is

Substitute in the values

If the data is given in a different form, it may not be so obvious what the midpoints of each group is

x
f	2	5	7	3

How do you find the midpoint of the group 1-9? All that you need do is find the mean of the two boundaries. That is, add them together and divide by 2.

The same procedure is used for all the midpoints, although you will not need to write the calculations down

For the second group.

The remaining groups will just be 10 higher

The table now becomes:

x
f	2	5	7	3
m	5	14.5	24.5	34.5
mf	10	72.5	171.5	103.5

So the mean is

Substitute in the values.

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