Comparing distributions Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Comparing distributions questions. See exactly how to solve problems on comparing distributions, mean, interquartile range, interpretation in context.

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?

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
Shop A and Shop B each recorded their daily sales, in pounds. The daily sales for Shop A were: 66,40,43,37,8166, 40, 43, 37, 81. Work out the mean of the daily sales for Shop A.

Worked solution

  1. Add up all the daily sales

    total=66+40+43+37+81=267\text{total} = 66 + 40 + 43 + 37 + 81 = 267

    The mean uses every value, so the first job is the total of all 5 daily sales.

  2. Work out the mean

    mean of Shop A=66+40+43+37+815=2675=53.4\text{mean of Shop A} = \frac{66 + 40 + 43 + 37 + 81}{5} = \frac{267}{5} = 53.4

    Putting the numbers into the definition gives a mean of 53.453.4 pounds.

  3. State the answer

    mean=53.4 pounds\text{mean} = 53.4 \text{ pounds}

    The mean of the daily sales for Shop A is 53.453.4 pounds.

Answer
mean=53.4 pounds\text{mean} = 53.4 \text{ pounds}
Question 3
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 44 and the range is 22. For Team B the median of the goals per match is 33 and the range is 33. Compare the two sets of goals per match using the median and the range. Which statement is the best comparison?

Worked solution

  1. Compare the medians of the two sets

    median: 4>3\text{median}: \ 4 > 3

    The median for Team A is 44 and for Team B it is 33, 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 ranges of the two sets

    range: 2<3\text{range}: \ 2 < 3

    The range for Team A is 22 and for Team B it is 33, so the 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: 4 vs 3⇒higher;range: 2 vs 3⇒smaller\text{median: } 4 \text{ vs } 3 \Rightarrow \text{higher}; \quad \text{range: } 2 \text{ vs } 3 \Rightarrow \text{smaller}

    The comparison that scores both marks is: the median for Team A is higher (44 compared with 33), so on average Team A had more goals per match than Team B; and the 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: 4 vs 3⇒higher;range: 2 vs 3⇒smaller\text{median: } 4 \text{ vs } 3 \Rightarrow \text{higher}; \quad \text{range: } 2 \text{ vs } 3 \Rightarrow \text{smaller}
Question 4
1 markeasy
The heights, in centimetres, of the plants in Field A and in Field B were measured. The heights for Field A were: 17,14,27,26,3417, 14, 27, 26, 34. Work out the median of the heights for Field A.

Worked solution

  1. Put the heights in order, smallest first

    Field A:14, 17, 26, 27, 34\text{Field A}: 14,\ 17,\ 26,\ 27,\ 34

    Ordering the data first is what makes the middle value, the quartiles and the largest and smallest values safe to read off.

  2. Work out the median

    median of Field A=the 3th of 5 ordered values=26\text{median of Field A} = \text{the } 3\text{th of 5 ordered values} = 26

    Putting the numbers into the definition gives a median of 2626 centimetres.

  3. State the answer

    median=26 centimetres\text{median} = 26 \text{ centimetres}

    The median of the heights for Field A is 2626 centimetres.

Answer
median=26 centimetres\text{median} = 26 \text{ centimetres}
Question 5
2 markseasy
The waiting times, in minutes, for buses on Route A and on Route B were recorded. For Route A the mode of the waiting times is 55 and the range is 77. For Route B the mode of the waiting times is 66 and the range is 55. Compare the two sets of waiting times using the mode and the range. Which statement is the best comparison?

Worked solution

  1. Compare the modes of the two sets

    mode: 5<6\text{mode}: \ 5 < 6

    The mode for Route A is 55 and for Route B it is 66, so the mode for Route A is lower. In context: on average Route A had shorter waiting times than Route B.

  2. Compare the ranges of the two sets

    range: 7>5\text{range}: \ 7 > 5

    The range for Route A is 77 and for Route B it is 55, so the range for Route A is larger. In context: the waiting times for Route A are more spread out — less consistent — than the waiting times for Route B.

  3. Write the comparison out in full

    mode: 5 vs 6⇒lower;range: 7 vs 5⇒larger\text{mode: } 5 \text{ vs } 6 \Rightarrow \text{lower}; \quad \text{range: } 7 \text{ vs } 5 \Rightarrow \text{larger}

    The comparison that scores both marks is: the mode for Route A is lower (55 compared with 66), so on average Route A had shorter waiting times than Route B; and the range for Route A is larger (77 compared with 55), so the waiting times for Route A are more spread out than the waiting times for Route B.

Answer
mode: 5 vs 6⇒lower;range: 7 vs 5⇒larger\text{mode: } 5 \text{ vs } 6 \Rightarrow \text{lower}; \quad \text{range: } 7 \text{ vs } 5 \Rightarrow \text{larger}

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