These problems use the mean, median, mode and range, including working backwards from an average and combining groups. A worked example shows how to start; hints are at the foot.
Worked example: how to start
Five numbers have a mean of $6$. Four of them are $3$, $7$, $8$ and $5$. Work out the fifth number.
Total $=$ mean $\times$ how many $=6\times5=30$.
The four known numbers add to $3+7+8+5=23$.
Fifth number $=30-23=7$.
1Four numbers have a mean of $6$. Three of them are $4$, $9$ and $5$. Work out the fourth number.[3 marks]
Answer:
2Here are seven numbers: $4,\ 9,\ 2,\ 7,\ 4,\ 8,\ 5$. Work out the median.[2 marks]
Answer:
3The mean of $5$ numbers is $11$. The mean of a different $3$ numbers is $19$. Work out the mean of all $8$ numbers.[3 marks]
Answer:
4The table shows goals scored in some matches: $0$ goals in $4$ matches, $1$ in $6$, $2$ in $3$, $3$ in $2$. Work out the mean number of goals per match.[3 marks]
Answer:
5A list of five numbers has mode $6$ and mean $6$. Four of the numbers are $6,\ 6,\ 9$ and $4$. Work out the fifth number.[3 marks]
Answer:
★Extension: Estimate the mean of this grouped data using midpoints: $0
Stuck? Hints (don't peek unless you need to)1. Total $=$ mean $\times$ how many, then subtract the three you know.2. Order them first, then take the middle of $7$ values.3. Find each total (mean × how many), add them, then divide by $8$.4. Total goals $=\sum(\text{goals}\times\text{matches})$; divide by the total number of matches.5. Use the mean to find the total; the mode of $6$ must still hold.
Solutions & mark scheme · Averages & Range
Total: 14 marks
Award the marks shown for each correct step; many of these have more than one valid route, so give method marks for any correct working.
1Four numbers have a mean of $6$. Three of them are $4$, $9$ and $5$. Work out the fourth number.[3]
Model solution
Total $=6\times4=24$.
Known three add to $4+9+5=18$.
Fourth $=24-18=6$.
Answer: $6$
Marks
✔1Total $=24$
✔1Known sum $=18$
✔1$=6$
2Here are seven numbers: $4,\ 9,\ 2,\ 7,\ 4,\ 8,\ 5$. Work out the median.[2]
Model solution
Ordered: $2,\ 4,\ 4,\ 5,\ 7,\ 8,\ 9$.
The $4$th value (the middle) is $5$.
Answer: $5$
Marks
✔1Orders the list
✔1Median $=5$
3The mean of $5$ numbers is $11$. The mean of a different $3$ numbers is $19$. Work out the mean of all $8$ numbers.[3]
Model solution
First total $=11\times5=55$.
Second total $=19\times3=57$.
Overall mean $=\dfrac{55+57}{8}=\dfrac{112}{8}=14$.
Answer: $14$
Marks
✔1Totals $55$ and $57$
✔1Sum $=112$ over $8$
✔1$=14$
4The table shows goals scored in some matches: $0$ goals in $4$ matches, $1$ in $6$, $2$ in $3$, $3$ in $2$. Work out the mean number of goals per match.[3]
Model solution
Total goals $=0(4)+1(6)+2(3)+3(2)=0+6+6+6=18$.
Total matches $=4+6+3+2=15$.
Mean $=\dfrac{18}{15}=1.2$.
Answer: $1.2$
Marks
✔1Total goals $=18$
✔1Total matches $=15$
✔1$=1.2$
5A list of five numbers has mode $6$ and mean $6$. Four of the numbers are $6,\ 6,\ 9$ and $4$. Work out the fifth number.[3]
Model solution
Total $=6\times5=30$.
Known four add to $6+6+9+4=25$.
Fifth $=30-25=5$ (and $6$ is still the most common, so the mode is $6$).
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