# QTS Numeracy Model Solutions – Test 2, Q15

In this series of blog posts, we look at some of the detailed methods that you could use to tackle the questions on the QTS Numeracy practice papers from the Department for Education.

In this post, we look at question 15 from practice test 2. The full practice paper can be downloaded here​.

## The Question

To inform her choice of reading materials, a primary teacher looked at the spread of reading ages in her class. The scatter graph shows the actual age and reading age of 21 pupils in the class. What is the range of reading ages for pupils in the class?

(a) 13 months; (b) 19 months; (c) 21 months

## Worked Solution

Range is one of the statistical measures that you will need to be able to find from a list of numbers, a table or a graph for your QTS Numeracy Test. In this case, you will need to work out the range from a scatter graph.

Unlike the three averages (mode, median and mean), the range doesn’t show you a typical value from the data, but instead it shows how spread out the data is. If a group of numbers have a big range, this means that there is a big difference between the highest and lowest values. A small range means that there isn’t much difference at all between the highest and lowest values.

In this question, you will need to find the difference between the highest reading age and the lowest reading age. To do this, you will need to subtract the lowest reading age from the highest reading age.

Range = Highest Value – Lowest Value

Each of the points on the scatter graph represents a different pupil, and you can use the scatter graph to find both the actual age (shown on the horizontal axis) and the reading age (shown on the vertical axis) of the pupil that the point represents. This question concerns the reading age, so you will be using the vertical axis. To find the reading age of a pupil, draw a horizontal line straight across from a point until it meets the vertical axis and read off the value that is shown on the vertical axis. This is the reading age of the pupil. The thick lines labelled 5, 6, 7 and 8 show the number of full years in the reading age, and the thinner lines between them show you the number of months in the reading age.

To work out the range, you will need to read off from the scatter graph the reading ages of the pupils with the very highest and very lowest reading ages.

You can see visually on the scatter graph which the highest and lowest reading ages are by finding the points that are closest to the top and closest to the bottom of the scatter graph. There are actually two points equally close to the top of the graph, meaning that there are two pupils that both have an equally high reading age. When you draw a horizontal line through these points, you find that it meets the vertical axis between the thick lines labelled 7 and 8. When you count up the thinner lines from the thick line labelled 7, your points are on the 7th such line, so the highest reading age is 7 years and 7 months.

For the lowest point, there is just one point that is closer to the bottom of the scatter graph than any of the other points. When you draw a horizontal line along from this point, you find it meets the axis between the thick lines labelled 5 and 6. When you count up from the thick line labelled 5, your point is on the 10th thin line, so the lowest reading age is 5 years and 10 months. (Note – Another way of doing this would be to count down from the thick line labelled 6. It is on the 2nd thick line before this, so the reading age will be 2 months less than 6 years, which as before is 5 years and 10 months).

Now you can find the range by working out the difference from 5 years and 10 months to 7 years and 7 months. You are working with years and months, so you can’t just subtract these values as you would decimals (remember that there are 12 months in a year, not 10). Instead, the best way of doing this is by counting up to whole numbers of years.

From 5 years and 10 months, you need to add on just 2 months to reach the next whole number of years (6 years). From here it will need another 12 months to increase this to 7 years. Finally, a further 7 months will be required to bring this up to 7 years and 7 months.

In total, this means the difference between the highest and lowest reading ages, and hence the range, is 2 + 12 + 7 = 21 months.

If you want to check this, you could simply count up the number of thin lines there are on the graph from the point representing the pupil with the lowest reading age to the pupil with the highest reading age. Again, you would get an answer of 21 months, so you can be confident in this answer.

Though the question was multiple choice, it is only once you have worked out an answer for yourself that you should take notice of the options given. You still need to go through the full process of reaching your answer from the data, and once you have obtained an answer, you are hoping that this answer is one of the options! In this case, the answer that we have worked out is the third of the options given, so this is the answer you would select.