In some situations, various values in the batch get repeated (there may be a limited number of different values that can occur, for example). It can be simpler to group the data and record the number of times with which each different value occurs. The number is called the frequency. The following example explores this possibility and comes up with an equivalent formula for calculating the mean of the batch.
Ten people were asked what size of household they lived in (that is, how many people lived in their household). They gave the following responses.
What is the mean household size?
To find the mean household size, add them all up and divide by ten.
Here is the list of numbers (in ascending order).
These data can be written in a table as follows.
| Size of household | Frequency (Number of responses) |
|---|---|
| 1 | 3 |
| 2 | 2 |
| 3 | 1 |
| 4 | 3 |
| 5 | 1 |
The number of responses is also called the frequency of response.
To calculate the mean size of these households, the total number of people in all of the households (the sum of the full list of numbers) must be divided by the total number of households. There are three households of size one so, when finding the total number of people in the households, one must be counted three times (1 × 3); similarly, size two must be counted twice (2 × 2), size three once (3 × 1), size four three times (4 × 3) and size five once (5 × 1). Instead of writing out and adding up the full list of numbers, it can be simpler and quicker to take these five products which, when added together, give the total number of people in these households.
Recall that ‘product? means two or more quantities multiplied together. This approach does not help much in this small batch, but can save a lot of time in larger batches.
Thus:
Then, if you add together the numbers of responses, you get the total number of households.
Dividing the total number of people in these households by the total number of households gives the mean.
An alternative way of doing the same calculation is to use a table.
| Size of household | frequency | size of household x frequency |
|---|---|---|
| 1 | 3 | 1 × 3 = 3 |
| 2 | 2 | 2 × 2 = 4 |
| 3 | 1 | 3 × 1 = 3 |
| 4 | 3 | 4 × 3 = 12 |
| 5 | 1 | 5 × 1 = 5 |
| Total | 10 | 27 |
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