### Overview Range and Averages

I find this section not too demanding. You have to learn some terms (range, mean, mode and median) and one or two techniques (for example, how to estimate the mean from a grouped frequency table).

There may be a Grade A question about the most appropriate average to use for a certain set of data. This is not as demanding as it seems and requires just a little common sense.

As previously, the best way for me to learn and memorise this subject is to set some questions, explain the approach and then detail the answer.

### Range, Different types of Average and Appropriate Averages

Question

A small primary school has a staff of 14 employees. The individual salaries of the staff are as follows:

£5,000, £18,000, £23,000, £5000, £16,000, £30,000, £175,000, £4,800, £24,000, £17,000, £21,000, £28,000, £13,000 and £6,000.

Calculate:-

- The range of the salaries.
- The mode of the salaries.
- The median of the salaries.
- The mean of the salaries.

Discuss which the most appropriate average to use for this set of data.

Approach

**Tip: Where you have a reasonably large number of values, say more than 5, it is useful to write the values down in ascending order (smallest to largest). This helps you to calculate the range, mode and median quickly and with less chance of making a mistake.**

The range of the data is calculated as :- Largest Value — Smallest Value

The mode (sometimes called the modal value) is the most frequent value.

The median is the middle value when all the data points are sorted in ascending order. There is a formula for this:

Median Value= (n+1)/2 n= number of values in the data set.

NB if you have an odd number of values this is quite straightforward as the median value will always be a whole number. However if you an even number of values the median value will always be x.5. In this case you add the two values either side of the median and divide by two. In this way you still get a value where the number of data points below the median is equal to the number of data points above the median. **I’m sure I’m making this sound far more difficult than it is! best to have a look at the answer below.**

The mean is the meanest average as it takes the longest to calculate. The formula is:-

Sum of all the values/Number of Values

The most appropriate average to use is the average that best represents the full range of values. If there are very small or very large values that drag the mean down or up then the median could be more appropriate. Similarly if the modal value is towards the bottom or top of a range then it is not likely to be the most appropriate average. Generally the larger the number of values and the more evenly they are distributed the more likely it is that the mean will be the most appropriate average.

Answer

Sorting the values into ascending order we have:-

£4,800, £5,000, £5,000, £6,000, £13,000, £16,000, £17,000, £18,000, £21,000, £23,000, £24,000, £28,000, £30,000, £175,000

1. The range of salaries =

£175,000 — £4,800 = £170,200

2. The mode of the salaries = £5,000. This is the only salary that is repeated.

3. The median of the salaries=

(Number of Values + 1)/2 = (14+1)÷2 =7.5^{th} value

7^{th} Value = £17,000, 8^{th} Value = £18,000

Median = (£17,000 + £18,000)/2 = £17,500

4. The mean of the salaries is:

Sum of all the salaries/Number of values = £385,800/15 = £25,720

The most appropriate average to use in this case is the median. The mode is not appropriate as only one value is repeated and that value is the second smallest in the range. The mean is not appropriate as it is skewed by the largest value of £175,000. Only three salaries are in excess of the mean whereas eleven salaries are below the mean. With the mode and mean averages at the low and high ends of the range, this leaves the median which tells you that half of the salaries are below £17,500 and half the salaries are above £17,500.