# QTS Numeracy Model Solutions – Test 2, Q17

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 17 from practice test 2. The full practice paper can be downloaded here​.

## The Question

A teacher calculated the mean points score achieved by pupils in an end of Key Stage 2 mathematics test.

There were 48 pupils in the year group.

The mean points score is given by the formula: What was the mean points score for the year group?

The total points for Levels 3 and 4 have been done for you.

## Worked Solution

This question is asking you to calculate the mean from the data given in a table. The mean is one of the three averages that you may be asked about in your QTS Numeracy Test, and each of the averages gives an idea of what the points score of a typical pupil would be.

Each of the three averages is worked out in a different way. The best way to think of the mean is like a group of people who have been out for dinner and decide to share the bill between them, even though each meal may have cost a different amount – it is splitting the total into a number of shares equivalent to the number of people in the group.

When the data is presented in a table, such as in this question, finding the mean can initially look quite complicated. It is important to be clear what each part of the table means and exactly what you are trying to calculate.

The first three columns of the table give you different pieces of information, and from this information you need to work out the values that go in the fourth column (the question tells you that this has already been done for levels 3 and 4).

The first column lists each of the levels in turn, and the second column tells you how many pupils achieved that level. There were 5 pupils that achieved level 3. There were 29 pupils that achieved level 4. There were 8 pupils that achieved level 5 and there were 6 pupils that achieved level 6.

The third column of the table tells you how many points were scored by each of the pupils that achieved that level. Achieving level 3 was worth 21 points. Level 4 is worth 27 points. Level 5 is worth 33 points and level 6 is worth 39 points.

To fill in the fourth column, you need to work out the ‘total points per level’, and you can do this by multiplying how many points the level is worth by the number of students who achieved that level.

For level 3, there were 5 pupils that each achieved 21 points, making a total of 5 × 21 = 105 points (which has already been filled in). For level 4, there were 29 pupils that achieved 27 points, making a total of 29 × 27 = 783 points (again, this has already been filled in). For level 5, there were 8 pupils that each achieved 33 points, making a total of 8 × 33 = 264 points. Finally, for level 6 there were 6 pupils that each achieved 39 points, making a total of 6 × 39 = 234 points.

You can now find the mean score by using the formula that is given in the question. The formula instructs you to divide the total points for the year group by the total number of pupils.

You are told in the question that the number of pupils is 48, so you just need to work out the total points for the year group. You can do this by adding together the total number of points for each of the levels.

105 + 783 + 264 + 234 = 1386

You can now work out the mean points score.

1386 ÷ 48 = 28.875

The answer that you have worked out is between 28 and 29, so you need to decide whether to round down to 28 or round up to 29. To do this, look at the digit immediately to the right of the decimal point. If the digit is a 5 or above then you round up. If it is a 4 or below then you need to round down.

In this case the digit immediately after the decimal point is a 8, so you need to round up to 29.