This method of calculating the mean may be summarised as follows.
The frequency of a household size is the number of responses corresponding to that size. The sum of the frequencies is the total number of households.
One use of symbols in mathematics is in providing a more compact way of writing complicated formulas in words. One way of abbreviating this formula is to use symbols to represent ‘household size? and ‘frequency?: for example, h for the size of a household and / for its frequency. Then the formula can be written as follows.
This is an example of using symbols to produce a compact record of a formula or procedure. Mathematicians sometimes use the first letter of a word as a symbol to help remember what the letter stands for.
This is a little shorter, but an abbreviation for the phrase ‘the sum of? would make it even shorter still. In fact, such a symbol is commonly used in mathematics: the Greek capital letter ‘sigma?, which is written Σ, and is used to mean ‘the sum of?. Furthermore, when using symbols to represent numbers, it is not necessary to include a multiplication sign between the numbers to be multiplied: hf can be written for h × f and so Σhf means ‘the sum of all the products h × f?. And similarly Σf means ‘the sum of all the frequencies f?. So the above formula for the mean household size may be written very concisely as follows.
This symbolic formula says exactly the same as the more wordy one: add together all the products (household size × frequency) and divide by the sum of all the frequencies.
Example 4 shows how to calculate the mean household size, using sigma notation. The letter h (for household) represents household size and f (for frequency) the number of responses corresponding to each household size.
You may find it useful to make a note in your Handbook of the meaning of the symbol Σ so that on a future occasion you can find it easily if you have forgotten.
| Size of household (h) | Number of responses (f) | Products hf |
|---|---|---|
| 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 |
|
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To calculate the mean, divide the sum of the hf products (Σhf) by the sum of the frequencies (Σf).
Check through this calculation and make sure you understand the main steps and the meanings of the symbolic forms Σhf and Σf
The ten people in Example 3 were also asked how many cars were available for use by the members of their household. Three of them said they had the use of no car, four of them said they had the use of one car, two said they had the use of two cars, and one had the use of three cars. Using the formula for the mean that uses frequencies, find the mean of the number of cars available to these people.
Denoting the number of cars by c and the frequencies by f, the data are as follows.
| Number (c) of cars | Number (f) of responses | Products (cf) |
|---|---|---|
| 0 | 3 | 0 |
| 1 | 4 | 4 |
| 2 | 2 | 4 |
| 3 | 1 | 3 |
| Σf = 10 | Σcf = 11 |
The mean number of cars is
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