Hard GCSE Averages and spread Questions

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 $5$.

Turn the first mean into a total

A mean of $17$ across $7$ numbers means those numbers add to $119$.

Count the numbers after one is removed

Taking one number away leaves $6$ numbers.

Turn the new mean into a total

The remaining $6$ numbers have a mean of $19$, so they add to $114$.

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: $119 - 114 = 5$.

State the answer

The number removed was $5$.

Check the answer

Removing $5$ from the total leaves $114$, and $114 \div 6 = 19$ — the mean the question gives.

Say why the mean moved the way it did

The value removed, $5$, was below the old mean of $17$. 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 $19$, which matches.

Set the problem up as an equation instead

Calling the removed number $x$, the remaining total is $119 - x$ shared between $6$ numbers.

Solve that equation

Multiplying up gives $119 - x = 114$, so $x = 5$.

Note the trap

The count drops from $7$ to $6$. Dividing the new total by $7$ instead of $6$ is the mistake this question is built to catch.

Note the shortcut worth knowing

Removing $x$ changed the mean of the others by $-2$ each, across $6$ numbers, so $x = 17 + 6 \times -2 = 5$. Same arithmetic, different route.

Restate the two totals side by side

Before: $7$ numbers totalling $119$. After: $6$ numbers totalling $114$.

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 $17$ would leave the mean unchanged. The mean did change, so the removed number cannot have been $17$ — and it was not.

Write the final answer

The number that was removed is $5$.

Turn the first mean into a total

A mean of $15$ across $9$ numbers means the $9$ numbers add to $135$. 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 $16.2$, so they add to $16.2 \times 10 = 162$.

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: $162 - 135 = 27$.

State the answer

The number added was $27$.

Check the answer

Adding $27$ to the old total gives $162$, and $162 \div 10 = 16.2$, exactly the new mean the question stated.

Say why the mean moved the way it did

The new number $27$ is above the old mean of $15$, so it pulls the mean up — and indeed the mean rose from $15$ to $16.2$.

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 $135 + x = 162$, so $x = 27$ — the same answer.

Note the shortcut worth knowing

The added value has to be the new mean itself, plus enough extra to lift each of the other $9$ numbers by the rise in the mean: $16.2 + 9 \times 1.2 = 27$. It is the same arithmetic seen from a different angle.

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.

Restate the two totals side by side

Before: $9$ numbers totalling $135$. After: $10$ numbers totalling $162$.

Say what this technique is for

Means cannot be added or subtracted, but totals can. Converting to totals, doing the arithmetic there, and converting back is the whole technique.

Check the answer is a sensible size

A common wrong answer is to write down the new mean $16.2$ itself. But one number equal to the new mean would leave the mean unchanged, and here the mean moved, so $27$ — not $16.2$ — is the number that was added.

Write the final answer

The number that was added is $27$.

Turn the first mean into a total

A mean of $17$ across $7$ numbers means the $7$ numbers add to $119$. You never need to know what they are individually.

Count the numbers after one more is added

Adding one more number makes $8$ numbers in total.

Turn the new mean into a total

The $8$ numbers now have a mean of $16.75$, so they add to $16.75 \times 8 = 134$.

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: $134 - 119 = 15$.

State the answer

The number added was $15$.

Check the answer

Adding $15$ to the old total gives $134$, and $134 \div 8 = 16.75$, exactly the new mean the question stated.

Say why the mean moved the way it did

The new number $15$ is below the old mean of $17$, so it pulls the mean down — and indeed the mean fell from $17$ to $16.75$.

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 $8$.

Solve that equation

Multiplying up gives $119 + x = 134$, so $x = 15$ — the same answer.

Note the shortcut worth knowing

The added value has to be the new mean itself, plus enough extra to lift each of the other $7$ numbers by the rise in the mean: $16.75 + 7 \times -0.25 = 15$. It is the same arithmetic seen from a different angle.

Note the trap

The number of values changed from $7$ to $8$. Using $7$ for both means is the mistake this question is built to catch.

Restate the two totals side by side

Before: $7$ numbers totalling $119$. After: $8$ numbers totalling $134$.

Say what this technique is for

Means cannot be added or subtracted, but totals can. Converting to totals, doing the arithmetic there, and converting back is the whole technique.

Check the answer is a sensible size

A common wrong answer is to write down the new mean $16.75$ itself. But one number equal to the new mean would leave the mean unchanged, and here the mean moved, so $15$ — not $16.75$ — is the number that was added.

Write the final answer

The number that was added is $15$.

Write down the rule the mean must obey

Even with an unknown frequency, the mean is still the total of the data divided by how many pieces of data there are. Write both of those in terms of $x$ and the rule becomes an equation.

Write the sum of the frequencies in terms of x

The known frequencies add to $22$, so there are $22 + x$ pieces of data altogether.

Write the total of the data in terms of x

The rows with known frequencies contribute $1 \times 8 + 3 \times 6 + 4 \times 8$, which is $58$ in total, and the unknown row contributes $2 \times x = 2x$.

Set the mean equal to the value given

The question says the mean is $2.5$, so the expression for the mean must equal $2.5$. That is an equation in $x$.

Clear the fraction

Multiply both sides by $\sum f$. Never try to cancel while $x$ sits in the denominator — clear it first.

Expand the bracket

Multiplying out the right-hand side gives $55 + 2.5x$.

Collect the x terms on one side

Subtract $2.5x$ from both sides and subtract $58$ from both sides, so all the $x$s are together.

Simplify both sides

The left-hand side becomes $2 - 2.5 = -0.5$ lots of $x$, and the right-hand side is $-3$.

Solve for x

Dividing gives $x = 6$. A frequency must be a whole number, and $6$ is one, which is a good sign that no slip has been made.

Check by putting the value back into the table

With $x = 6$ the frequencies add to $28$.

Check the total

With $x = 6$ the data totals $70$.

Check the mean comes out right

And $70 \div 28 = 2.5$, which is exactly the mean the question gave. The answer checks out.

Say why the answer had to be a whole number

A frequency counts things, so it can only be a whole number. A fractional answer here would mean an arithmetic slip somewhere, and is worth going back for.

Note the mistake to avoid

Forgetting that the unknown frequency also adds to $\sum f$ — dividing by $22$ instead of $22 + x$ — is the error this question is built to catch. The unknown appears on the top AND the bottom.

State the answer

The missing frequency is $x = 6$.