Calculating Means From Frequency Tables

Do you remember how to calculate the mean (or the average)? It's the sum of all the scores divided by the total number of scores:

Example

For example, to find the mean of the scores:

we

1. Find the sum of the scores (add up the scores): \(1 + 17 + 18 + 12 = 48\)
2. Divide by the number of scores (\(4\)) to give \(48 \div 4 = 12\)

to calculate the mean of \(12\).

Sometimes it isn't so simple as being given a list of scores. Our scores (or other data) might be presented in a frequency table like the one below. The frequency table lists each of the scores, together with its frequency (the number of times it occurs)

This table tells us that the score 5 occurred 6 times, the score 6 occurred 4 times, the score 7 occurred 3 times, and so on.

To find the average of these scores we could do it the hard way, by listing out all the scores, adding them together and dividing by the number of scores like this:

or we could use multiplication to total the scores: and add up the numbers in the frequency column to give the number of scores: Finally, we can calculate the mean:

Let's try another example.

Example

Christo is watching some old British comedies on TV, and laughing his head off. Sam thinks this is hilarious. He decides to work out the average number of times that Christo laughs at an episode. So, he draws up the following frequency table:

Now he needs to work out the mean number of laughing episodes. The answer is

So, the mean number of times that Christo laughs in an episode is 7.29 (correct to two decimal places).

Summation Notation

We can actually use notation to make our formulas more compact.

The symbol \(\Sigma\) means to add. So, we can write down the sum of our frequencies as \(\Sigma f\), where \(f\) runs over all the frequencies.

In the above example, \(\Sigma f\) would be:

We can also use this notation to add up the products of our scores and their frequencies like this:

Finally, we use the notation \(\overline{x}\) to stand for the mean, so we have the formula:

We can also add columns and rows to our table to keep track of our calculations as follows:

And then, we can quickly calculate That's the most efficient way to do it, Sam!