GCSE Averages and spread Practice Questions

Add up all of the values

The mean shares the total out equally, so the first job is the total. Adding the $6$ values gives $69$.

Count how many values there are

There are $6$ values in the list, so the total is shared between $6$.

Divide the total by the number of values

The mean is $69 \div 6 = 11.5$.

Put the values in order so the gaps are visible

An outlier stands apart from the rest of the data, and ordering the list is what makes a gap easy to see.

Compare each value with the rest of the list

Every value except one sits in the narrow band from $28$ to $40$.

Apply the rule given in the question

The nearest other value is $28$, and $3$ is $25$ below it. That is more than $20$ away from every other value, so $3$ is the outlier.

Turn the mean back into a total

The mean is the total shared between the values, so the total is the mean multiplied back up: $7.2 \times 5 = 36$. Recovering the total is the key move in every reverse-mean question.

Add up the values you are given

The $4$ known values add to $29$.

Subtract to find the missing value

The missing value is whatever is left of the total: $36 - 29 = 7$.

Check by working the mean out again

Putting $7$ back into the list gives a total of $36$, and $36 \div 5 = 7.2$, which is the mean the question stated. The answer is right.

Note the structure of the method

Every "find the missing value" question is the same three moves: mean to total, subtract what you know, and what is left is the answer.

State the answer

The missing number is $7$.

Turn the first mean into a total

A mean of $13$ across $9$ numbers means the $9$ numbers add to $117$. You never need to know what they are individually.

Count the numbers after one more is added

Adding one more number makes $10$ numbers in total.

Turn the new mean into a total

The $10$ numbers now have a mean of $12.4$, so they add to $12.4 \times 10 = 124$.

Find the difference between the two totals

The only thing that changed was the extra number, so the extra number IS the change in the total: $124 - 117 = 7$.

Say why the mean moved the way it did

The new number $7$ is below the old mean of $13$, so it pulls the mean down — and indeed the mean fell from $13$ to $12.4$.

Check the answer

Adding $7$ to the old total gives $124$, and $124 \div 10 = 12.4$, exactly the new mean the question stated.

Set the problem up as an equation instead

Calling the added number $x$, the new mean is the old total plus $x$, all divided by $10$.

Solve that equation

Multiplying up gives $117 + x = 124$, so $x = 7$ — the same answer.

Note the trap

The number of values changed from $9$ to $10$. Using $9$ for both means is the mistake this question is built to catch.

State the answer

The number added was $7$.

Turn the first mean into a total

A mean of $7$ across $6$ numbers means those numbers add to $42$.

Count the numbers after one is removed

Taking one number away leaves $5$ numbers.

Turn the new mean into a total

The remaining $5$ numbers have a mean of $7.4$, so they add to $37$.

Find the difference between the two totals

The only thing that left the list was the removed number, so it is exactly the drop in the total: $42 - 37 = 5$.

State the answer

The number removed was $5$.

Check the answer

Removing $5$ from the total leaves $37$, and $37 \div 5 = 7.4$ — the mean the question gives.

Say why the mean moved the way it did

The value removed, $5$, was below the old mean of $7$. Taking an above-average value out pulls the mean down, and taking a below-average value out pushes it up. Here the mean rose to $7.4$, which matches.

Set the problem up as an equation instead

Calling the removed number $x$, the remaining total is $42 - x$ shared between $5$ numbers.

Solve that equation

Multiplying up gives $42 - x = 37$, so $x = 5$.

Note the trap

The count drops from $6$ to $5$. Dividing the new total by $6$ instead of $5$ is the mistake this question is built to catch.

Note the shortcut worth knowing

Removing $x$ changed the mean of the others by $-0.4$ each, across $5$ numbers, so $x = 7 + 5 \times -0.4 = 5$. Same arithmetic, different route.

Restate the two totals side by side

Before: $6$ numbers totalling $42$. After: $5$ numbers totalling $37$.

Say what this technique is for

Totals can be added and subtracted; means cannot. Convert to totals, do the arithmetic, convert back.

Check the answer is a sensible size

Removing a number equal to the old mean $7$ would leave the mean unchanged. The mean did change, so the removed number cannot have been $7$ — and it was not.

Write the final answer

The number that was removed is $5$.