Hard GCSE Comparing distributions Questions

Challenging, exam-style GCSE Comparing distributions questions with worked solutions. Stretch yourself on the hardest comparing distributions, median, range, interpretation in context problems.

comparing distributionsmedianrangeinterpretation in contextmeasures of centre and spreadmeasure of centre
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
Shop A and Shop B each recorded their daily sales, in pounds. The box plots show the daily sales for Shop A and for Shop B. Compare the two sets of daily sales using the median and the interquartile range. Which statement is the best comparison?
Show worked solution

Worked solution

  1. Say what a full comparison has to contain

    compare an average  +  compare the spread\text{compare an average} \; + \; \text{compare the spread}

    Comparison questions carry one mark for comparing an average and one mark for comparing the spread, and each comparison only scores if it is written in context — about the daily sales — rather than left as a bare pair of numbers. Here the average is the median and the spread is the interquartile range.

  2. Read the five-number summary off each box plot

    Shop A: min=39, Q1=45, median=53, Q3=67, max=74Shop B: min=21, Q1=29, median=37, Q3=47, max=49\text{Shop A}: \ \text{min} = 39, \ Q_1 = 45, \ \text{median} = 53, \ Q_3 = 67, \ \text{max} = 74 \\ \text{Shop B}: \ \text{min} = 21, \ Q_1 = 29, \ \text{median} = 37, \ Q_3 = 47, \ \text{max} = 49

    The ends of the whiskers give the smallest and the largest value, the two ends of the box give the lower and upper quartiles, and the line drawn inside the box gives the median.

  3. Find the median of the daily sales for Shop A

    median of Shop A=the line inside the box=53\text{median of Shop A} = \text{the line inside the box} = 53

    The median is the middle value once the data are in order. Half the values are below it and half are above it.

  4. Find the median of the daily sales for Shop B

    median of Shop B=the line inside the box=37\text{median of Shop B} = \text{the line inside the box} = 37

    The same method applied to Shop B gives a median of 3737 pounds, which is the number that will be compared with Shop A.

  5. Compare the two medians

    median: 53>37\text{median}: \ 53 > 37

    The median for Shop A is 5353 and the median for Shop B is 3737, so the median for Shop A is the higher of the two.

  6. Say what the comparison of averages means in context

    so on average Shop A had higher daily sales than Shop B\text{so on average Shop A had higher daily sales than Shop B}

    This is the sentence the mark is actually for. "The median is higher" on its own is just a number; saying that on average Shop A had higher daily sales than Shop B says what the number means about the daily sales.

  7. Find the interquartile range of the daily sales for Shop A

    IQR of Shop A=Q3−Q1=67−45=22\text{IQR of Shop A} = Q_3 - Q_1 = 67 - 45 = 22

    The interquartile range is the upper quartile minus the lower quartile: the spread of the middle half of the data. A larger IQR means the middle half is more spread out.

  8. Find the interquartile range of the daily sales for Shop B

    IQR of Shop B=Q3−Q1=47−29=18\text{IQR of Shop B} = Q_3 - Q_1 = 47 - 29 = 18

    The interquartile range of the daily sales for Shop B is 1818 pounds. Now both measures of spread are known, so they can be compared.

  9. Compare the two interquartile ranges

    interquartile range: 22>18\text{interquartile range}: \ 22 > 18

    The interquartile range for Shop A is 2222 and the interquartile range for Shop B is 1818, so the interquartile range for Shop A is the larger of the two.

  10. Say what the comparison of spread means in context

    so the daily sales for Shop A are more spread out than the daily sales for Shop B\text{so the daily sales for Shop A are more spread out than the daily sales for Shop B}

    A larger interquartile range means the values are more spread out, so the daily sales for Shop A are less consistent than the daily sales for Shop B. Spread is about consistency, not about who did better.

  11. Rule out the statements that compare only one thing

    centre only  or  spread only⇒1 mark lost\text{centre only} \; \text{or} \; \text{spread only} \Rightarrow \text{1 mark lost}

    A statement that only compares the median, or only compares the interquartile range, is true but incomplete: it answers half the question and scores half the marks. A full comparison needs both.

  12. Rule out the statement that only quotes the figures

    numbers≠comparison\text{numbers} \ne \text{comparison}

    "The median for Shop A is 5353 and for Shop B it is 3737" repeats the data back. It never says which is bigger and it never says what that means about the daily sales, so it earns nothing.

  13. Rule out the statement that claims a cause

    correlation≠cause\text{correlation} \ne \text{cause}

    The data can show that Shop A was higher on average and more spread out. It cannot show WHY. A statement that ends "this proves that Shop A is in a better location than Shop B" goes beyond the data and is not a valid comparison.

  14. Check that each comparison points the right way

    median: 53>37interquartile range: 22>18\text{median}: \ 53 > 37 \quad \text{interquartile range}: \ 22 > 18

    A common slip is to compare the numbers correctly and then write the sentence backwards. Check both: 53>3753 > 37 so the median for Shop A really is higher, and 22>1822 > 18 so the interquartile range for Shop A really is larger. A larger interquartile range means more spread out, never the other way round.

  15. Write the comparison out in full

    median: 53 vs 37⇒higher;interquartile range: 22 vs 18⇒larger\text{median: } 53 \text{ vs } 37 \Rightarrow \text{higher}; \quad \text{interquartile range: } 22 \text{ vs } 18 \Rightarrow \text{larger}

    The comparison that scores both marks is: the median for Shop A is higher (5353 compared with 3737), so on average Shop A had higher daily sales than Shop B; and the interquartile range for Shop A is larger (2222 compared with 1818), so the daily sales for Shop A are more spread out than the daily sales for Shop B.

Answer
median: 53 vs 37⇒higher;interquartile range: 22 vs 18⇒larger\text{median: } 53 \text{ vs } 37 \Rightarrow \text{higher}; \quad \text{interquartile range: } 22 \text{ vs } 18 \Rightarrow \text{larger}
Question 2
6 markschallenging
The waiting times, in minutes, for buses on Route A and on Route B were recorded. The box plots show the waiting times for Route A and for Route B. Compare the two sets of waiting times using the median and the interquartile range. Which statement is the best comparison?
Show worked solution

Worked solution

  1. Say what a full comparison has to contain

    compare an average  +  compare the spread\text{compare an average} \; + \; \text{compare the spread}

    Comparison questions carry one mark for comparing an average and one mark for comparing the spread, and each comparison only scores if it is written in context — about the waiting times — rather than left as a bare pair of numbers. Here the average is the median and the spread is the interquartile range.

  2. Read the five-number summary off each box plot

    Route A: min=5, Q1=7, median=9, Q3=11, max=14Route B: min=5, Q1=8, median=10, Q3=14, max=18\text{Route A}: \ \text{min} = 5, \ Q_1 = 7, \ \text{median} = 9, \ Q_3 = 11, \ \text{max} = 14 \\ \text{Route B}: \ \text{min} = 5, \ Q_1 = 8, \ \text{median} = 10, \ Q_3 = 14, \ \text{max} = 18

    The ends of the whiskers give the smallest and the largest value, the two ends of the box give the lower and upper quartiles, and the line drawn inside the box gives the median.

  3. Find the median of the waiting times for Route A

    median of Route A=the line inside the box=9\text{median of Route A} = \text{the line inside the box} = 9

    The median is the middle value once the data are in order. Half the values are below it and half are above it.

  4. Find the median of the waiting times for Route B

    median of Route B=the line inside the box=10\text{median of Route B} = \text{the line inside the box} = 10

    The same method applied to Route B gives a median of 1010 minutes, which is the number that will be compared with Route A.

  5. Compare the two medians

    median: 9<10\text{median}: \ 9 < 10

    The median for Route A is 99 and the median for Route B is 1010, so the median for Route A is the lower of the two.

  6. Say what the comparison of averages means in context

    so on average Route A had shorter waiting times than Route B\text{so on average Route A had shorter waiting times than Route B}

    This is the sentence the mark is actually for. "The median is lower" on its own is just a number; saying that on average Route A had shorter waiting times than Route B says what the number means about the waiting times.

  7. Find the interquartile range of the waiting times for Route A

    IQR of Route A=Q3−Q1=11−7=4\text{IQR of Route A} = Q_3 - Q_1 = 11 - 7 = 4

    The interquartile range is the upper quartile minus the lower quartile: the spread of the middle half of the data. A larger IQR means the middle half is more spread out.

  8. Find the interquartile range of the waiting times for Route B

    IQR of Route B=Q3−Q1=14−8=6\text{IQR of Route B} = Q_3 - Q_1 = 14 - 8 = 6

    The interquartile range of the waiting times for Route B is 66 minutes. Now both measures of spread are known, so they can be compared.

  9. Compare the two interquartile ranges

    interquartile range: 4<6\text{interquartile range}: \ 4 < 6

    The interquartile range for Route A is 44 and the interquartile range for Route B is 66, so the interquartile range for Route A is the smaller of the two.

  10. Say what the comparison of spread means in context

    so the waiting times for Route A are less spread out than the waiting times for Route B\text{so the waiting times for Route A are less spread out than the waiting times for Route B}

    A smaller interquartile range means the values are less spread out, so the waiting times for Route A are more consistent than the waiting times for Route B. Spread is about consistency, not about who did better.

  11. Rule out the statements that compare only one thing

    centre only  or  spread only⇒1 mark lost\text{centre only} \; \text{or} \; \text{spread only} \Rightarrow \text{1 mark lost}

    A statement that only compares the median, or only compares the interquartile range, is true but incomplete: it answers half the question and scores half the marks. A full comparison needs both.

  12. Rule out the statement that only quotes the figures

    numbers≠comparison\text{numbers} \ne \text{comparison}

    "The median for Route A is 99 and for Route B it is 1010" repeats the data back. It never says which is bigger and it never says what that means about the waiting times, so it earns nothing.

  13. Rule out the statement that claims a cause

    correlation≠cause\text{correlation} \ne \text{cause}

    The data can show that Route A was lower on average and less spread out. It cannot show WHY. A statement that ends "this proves that the buses on Route A are older than the buses on Route B" goes beyond the data and is not a valid comparison.

  14. Check that each comparison points the right way

    median: 9<10interquartile range: 4<6\text{median}: \ 9 < 10 \quad \text{interquartile range}: \ 4 < 6

    A common slip is to compare the numbers correctly and then write the sentence backwards. Check both: 9<109 < 10 so the median for Route A really is lower, and 4<64 < 6 so the interquartile range for Route A really is smaller. A smaller interquartile range means less spread out, never the other way round.

  15. Write the comparison out in full

    median: 9 vs 10⇒lower;interquartile range: 4 vs 6⇒smaller\text{median: } 9 \text{ vs } 10 \Rightarrow \text{lower}; \quad \text{interquartile range: } 4 \text{ vs } 6 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the median for Route A is lower (99 compared with 1010), so on average Route A had shorter waiting times than Route B; and the interquartile range for Route A is smaller (44 compared with 66), so the waiting times for Route A are less spread out than the waiting times for Route B.

Answer
median: 9 vs 10⇒lower;interquartile range: 4 vs 6⇒smaller\text{median: } 9 \text{ vs } 10 \Rightarrow \text{lower}; \quad \text{interquartile range: } 4 \text{ vs } 6 \Rightarrow \text{smaller}
Question 3
6 markschallenging
Class A and Class B took the same test. The box plots show the test scores for Class A and for Class B. Compare the two sets of test scores using the median and the interquartile range. Which statement is the best comparison?
Show worked solution

Worked solution

  1. Say what a full comparison has to contain

    compare an average  +  compare the spread\text{compare an average} \; + \; \text{compare the spread}

    Comparison questions carry one mark for comparing an average and one mark for comparing the spread, and each comparison only scores if it is written in context — about the test scores — rather than left as a bare pair of numbers. Here the average is the median and the spread is the interquartile range.

  2. Read the five-number summary off each box plot

    Class A: min=3, Q1=8, median=10, Q3=14, max=18Class B: min=6, Q1=9, median=11, Q3=14, max=18\text{Class A}: \ \text{min} = 3, \ Q_1 = 8, \ \text{median} = 10, \ Q_3 = 14, \ \text{max} = 18 \\ \text{Class B}: \ \text{min} = 6, \ Q_1 = 9, \ \text{median} = 11, \ Q_3 = 14, \ \text{max} = 18

    The ends of the whiskers give the smallest and the largest value, the two ends of the box give the lower and upper quartiles, and the line drawn inside the box gives the median.

  3. Find the median of the test scores for Class A

    median of Class A=the line inside the box=10\text{median of Class A} = \text{the line inside the box} = 10

    The median is the middle value once the data are in order. Half the values are below it and half are above it.

  4. Find the median of the test scores for Class B

    median of Class B=the line inside the box=11\text{median of Class B} = \text{the line inside the box} = 11

    The same method applied to Class B gives a median of 1111 marks, which is the number that will be compared with Class A.

  5. Compare the two medians

    median: 10<11\text{median}: \ 10 < 11

    The median for Class A is 1010 and the median for Class B is 1111, so the median for Class A is the lower of the two.

  6. Say what the comparison of averages means in context

    so on average Class A had lower test scores than Class B\text{so on average Class A had lower test scores than Class B}

    This is the sentence the mark is actually for. "The median is lower" on its own is just a number; saying that on average Class A had lower test scores than Class B says what the number means about the test scores.

  7. Find the interquartile range of the test scores for Class A

    IQR of Class A=Q3−Q1=14−8=6\text{IQR of Class A} = Q_3 - Q_1 = 14 - 8 = 6

    The interquartile range is the upper quartile minus the lower quartile: the spread of the middle half of the data. A larger IQR means the middle half is more spread out.

  8. Find the interquartile range of the test scores for Class B

    IQR of Class B=Q3−Q1=14−9=5\text{IQR of Class B} = Q_3 - Q_1 = 14 - 9 = 5

    The interquartile range of the test scores for Class B is 55 marks. Now both measures of spread are known, so they can be compared.

  9. Compare the two interquartile ranges

    interquartile range: 6>5\text{interquartile range}: \ 6 > 5

    The interquartile range for Class A is 66 and the interquartile range for Class B is 55, so the interquartile range for Class A is the larger of the two.

  10. Say what the comparison of spread means in context

    so the test scores for Class A are more spread out than the test scores for Class B\text{so the test scores for Class A are more spread out than the test scores for Class B}

    A larger interquartile range means the values are more spread out, so the test scores for Class A are less consistent than the test scores for Class B. Spread is about consistency, not about who did better.

  11. Rule out the statements that compare only one thing

    centre only  or  spread only⇒1 mark lost\text{centre only} \; \text{or} \; \text{spread only} \Rightarrow \text{1 mark lost}

    A statement that only compares the median, or only compares the interquartile range, is true but incomplete: it answers half the question and scores half the marks. A full comparison needs both.

  12. Rule out the statement that only quotes the figures

    numbers≠comparison\text{numbers} \ne \text{comparison}

    "The median for Class A is 1010 and for Class B it is 1111" repeats the data back. It never says which is bigger and it never says what that means about the test scores, so it earns nothing.

  13. Rule out the statement that claims a cause

    correlation≠cause\text{correlation} \ne \text{cause}

    The data can show that Class A was lower on average and more spread out. It cannot show WHY. A statement that ends "this proves that the pupils in Class A are cleverer than the pupils in Class B" goes beyond the data and is not a valid comparison.

  14. Check that each comparison points the right way

    median: 10<11interquartile range: 6>5\text{median}: \ 10 < 11 \quad \text{interquartile range}: \ 6 > 5

    A common slip is to compare the numbers correctly and then write the sentence backwards. Check both: 10<1110 < 11 so the median for Class A really is lower, and 6>56 > 5 so the interquartile range for Class A really is larger. A larger interquartile range means more spread out, never the other way round.

  15. Write the comparison out in full

    median: 10 vs 11⇒lower;interquartile range: 6 vs 5⇒larger\text{median: } 10 \text{ vs } 11 \Rightarrow \text{lower}; \quad \text{interquartile range: } 6 \text{ vs } 5 \Rightarrow \text{larger}

    The comparison that scores both marks is: the median for Class A is lower (1010 compared with 1111), so on average Class A had lower test scores than Class B; and the interquartile range for Class A is larger (66 compared with 55), so the test scores for Class A are more spread out than the test scores for Class B.

Answer
median: 10 vs 11⇒lower;interquartile range: 6 vs 5⇒larger\text{median: } 10 \text{ vs } 11 \Rightarrow \text{lower}; \quad \text{interquartile range: } 6 \text{ vs } 5 \Rightarrow \text{larger}
Question 4
5 markschallenging
Team A and Team B each played the same number of matches. The goals per match for Team A were: 6,1,4,2,36, 1, 4, 2, 3. So far the goals per match for Team B were: 3,6,3,03, 6, 3, 0. Team B will record one more value. Work out the value Team B must record so that the mean of the goals per match for Team B is equal to the mean of the goals per match for Team A.
Show worked solution

Worked solution

  1. Say what the question is really asking

    mean of B=mean of A\text{mean of } B = \text{mean of } A

    One more value is going to be added to Team B. It has to be chosen so that the two sets end up with the same average — so the first job is to find the average Team B has to hit.

  2. Put the goals per match for Team A in order

    Team A:1, 2, 3, 4, 6\text{Team A}: 1,\ 2,\ 3,\ 4,\ 6

    Ordering is not needed for the mean, but it makes the data easy to check and will be needed for the spread later.

  3. Add up the goals per match for Team A

    totalA=6+1+4+2+3=16\text{total}_{A} = 6 + 1 + 4 + 2 + 3 = 16

    The mean uses every value, so all 5 of them are added.

  4. Count the goals per match for Team A

    nA=5n_{A} = 5

    There are 55 values for Team A.

  5. Work out the mean for Team A

    meanA=165=3.2\text{mean}_{A} = \frac{16}{5} = 3.2

    The mean of the goals per match for Team A is 3.23.2 goals. This is the target Team B has to reach.

  6. Count how many values Team B will have

    nB=4+1=5n_{B} = 4 + 1 = 5

    Team B has 44 values so far and one more is being added, so the mean will be worked out over 55 values.

  7. Work out the total Team B must reach

    totalB=3.2×5=16\text{total}_{B} = 3.2 \times 5 = 16

    For Team B to have a mean of 3.23.2 over 55 values, those values must add up to 1616 goals.

  8. Add up the goals per match Team B has so far

    3+6+3+0=123 + 6 + 3 + 0 = 12

    The 44 values already recorded for Team B come to 1212 goals.

  9. Work out the value that must be added

    missing value=16−12=4\text{missing value} = 16 - 12 = 4

    The gap between the total Team B needs and the total it already has is the value that must be recorded next: 44 goals.

  10. Check by working out the new mean

    12+45=165=3.2\frac{12 + 4}{5} = \frac{16}{5} = 3.2

    The new mean for Team B is 3.23.2 goals, the same as for Team A, so the value is right.

  11. Write the complete set of goals per match for Team B in order

    Team B:0, 3, 3, 4, 6\text{Team B}: 0,\ 3,\ 3,\ 4,\ 6

    With the extra value added the whole of the second distribution is known, so its spread can be compared too.

  12. Compare the spreads now that the averages are equal

    rangeA=5,rangeB=6\text{range}_{A} = 5, \quad \text{range}_{B} = 6

    This is the point of the question. The two sets now have the SAME mean, so the average tells you nothing at all about which is which — and the goals per match for Team B are still more spread out than the goals per match for Team A. Centre and spread are genuinely different things.

  13. Say what that shows about comparing distributions

    same centre  ⇏  same distribution\text{same centre} \;\not\Rightarrow\; \text{same distribution}

    Two data sets can have identical means and still be nothing alike. That is exactly why a GCSE comparison always asks for a measure of spread as well as an average, both interpreted in context.

  14. Note the mistake to avoid

    missing value≠mean of A\text{missing value} \ne \text{mean of } A

    The value to add is almost never the mean itself. It is only the mean if the values Team B already has happen to average out to the target. Always go through the totals.

  15. State the answer

    value needed=4 goals\text{value needed} = 4 \text{ goals}

    Team B must record a value of 44 goals.

Answer
value needed=4 goals\text{value needed} = 4 \text{ goals}
Question 5
6 markschallenging
Group A and Group B each took a typing test. Their typing speeds, in words per minute, were recorded. The box plots show the typing speeds for Group A and for Group B. Compare the two sets of typing speeds using the median and the interquartile range. Which statement is the best comparison?
Show worked solution

Worked solution

  1. Say what a full comparison has to contain

    compare an average  +  compare the spread\text{compare an average} \; + \; \text{compare the spread}

    Comparison questions carry one mark for comparing an average and one mark for comparing the spread, and each comparison only scores if it is written in context — about the typing speeds — rather than left as a bare pair of numbers. Here the average is the median and the spread is the interquartile range.

  2. Read the five-number summary off each box plot

    Group A: min=48, Q1=50, median=54, Q3=61, max=72Group B: min=25, Q1=29, median=41, Q3=47, max=56\text{Group A}: \ \text{min} = 48, \ Q_1 = 50, \ \text{median} = 54, \ Q_3 = 61, \ \text{max} = 72 \\ \text{Group B}: \ \text{min} = 25, \ Q_1 = 29, \ \text{median} = 41, \ Q_3 = 47, \ \text{max} = 56

    The ends of the whiskers give the smallest and the largest value, the two ends of the box give the lower and upper quartiles, and the line drawn inside the box gives the median.

  3. Find the median of the typing speeds for Group A

    median of Group A=the line inside the box=54\text{median of Group A} = \text{the line inside the box} = 54

    The median is the middle value once the data are in order. Half the values are below it and half are above it.

  4. Find the median of the typing speeds for Group B

    median of Group B=the line inside the box=41\text{median of Group B} = \text{the line inside the box} = 41

    The same method applied to Group B gives a median of 4141 words per minute, which is the number that will be compared with Group A.

  5. Compare the two medians

    median: 54>41\text{median}: \ 54 > 41

    The median for Group A is 5454 and the median for Group B is 4141, so the median for Group A is the higher of the two.

  6. Say what the comparison of averages means in context

    so on average Group A had faster typing speeds than Group B\text{so on average Group A had faster typing speeds than Group B}

    This is the sentence the mark is actually for. "The median is higher" on its own is just a number; saying that on average Group A had faster typing speeds than Group B says what the number means about the typing speeds.

  7. Find the interquartile range of the typing speeds for Group A

    IQR of Group A=Q3−Q1=61−50=11\text{IQR of Group A} = Q_3 - Q_1 = 61 - 50 = 11

    The interquartile range is the upper quartile minus the lower quartile: the spread of the middle half of the data. A larger IQR means the middle half is more spread out.

  8. Find the interquartile range of the typing speeds for Group B

    IQR of Group B=Q3−Q1=47−29=18\text{IQR of Group B} = Q_3 - Q_1 = 47 - 29 = 18

    The interquartile range of the typing speeds for Group B is 1818 words per minute. Now both measures of spread are known, so they can be compared.

  9. Compare the two interquartile ranges

    interquartile range: 11<18\text{interquartile range}: \ 11 < 18

    The interquartile range for Group A is 1111 and the interquartile range for Group B is 1818, so the interquartile range for Group A is the smaller of the two.

  10. Say what the comparison of spread means in context

    so the typing speeds for Group A are less spread out than the typing speeds for Group B\text{so the typing speeds for Group A are less spread out than the typing speeds for Group B}

    A smaller interquartile range means the values are less spread out, so the typing speeds for Group A are more consistent than the typing speeds for Group B. Spread is about consistency, not about who did better.

  11. Rule out the statements that compare only one thing

    centre only  or  spread only⇒1 mark lost\text{centre only} \; \text{or} \; \text{spread only} \Rightarrow \text{1 mark lost}

    A statement that only compares the median, or only compares the interquartile range, is true but incomplete: it answers half the question and scores half the marks. A full comparison needs both.

  12. Rule out the statement that only quotes the figures

    numbers≠comparison\text{numbers} \ne \text{comparison}

    "The median for Group A is 5454 and for Group B it is 4141" repeats the data back. It never says which is bigger and it never says what that means about the typing speeds, so it earns nothing.

  13. Rule out the statement that claims a cause

    correlation≠cause\text{correlation} \ne \text{cause}

    The data can show that Group A was higher on average and less spread out. It cannot show WHY. A statement that ends "this proves that the people in Group A practise typing more than the people in Group B" goes beyond the data and is not a valid comparison.

  14. Check that each comparison points the right way

    median: 54>41interquartile range: 11<18\text{median}: \ 54 > 41 \quad \text{interquartile range}: \ 11 < 18

    A common slip is to compare the numbers correctly and then write the sentence backwards. Check both: 54>4154 > 41 so the median for Group A really is higher, and 11<1811 < 18 so the interquartile range for Group A really is smaller. A smaller interquartile range means less spread out, never the other way round.

  15. Write the comparison out in full

    median: 54 vs 41⇒higher;interquartile range: 11 vs 18⇒smaller\text{median: } 54 \text{ vs } 41 \Rightarrow \text{higher}; \quad \text{interquartile range: } 11 \text{ vs } 18 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the median for Group A is higher (5454 compared with 4141), so on average Group A had faster typing speeds than Group B; and the interquartile range for Group A is smaller (1111 compared with 1818), so the typing speeds for Group A are less spread out than the typing speeds for Group B.

Answer
median: 54 vs 41⇒higher;interquartile range: 11 vs 18⇒smaller\text{median: } 54 \text{ vs } 41 \Rightarrow \text{higher}; \quad \text{interquartile range: } 11 \text{ vs } 18 \Rightarrow \text{smaller}

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