Teach Me Maths

Comparing Distributions

Question 11 mark

Two classes took the same test.

Class A: median score 34

Class B: median score 41

Which statement is correct?

Hint

The median tells you about a typical score.

Worked solution

Two classes took the same test.

Class A: median score 34

Class B: median score 41

Which statement is correct?

  1. On average, Class A got higher scores than Class B.
  2. On average, Class B got higher scores than Class A.
  3. Every student in Class B scored more than every student in Class A.
  4. The scores in Class B were more consistent.
  1. The median is an average, so it shows a typical score.
  2. 41 is greater than 34.
  3. So on average Class B got higher scores than Class A.

Question 21 mark

Two machines fill bags of rice.

Machine P: range of masses 12 g

Machine Q: range of masses 5 g

Which statement is correct?

Hint

A smaller range means the values are closer together.

Worked solution

Two machines fill bags of rice.

Machine P: range of masses 12 g

Machine Q: range of masses 5 g

Which statement is correct?

  1. The masses from Machine P are more consistent, because 12 is bigger than 5.
  2. On average, Machine P puts more rice in each bag.
  3. The masses from Machine Q are more consistent, because its range is smaller.
  4. Machine Q puts 7 g less rice in each bag.
  1. The range measures how spread out the masses are.
  2. Machine Q has the smaller range, 5 g.
  3. So the masses of Machine Q's bags are more consistent.

Question 32 marks

Ali and Ben each ran 5 races. Here are their times, in seconds.

Ali: 14, 17, 15, 13, 16

Ben: 15, 19, 12, 18, 16

(a)

Find the median of Ali's times.

1 mark

(b)

Find the median of Ben's times.

1 mark

Hint

Put each list in order before finding the middle value.

Worked solution

Ali and Ben each ran 5 races. Here are their times, in seconds.

Ali: 14, 17, 15, 13, 16

Ben: 15, 19, 12, 18, 16

Part (a)

Find the median of Ali's times.

  1. In order: 13,\ 14,\ 15,\ 16,\ 17.
  2. The middle value is 15 seconds.

Part (b)

Find the median of Ben's times.

  1. In order: 12,\ 15,\ 16,\ 18,\ 19.
  2. The middle value is 16 seconds.
  3. Ali's median is lower, so on average Ali was faster.

Question 41 mark

Two bus routes go from a town to a school. Here is some information about the journey times.

Route A: median 24 minutes, range 6 minutes

Route B: median 19 minutes, range 15 minutes

Which statement correctly compares the two routes?

Hint

A shorter time is quicker, and a smaller range is more consistent.

Worked solution

Two bus routes go from a town to a school. Here is some information about the journey times.

Route A: median 24 minutes, range 6 minutes

Route B: median 19 minutes, range 15 minutes

Which statement correctly compares the two routes?

  1. On average Route A is quicker, and Route A is more consistent.
  2. On average Route A is quicker, and Route B is more consistent.
  3. On average Route B is quicker, and Route B is more consistent.
  4. On average Route B is quicker, and Route A is more consistent.
  1. Route B has the lower median, so on average its journeys are quicker.
  2. Route A has the smaller range, so its journey times are more consistent.
  3. Answer: on average Route B is quicker, and Route A is more consistent.

Question 52 marks

The table gives information about the heights, in cm, of a group of girls and a group of boys.

Work out how much greater the range of the girls' heights is than the range of the boys' heights.

Hint

Range = greatest height − least height, for each group.

Worked solution

The table gives information about the heights, in cm, of a group of girls and a group of boys.

Work out how much greater the range of the girls' heights is than the range of the boys' heights.

  1. Girls' range: 171-148=23 cm.
  2. Boys' range: 168-151=17 cm.
  3. 23-17=6
  4. Answer: 6 cm. The girls' heights are more spread out.

Question 61 mark

Two groups of students did the same puzzle.

Group X: mean time 8.4 minutes

Group Y: mean time 6.9 minutes

Which statement is correct?

Hint

A smaller time means faster.

Worked solution

Two groups of students did the same puzzle.

Group X: mean time 8.4 minutes

Group Y: mean time 6.9 minutes

Which statement is correct?

  1. On average, Group X did the puzzle faster, because 8.4 is bigger than 6.9
  2. On average, Group Y did the puzzle faster, because its mean time is smaller.
  3. Every student in Group Y was faster than every student in Group X.
  4. Group Y's times were more consistent than Group X's times.
  1. The mean is an average, so it shows a typical time.
  2. Group Y's mean time, 6.9 minutes, is less than Group X's, 8.4 minutes.
  3. So on average Group Y did the puzzle faster.
  4. The means say nothing about every student, or about how spread out the times are.

Question 73 marks

Amy and Bilal each played 5 basketball games. Here are the points they scored.

Amy: 12, 18, 15, 20, 14

Bilal: 9, 22, 16, 11, 19

(a)

Work out the range of Amy's points.

1 mark

(b)

Work out the range of Bilal's points.

1 mark

(c)

Whose points were more consistent?

1 mark

Hint

Range = highest value − lowest value.

Worked solution

Amy and Bilal each played 5 basketball games. Here are the points they scored.

Amy: 12, 18, 15, 20, 14

Bilal: 9, 22, 16, 11, 19

Part (a)

Work out the range of Amy's points.

  1. Highest 20, lowest 12.
  2. 20-12=8

Part (b)

Work out the range of Bilal's points.

  1. Highest 22, lowest 9.
  2. 22-9=13

Part (c)

Whose points were more consistent?

  1. Amy, because her range is smaller.
  2. Bilal, because his range is bigger.
  3. Bilal, because he scored the most points in one game.
  4. Neither, because they both played 5 games.
  1. Amy's range, 8, is smaller than Bilal's, 13.
  2. So Amy's points were more consistent.

Question 81 mark

Here is some information about the marks of two classes in a test.

Class A: median 45, range 12

Class B: median 52, range 20

Which of these is a correct full comparison of the marks?

Hint

A full comparison compares an average and compares a measure of spread, in context.

Worked solution

Here is some information about the marks of two classes in a test.

Class A: median 45, range 12

Class B: median 52, range 20

Which of these is a correct full comparison of the marks?

  1. Class A had a median of 45 and a range of 12. Class B had a median of 52 and a range of 20.
  2. On average Class B scored higher, because its median is higher, and Class B's marks were more consistent, because its range is bigger.
  3. On average Class B scored higher, because its median is higher, and Class A's marks were more consistent, because its range is smaller.
  4. On average Class B scored higher, because its median is higher.
  1. A full comparison compares the averages and compares the spreads, and says what they mean.
  2. Medians: 52>45, so on average Class B scored higher.
  3. Ranges: 12<20, so Class A's marks were more consistent (a smaller range means more consistent).
  4. The first statement only quotes the values, and the last one leaves out the spread.

Question 91 mark

Here is some information about the scores of two groups in a quiz.

Group A: mean 34, range 25

Group B: mean 39, range 12

Which of these statements is not correct?

Hint

The range tells you about spread, not about which group scored more.

Worked solution

Here is some information about the scores of two groups in a quiz.

Group A: mean 34, range 25

Group B: mean 39, range 12

Which of these statements is not correct?

  1. On average, Group B scored higher than Group A.
  2. Group A's scores were more spread out than Group B's.
  3. Group A scored higher on average, because its range is bigger.
  4. Group B's scores were more consistent than Group A's.
  1. Group B's mean, 39, is higher, so on average Group B scored higher.
  2. Group A's range, 25, is bigger, so its scores were more spread out and Group B's were more consistent.
  3. A bigger range does not mean a higher average.
  4. Answer: "Group A scored higher on average, because its range is bigger" is not correct.

Question 102 marks

The table gives information about the masses of apples picked at two farms.

(a)

Which statement about the average masses is correct?

1 mark

(b)

Which statement about the spread of the masses is correct?

1 mark

Hint

The mean compares the average mass. The range compares how spread out the masses are.

Worked solution

The table gives information about the masses of apples picked at two farms.

Part (a)

Which statement about the average masses is correct?

  1. On average, Farm B's apples were heavier, because its range is smaller.
  2. On average, Farm A's apples were heavier, because its mean is bigger.
  3. Every apple from Farm A was heavier than every apple from Farm B.
  4. On average, the apples from the two farms had the same mass.
  1. Farm A's mean, 152 g, is more than Farm B's, 147 g.
  2. So on average Farm A's apples were heavier.

Part (b)

Which statement about the spread of the masses is correct?

  1. Farm A's masses were more consistent, because its range is bigger.
  2. Farm A's masses were more consistent, because its mean is bigger.
  3. The masses at the two farms were equally consistent.
  4. Farm B's masses were more consistent, because its range is smaller.
  1. Farm B's range, 18 g, is smaller than Farm A's, 30 g.
  2. So the masses of Farm B's apples were more consistent.

Question 113 marks

The stem and leaf diagram shows the number of press-ups done in one minute by each of 15 members of Club A.

For 15 members of Club B, the median was 31 press-ups and the range was 22 press-ups.

(a)

Find the median for Club A.

1 mark

(b)

Work out the range for Club A.

1 mark

(c)

Which statement correctly compares the two clubs?

1 mark

Hint

With 15 values, the median is the 8th value.

Worked solution

The stem and leaf diagram shows the number of press-ups done in one minute by each of 15 members of Club A.

For 15 members of Club B, the median was 31 press-ups and the range was 22 press-ups.

Part (a)

Find the median for Club A.

  1. With 15 values, the median is the 8th value.
  2. Counting through the diagram: 15, 18, 19, 22, 24, 24, 27, 28.
  3. The median is 28 press-ups.

Part (b)

Work out the range for Club A.

  1. Largest 45, smallest 15.
  2. 45-15=30 press-ups.

Part (c)

Which statement correctly compares the two clubs?

  1. On average Club A did more press-ups, and Club A's numbers were more consistent.
  2. On average Club A did more press-ups, and Club B's numbers were more consistent.
  3. On average Club B did more press-ups, and Club B's numbers were more consistent.
  4. On average Club B did more press-ups, and Club A's numbers were more consistent.
  1. Medians: Club B 31, Club A 28, so on average Club B did more press-ups.
  2. Ranges: Club B 22, Club A 30, so Club B's numbers were more consistent.

Question 124 marks

The table shows the number of goals Team Q scored in each of 20 matches.

Team P also played 20 matches. Team P's mean was 1.85 goals per match.

(a)

Work out the mean number of goals per match for Team Q.

3 marks

(b)

Which statement is correct?

1 mark

Hint

Multiply each number of goals by its frequency, add, then divide by the number of matches.

Worked solution

The table shows the number of goals Team Q scored in each of 20 matches.

Team P also played 20 matches. Team P's mean was 1.85 goals per match.

Part (a)

Work out the mean number of goals per match for Team Q.

  1. 0\times6=0, 1\times4=4, 2\times5=10, 3\times3=9, 4\times2=8
  2. Total goals: 0+4+10+9+8=31
  3. 31\div20=1.55

Part (b)

Which statement is correct?

  1. On average, Team P scored more goals per match than Team Q.
  2. On average, Team Q scored more goals per match than Team P.
  3. Team P scored more goals than Team Q in every match.
  4. The teams cannot be compared, because a mean of 1.85 goals is impossible.
  1. Team P's mean, 1.85, is more than Team Q's, 1.55.
  2. So on average Team P scored more goals per match.

Question 132 marks

Six people in Group A and six people in Group B each walked round a park. Here are their times, in minutes.

Group A: 21, 22, 22, 23, 24, 46

Group B: 20, 23, 25, 27, 29, 30

Which statement is best?

Hint

Work out both ranges, then look at Group A without the 46.

Worked solution

Six people in Group A and six people in Group B each walked round a park. Here are their times, in minutes.

Group A: 21, 22, 22, 23, 24, 46

Group B: 20, 23, 25, 27, 29, 30

Which statement is best?

  1. Group A's times are more spread out, because all six of its times are far apart.
  2. Group B's range is bigger, so Group B's times are more spread out.
  3. Group A's range is bigger only because of one very slow time; its other times are close together.
  4. Group A's range is bigger, so every Group A time is slower than every Group B time.
  1. Group A's range: 46-21=25 minutes. Group B's range: 30-20=10 minutes.
  2. Without the 46, Group A's times go from 21 to 24, a range of only 3 minutes.
  3. So Group A's large range comes from one very slow time.
  4. Answer: Group A's range is bigger only because of one very slow time.

Question 141 mark

Two bakers make loaves of bread.

Baker X: median mass 798 g, range 6 g

Baker Y: median mass 806 g, range 15 g

Kim says, "Baker X's loaves are more consistent, so Baker X's loaves are heavier on average."

Which of these is correct?

Hint

Use the median for the average and the range for consistency, separately.

Worked solution

Two bakers make loaves of bread.

Baker X: median mass 798 g, range 6 g

Baker Y: median mass 806 g, range 15 g

Kim says, "Baker X's loaves are more consistent, so Baker X's loaves are heavier on average."

Which of these is correct?

  1. Kim is right: Baker X's loaves are more consistent, and a smaller range means heavier loaves.
  2. Kim is wrong: Baker X's loaves are more consistent, but Baker Y's loaves are heavier on average.
  3. Kim is wrong: Baker Y's loaves are heavier on average, and they are also more consistent.
  4. Kim is wrong: Baker X's loaves are less consistent, but they are heavier on average.
  1. Baker X has the smaller range, so Baker X's loaves are more consistent.
  2. Baker Y has the bigger median, 806 g, so Baker Y's loaves are heavier on average.
  3. Being more consistent does not make the loaves heavier, so Kim is wrong.

Question 154 marks

Here are the numbers of customers at two shops on each of 7 days.

Shop A: 42, 55, 38, 61, 47, 52, 40

Shop B: 50, 44, 58, 49, 53, 47, 51

(a)

How much greater is the median for Shop B than the median for Shop A?

2 marks

(b)

Which statement correctly compares the two shops?

2 marks

Hint

Put each list in order before finding the median and the range.

Worked solution

Here are the numbers of customers at two shops on each of 7 days.

Shop A: 42, 55, 38, 61, 47, 52, 40

Shop B: 50, 44, 58, 49, 53, 47, 51

Part (a)

How much greater is the median for Shop B than the median for Shop A?

  1. Shop A in order: 38, 40, 42, 47, 52, 55, 61. Median 47.
  2. Shop B in order: 44, 47, 49, 50, 51, 53, 58. Median 50.
  3. 50-47=3

Part (b)

Which statement correctly compares the two shops?

  1. On average Shop A had more customers, and Shop A's numbers were more consistent.
  2. On average Shop B had more customers, and Shop B's numbers were more consistent.
  3. On average Shop B had more customers, and Shop A's numbers were more consistent.
  4. On average Shop A had more customers, and Shop B's numbers were more consistent.
  1. Shop B's median is higher, so on average Shop B had more customers.
  2. Ranges: Shop A 61-38=23, Shop B 58-44=14.
  3. Shop B's range is smaller, so its numbers of customers were more consistent.

Question 16Challenge3 marks

Two classes took the same test.

Class X: lowest mark 37, median 52 marks, highest mark 67

Class Y: median 58 marks, range 18 marks

The teacher finds a mistake in the marking, and adds 4 marks to every score in Class X.

(a)

Write down the new median for Class X.

1 mark

(b)

Work out the new range for Class X.

1 mark

(c)

Which statement is now correct?

1 mark

Hint

Adding the same amount to every value moves all the values up together.

Worked solution

Two classes took the same test.

Class X: lowest mark 37, median 52 marks, highest mark 67

Class Y: median 58 marks, range 18 marks

The teacher finds a mistake in the marking, and adds 4 marks to every score in Class X.

Part (a)

Write down the new median for Class X.

  1. Every mark goes up by 4, so the middle mark goes up by 4.
  2. 52+4=56

Part (b)

Work out the new range for Class X.

  1. The new lowest mark is 37+4=41 and the new highest mark is 67+4=71.
  2. New range: 71-41=30 marks.
  3. (The range has not changed, because every mark went up by the same amount.)

Part (c)

Which statement is now correct?

  1. On average Class X now did better, and Class X's marks are more consistent.
  2. On average Class X now did better, but Class Y's marks are more consistent.
  3. The two classes now have the same median.
  4. On average Class Y still did better, and Class Y's marks are still more consistent.
  1. Medians: Class Y 58, Class X 56, so Class Y still did better on average.
  2. Ranges: Class Y 18, Class X 30, so Class Y's marks are still more consistent.

Question 17Challenge3 marks

Here are the heights, in cm, of the players in two teams.

Team A: 162, 168, 171, 175, 180

Team B: 158, 165, 167, 172, 174

(a)

How much greater is the median height of Team A than the median height of Team B?

2 marks

(b)

Jo says, "Every player in Team A is taller than every player in Team B." Which of these is correct?

1 mark

Hint

The median is the middle height in each team.

Worked solution

Here are the heights, in cm, of the players in two teams.

Team A: 162, 168, 171, 175, 180

Team B: 158, 165, 167, 172, 174

Part (a)

How much greater is the median height of Team A than the median height of Team B?

  1. Team A's median is the 3rd height: 171 cm.
  2. Team B's median is the 3rd height: 167 cm.
  3. 171-167=4 cm.

Part (b)

Jo says, "Every player in Team A is taller than every player in Team B." Which of these is correct?

  1. Jo is right, because Team A's median is higher.
  2. Jo is wrong: Team A's shortest player is shorter than some players in Team B.
  3. Jo is wrong, because Team B's range is bigger than Team A's.
  4. Jo is right, because Team A's tallest player is taller than Team B's tallest player.
  1. Team A is taller on average, but that does not tell you about every player.
  2. Team A's shortest player is 162 cm, which is shorter than four of the Team B players.

Question 18Challenge4 marks

Ellie and Priya are high jumpers. Here are the heights, in metres, of their last five jumps.

Ellie: 1.30, 1.52, 1.50, 1.36, 1.52

Priya: 1.42, 1.45, 1.40, 1.44, 1.44

The mean of Priya's jumps is 1.43 m and the range is 0.05 m.

(a)

Work out the mean of Ellie's jumps.

2 marks

(b)

Work out the range of Ellie's jumps.

1 mark

(c)

In a competition, the coach wants the jumper who is most likely to clear 1.40 m with one jump. Who should she choose?

1 mark

Hint

Find Ellie's mean and range, then compare them with Priya's.

Worked solution

Ellie and Priya are high jumpers. Here are the heights, in metres, of their last five jumps.

Ellie: 1.30, 1.52, 1.50, 1.36, 1.52

Priya: 1.42, 1.45, 1.40, 1.44, 1.44

The mean of Priya's jumps is 1.43 m and the range is 0.05 m.

Part (a)

Work out the mean of Ellie's jumps.

  1. 1.30+1.52+1.50+1.36+1.52=7.20
  2. 7.20\div5=1.44 m

Part (b)

Work out the range of Ellie's jumps.

  1. 1.52-1.30=0.22 m

Part (c)

In a competition, the coach wants the jumper who is most likely to clear 1.40 m with one jump. Who should she choose?

  1. Ellie, because her mean is higher and her best jump is the highest of all.
  2. Ellie, because her range is bigger, so she can jump higher.
  3. Priya, because her range is much smaller and all her jumps were at least 1.40 m.
  4. Either of them, because their means are almost the same.
  1. Ellie's mean, 1.44 m, is only a little more than Priya's, 1.43 m.
  2. Priya's range, 0.05 m, is much smaller than Ellie's, 0.22 m, so Priya is much more consistent.
  3. All of Priya's jumps were at least 1.40 m, but two of Ellie's were lower. Choose Priya.

Question 19Challenge3 marks

Sam wants to know whether Year 11 students at his school spend longer on homework each night than Year 7 students. He asks 4 students from each year. Here are their times, in minutes.

Year 7: 15, 35, 25, 30

Year 11: 45, 30, 60, 50

(a)

How much greater is the median for Year 11 than the median for Year 7?

2 marks

(b)

Sam says, "This shows that Year 11 students at my school spend longer on homework than Year 7 students." Which comment is best?

1 mark

Hint

With 4 values, the median is halfway between the 2nd and 3rd values.

Worked solution

Sam wants to know whether Year 11 students at his school spend longer on homework each night than Year 7 students. He asks 4 students from each year. Here are their times, in minutes.

Year 7: 15, 35, 25, 30

Year 11: 45, 30, 60, 50

Part (a)

How much greater is the median for Year 11 than the median for Year 7?

  1. Year 7 in order: 15, 25, 30, 35. Median \frac{25+30}{2}=27.5.
  2. Year 11 in order: 30, 45, 50, 60. Median \frac{45+50}{2}=47.5.
  3. 47.5-27.5=20 minutes.

Part (b)

Sam says, "This shows that Year 11 students at my school spend longer on homework than Year 7 students." Which comment is best?

  1. Sam is right, because Year 11's median is higher than Year 7's median.
  2. Sam cannot be sure, because he only asked 4 students from each year.
  3. Sam is wrong, because Year 11's range is bigger, so the times cannot be compared.
  4. Sam is wrong, because the medians are not whole numbers, so they cannot be used.
  1. The medians suggest Year 11 spend longer, but Sam only asked 4 students from each year.
  2. That is too few to be sure about all the students in the school.

Question 20Challenge5 marks

Two companies deliver parcels. Here are the delivery times, in hours, for 8 parcels from each company.

Company A: 26, 30, 28, 31, 27, 29, 30, 27

Company B: 20, 35, 24, 38, 22, 33, 25, 39

(a)

Work out the mean delivery time for Company A.

2 marks

(b)

Work out the mean delivery time for Company B.

2 marks

(c)

Which conclusion is correct?

1 mark

Hint

Work out the mean and the range for each company.

Worked solution

Two companies deliver parcels. Here are the delivery times, in hours, for 8 parcels from each company.

Company A: 26, 30, 28, 31, 27, 29, 30, 27

Company B: 20, 35, 24, 38, 22, 33, 25, 39

Part (a)

Work out the mean delivery time for Company A.

  1. Total: 26+30+28+31+27+29+30+27=228
  2. 228\div8=28.5 hours

Part (b)

Work out the mean delivery time for Company B.

  1. Total: 20+35+24+38+22+33+25+39=236
  2. 236\div8=29.5 hours

Part (c)

Which conclusion is correct?

  1. Company A is quicker on average and more consistent.
  2. Company B is quicker, because it delivered one parcel in only 20 hours.
  3. Company A is quicker on average, but Company B is more consistent.
  4. Company B is quicker on average and more consistent.
  1. Means: Company A 28.5 hours, Company B 29.5 hours, so Company A is quicker on average.
  2. Ranges: Company A 31-26=5 hours, Company B 39-20=19 hours, so Company A is more consistent.

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