GCSE Comparing distributions Practice Questions

Free GCSE Comparing distributions practice questions with full step-by-step worked solutions. Covers comparing distributions, mean, interquartile range, interpretation in context. Practise exam-style problems and check your method.

comparing distributionsmeaninterquartile rangeinterpretation in contextmeasures of centre and spreadmeasure of centre
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
Class A and Class B took the same test. For Class A the mean of the test scores is 1212 and the interquartile range is 77. For Class B the mean of the test scores is 99 and the interquartile range is 55. Compare the two sets of test scores using the mean and the interquartile range. Which statement is the best comparison?
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Worked solution

  1. Compare the means of the two sets

    mean: 12>9\text{mean}: \ 12 > 9

    The mean for Class A is 1212 and for Class B it is 99, so the mean for Class A is higher. In context: on average Class A had higher test scores than Class B.

  2. Compare the interquartile ranges of the two sets

    interquartile range: 7>5\text{interquartile range}: \ 7 > 5

    The interquartile range for Class A is 77 and for Class B it is 55, so the interquartile range for Class A is larger. In context: the test scores for Class A are more spread out — less consistent — than the test scores for Class B.

  3. Write the comparison out in full

    mean: 12 vs 9⇒higher;interquartile range: 7 vs 5⇒larger\text{mean: } 12 \text{ vs } 9 \Rightarrow \text{higher}; \quad \text{interquartile range: } 7 \text{ vs } 5 \Rightarrow \text{larger}

    The comparison that scores both marks is: the mean for Class A is higher (1212 compared with 99), so on average Class A had higher test scores than Class B; and the interquartile range for Class A is larger (77 compared with 55), so the test scores for Class A are more spread out than the test scores for Class B.

Answer
mean: 12 vs 9⇒higher;interquartile range: 7 vs 5⇒larger\text{mean: } 12 \text{ vs } 9 \Rightarrow \text{higher}; \quad \text{interquartile range: } 7 \text{ vs } 5 \Rightarrow \text{larger}
Question 2
2 markseasy
Team A and Team B each played the same number of matches. For Team A the median of the goals per match is 55 and the interquartile range is 22. For Team B the median of the goals per match is 22 and the interquartile range is 33. Compare the two sets of goals per match using the median and the interquartile range. Which statement is the best comparison?
Show worked solution

Worked solution

  1. Compare the medians of the two sets

    median: 5>2\text{median}: \ 5 > 2

    The median for Team A is 55 and for Team B it is 22, so the median for Team A is higher. In context: on average Team A had more goals per match than Team B.

  2. Compare the interquartile ranges of the two sets

    interquartile range: 2<3\text{interquartile range}: \ 2 < 3

    The interquartile range for Team A is 22 and for Team B it is 33, so the interquartile range for Team A is smaller. In context: the goals per match for Team A are less spread out — more consistent — than the goals per match for Team B.

  3. Write the comparison out in full

    median: 5 vs 2⇒higher;interquartile range: 2 vs 3⇒smaller\text{median: } 5 \text{ vs } 2 \Rightarrow \text{higher}; \quad \text{interquartile range: } 2 \text{ vs } 3 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the median for Team A is higher (55 compared with 22), so on average Team A had more goals per match than Team B; and the interquartile range for Team A is smaller (22 compared with 33), so the goals per match for Team A are less spread out than the goals per match for Team B.

Answer
median: 5 vs 2⇒higher;interquartile range: 2 vs 3⇒smaller\text{median: } 5 \text{ vs } 2 \Rightarrow \text{higher}; \quad \text{interquartile range: } 2 \text{ vs } 3 \Rightarrow \text{smaller}
Question 3
2 marksintermediate
Class A and Class B took the same test. The dot plot shows the test scores for Class A and for Class B. Compare the two sets of test scores using the mean and the range. Which statement is the best comparison?
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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 mean and the spread is the range.

  2. Find the mean of each set of test scores

    mean of Class A=7+14+14+4+13+10+6+118=798=9.875mean of Class B=5+14+9+10+3+7+7+38=588=7.25\text{mean of Class A} = \frac{7 + 14 + 14 + 4 + 13 + 10 + 6 + 11}{8} = \frac{79}{8} = 9.875 \\ \text{mean of Class B} = \frac{5 + 14 + 9 + 10 + 3 + 7 + 7 + 3}{8} = \frac{58}{8} = 7.25

    The mean is the total of the values divided by how many values there are. It uses every value, so one very large or very small value pulls it along. Working it out for both sets gives 9.8759.875 for Class A and 7.257.25 for Class B.

  3. Compare the means and say what that means

    mean: 9.875>7.25\text{mean}: \ 9.875 > 7.25

    9.875>7.259.875 > 7.25, so the mean for Class A is higher. Written in context: on average Class A had higher test scores than Class B. That sentence, not the inequality, is what earns the mark.

  4. Find the range of each set of test scores

    range of Class A=14−4=10range of Class B=14−3=11\text{range of Class A} = 14 - 4 = 10 \\ \text{range of Class B} = 14 - 3 = 11

    The range is the largest value minus the smallest value. A larger range means the data are more spread out, so less consistent. Working it out for both sets gives 1010 for Class A and 1111 for Class B.

  5. Compare the ranges and say what that means

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

    10<1110 < 11, so the range for Class A is smaller. Written in context: the test scores for Class A are less spread out than the test scores for Class B, so Class A is more consistent.

  6. Write the comparison out in full

    mean: 9.875 vs 7.25⇒higher;range: 10 vs 11⇒smaller\text{mean: } 9.875 \text{ vs } 7.25 \Rightarrow \text{higher}; \quad \text{range: } 10 \text{ vs } 11 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the mean for Class A is higher (9.8759.875 compared with 7.257.25), so on average Class A had higher test scores than Class B; and the range for Class A is smaller (1010 compared with 1111), so the test scores for Class A are less spread out than the test scores for Class B.

Answer
mean: 9.875 vs 7.25⇒higher;range: 10 vs 11⇒smaller\text{mean: } 9.875 \text{ vs } 7.25 \Rightarrow \text{higher}; \quad \text{range: } 10 \text{ vs } 11 \Rightarrow \text{smaller}
Question 4
4 markshard
The monthly rainfall totals, in millimetres, were recorded for Town A and for Town B. The box plots show the rainfall totals for Town A and for Town B. Compare the two sets of rainfall totals 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 rainfall totals — 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

    Town A: min=18, Q1=24, median=27, Q3=39, max=44Town B: min=11, Q1=23, median=37, Q3=57, max=60\text{Town A}: \ \text{min} = 18, \ Q_1 = 24, \ \text{median} = 27, \ Q_3 = 39, \ \text{max} = 44 \\ \text{Town B}: \ \text{min} = 11, \ Q_1 = 23, \ \text{median} = 37, \ Q_3 = 57, \ \text{max} = 60

    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 rainfall totals for Town A

    median of Town A=the line inside the box=27\text{median of Town A} = \text{the line inside the box} = 27

    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 rainfall totals for Town B

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

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

  5. Compare the medians and say what that means

    median: 27<37\text{median}: \ 27 < 37

    27<3727 < 37, so the median for Town A is lower. Written in context: on average Town A had lower rainfall totals than Town B. That sentence, not the inequality, is what earns the mark.

  6. Find the interquartile range of the rainfall totals for Town A

    IQR of Town A=Q3−Q1=39−24=15\text{IQR of Town A} = Q_3 - Q_1 = 39 - 24 = 15

    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.

  7. Find the interquartile range of the rainfall totals for Town B

    IQR of Town B=Q3−Q1=57−23=34\text{IQR of Town B} = Q_3 - Q_1 = 57 - 23 = 34

    The interquartile range of the rainfall totals for Town B is 3434 millimetres. Now both measures of spread are known, so they can be compared.

  8. Compare the interquartile ranges and say what that means

    interquartile range: 15<34\text{interquartile range}: \ 15 < 34

    15<3415 < 34, so the interquartile range for Town A is smaller. Written in context: the rainfall totals for Town A are less spread out than the rainfall totals for Town B, so Town A is more consistent.

  9. Rule out the statement that claims a cause

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

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

  10. Write the comparison out in full

    median: 27 vs 37⇒lower;interquartile range: 15 vs 34⇒smaller\text{median: } 27 \text{ vs } 37 \Rightarrow \text{lower}; \quad \text{interquartile range: } 15 \text{ vs } 34 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the median for Town A is lower (2727 compared with 3737), so on average Town A had lower rainfall totals than Town B; and the interquartile range for Town A is smaller (1515 compared with 3434), so the rainfall totals for Town A are less spread out than the rainfall totals for Town B.

Answer
median: 27 vs 37⇒lower;interquartile range: 15 vs 34⇒smaller\text{median: } 27 \text{ vs } 37 \Rightarrow \text{lower}; \quad \text{interquartile range: } 15 \text{ vs } 34 \Rightarrow \text{smaller}
Question 5
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}

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