Numeracy Skills

2.3. Averages (part 1)

Having gathered the relevant information, let us say that the islanders of Baal, being an inquisitive race, wish to know more.

Most importantly, they wish to know the average yearly rate of increase in the talking cat population, believing this will help them predict the rate at which this species might reasonably be expected to continue reproducing.

There would be three ways of calculating such an increase, only two of which need concern us here.

The first of these is the mean average which is calculated by adding up all the values in a set, and then dividing this new number by the original number of values.

The second is the median average which is achieved by listing the values in order (small to large), and then identifying the middle value in this sequence.

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Each method suits a specific purpose. If we wished to calculate, for example, what an average salary was for a specific company, then before we decided which method to use, we would first need to know about the range of salaries paid out.

Let us say there are ten employees, nine of whom earn £25,000 a year and one, the director, who earns £150,000. Appealing to the mean would be rather unhelpful here as it would generate an average income of £37,500 per person:

(9 x 25,000) + (1 x 150,000) /10 = 37,500

a figure clearly unrepresentative of the majority of employees.

Using the median, however, we can see that a much fairer calculation can be arrived at:

25k, 25k, 25k, 25k, 25k, 25k, 25k, 25k, 25k, 150k

The mid-point here would be £25,000 - values 5 & 6.

If these are different, then the median will be the mid-point between the two values and this would be a reasonable representation of the general level of pay.

If we were to change the example, however, and say that two of the employees were on £5,000; two on £6,000; two on £25,000; one on £70,000; one on £100,000 and the final two on £250,000, here the median would still generate an average wage value of £25,000:

5k, 5k, 6k, 6k, 25k, 25k, 70k, 100k, 250k, 250k

which would clearly be disproportionate as it represents the wage value of only two employees and bears no relation whatsoever to those of the other eight.

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Applying this to the situation that confronts the islanders of Baal, in order to calculate the mean and median yearly rate of increase of the talking-cat population, we first need to extract the raw figures of this increase - i.e. by how much the cat population has increased each year.

And once we have this, calculating the mean average increase is straightforward:

(100 + 100 + 200 + 500 + 1,000 + 2,000 + 4,000 + 8,000 + 16,000) / 9 = 3,544

And so too the median:

100, 100, 200, 500, 1,000, 2,000, 4,000, 8,000, 16,000

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Quick Activity

Before moving on, have a go at answering the questions in this worksheet.