Averages Problems

An averages problem gives you an average and asks you to use it: to find a missing value, to see what happens when a value is added or taken away, or to find numbers that fit. Most of them use the mean, and the key to the mean is the total.

In the exam you are usually asked to find a missing value from a mean, the mean of what is left when a value is taken away, the mean of one group from the mean of everyone, hidden numbers from their averages, or a missing value from a ratio of two means. You may also be asked how a mistake changes a mean, or which average is best. These questions are worth between 2 and 5 marks, and most come in the second half of the paper.

You need to be able to find the mode, median, mean and range of a list first. This page goes further and works backwards from them.

The mean is the value each item would have if the total were shared out equally. The bars 3, 7, 2 and 4 add up to 16. Shared equally between 4 bars, each would be 16 \div 4 = 4

Turn the mean round to find the total:

\text{total} = \text{mean} \times \text{number of values}

So if you know any two of the total, the mean and the number of values, you can find the third. If 5 friends spend £40 altogether, the mean is 40 \div 5 = 8, so £8 each. If you know the total, £40, and the mean, £8, the number of friends is 40 \div 8 = 5

Before you divide, make the units match. A time such as 3.5 hours is 3 hours 30 minutes, because half an hour is 30 minutes. To change hours to minutes, multiply by 60

When a value is added to a list, or taken away, work with the totals, not the means. Find the total before and the total after. The difference between them is the value that was added or taken away.

To find the mean of the values that are left, subtract the value from the total, then divide by the new number of values. There is one value fewer, so do not divide by the old number.

A new value pulls the mean towards it. A value bigger than the mean makes the mean go up, a smaller one makes it go down, and a value equal to the mean leaves the mean the same.

If a value was written down wrongly, correct the total, then divide again. For example, 4 scores have a mean of 15, so their total is 60. If a score of 12 should have been 20, the total is 8 more, 68, and the correct mean is 68 \div 4 = 17

To combine two groups, add their totals and add their numbers of values. Then divide the total by the number of values.

Adding the two means and halving gives (5 + 10) \div 2 = 7.5, which is wrong. Group B has more values, so the mean of everyone is nearer to Group B's mean.

It also works backwards. If you know the mean of everyone and the mean of one group, find both totals and subtract to get the other group's total. Then divide by the number of values in that group. For example, 10 values with a mean of 8 have a total of 80. If 4 of them have a mean of 5, their total is 20, so the other 6 have a total of 80 - 20 = 60 and a mean of 60 \div 6 = 10

Check: the mean of everyone always lies between the two groups' means.

Some questions give you the averages and ask for the numbers. Write the numbers as boxes in order of size, then fill them in one fact at a time.

• Median: goes in the middle box.
• Mode: appears more often than any other number, so write it in at least two boxes, next to each other.
• Range: the largest number is the smallest number plus the range.
• Mean: the total is the mean times how many numbers there are. The last box is the total subtract all the others.

Some questions give a ratio of two means. Work out the mean you can, use the ratio to find the other mean, then turn that mean into a total. The missing value is that total subtract the values you know.

When you have finished, check every fact against your numbers, and check they are still in order of size.

If just one value changes, the total changes, so the mean changes. The median and the mode may not. In 2,\ 3,\ 3,\ 5,\ 9, changing the 9 to 30 moves the mean from 4.4 to 8.6, but the median and the mode are both still 3

A mean always lies between the smallest value and the largest value. So if no parcel weighs more than 5 kg, the mean mass of the parcels cannot be more than 5 kg. Use this to explain why a mean someone gives is impossible.

Which average to use:

• Mode: the most common value. Use it for data that is not numbers, such as favourite colours, or when the most popular value matters, such as the shoe size a shop sells most of.
• Median: the middle value. Use it when one or two values are much bigger or smaller than the rest, because they do not pull the median.
• Mean: uses every value. Use it when there are no values far from the rest.

For example, 5 journeys to work take 10, 12, 12, 15 and 71 minutes. The mean is 120 \div 5 = 24 minutes, longer than four of the five journeys, so the median, 12 minutes, is the better average here.

• Adding two means and halving. For two groups, find each group's total, add the totals, then divide by the total number of values.
• Not finding the total first. Multiply the mean by the number of values to get the total, before anything else.
• Using the wrong number of values. Count how many values there are before you multiply or divide: when values are taken away, divide by the number that are left.
• Giving the total instead of the answer. Write the value the question asks for on the answer line, not the total or the mean.
• Treating 0.3 hours as 30 minutes. Multiply hours by 60: 0.3 hours is 18 minutes.
• Trying numbers until the mean works. Guessing gets all the marks or none, so work from the total instead.