The mean

The fair share

Five friends scored $2, 0, 3, 1$ and $4$ goals in five matches. If every match had the same number of goals, with the same total, how many would it be? That number is the mean.

Here the total is $2 + 0 + 3 + 1 + 4 = 10$, and there are $5$ values, so the mean is $10 \div 5 = 2$ goals.

The match with $0$ goals is counted too. A zero is a real value, and it belongs in the count. If we left it out and divided by $4$, we would get $2.5$, which is wrong.

The balance point

The mean is the balance point of the data. The values below it, 0 and 1, are 2 and 1 away from 2. The values above it, 3 and 4, are 1 and 2 away. The two sides balance, as a seesaw does when the weights on both sides are equal.

You should find that the two readouts for the distances are always equal. Move one dot far to the right, and the mean follows it, until the two sides balance again.

Try it in Python

The next program finds the mean of the goals, and then adds up the distances from the mean.

The distances always add up to $0$. That is true for every list, and it is why the mean is the balance point.

When some values repeat

If a value appears several times, we can multiply instead of adding it again and again. In the marks $2, 2, 2, 5, 5$, the value $2$ appears $3$ times and $5$ appears $2$ times. The total is $3 \times 2 + 2 \times 5 = 16$, and the mean is $16 \div 5 = 3.2$.

A useful trick

If you know the mean and the count, you know the total:

This helps you find a missing value. A student has four tests with a mean of 68 marks and wants a mean of 70 over five tests.

• The total needed is $5 \times 70 = 350$ marks.
• The first four give $4 \times 68 = 272$ marks.
• The fifth mark must be $350 - 272 = 78$.

Practise with new numbers

Common mistakes

• Leaving out a zero. A zero is a value and must be counted.
• Dividing by the wrong count. Count the values, and divide by that number.
• Taking the mean of two means. If the groups have different sizes, the mean of the means is not the mean of all the values.
• Thinking that the mean is always one of the values. It need not be. The mean of $1$ and $2$ is $1.5$.

Remember these

• Mean = total divided by how many values.
• The mean is the balance point of the data.
• The distances from the mean add up to $0$.
• Total = mean times count.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the mean of $4, 8$ and $12$?

2. Five numbers have a mean of $6$. What is their total?

3. The mean of $3, 5, 7$ and $x$ is $6$. Find $x$.

4. Four students have a mean mark of $50$, and a fifth student has $100$. What is the mean of all five?