Averages & spread

Here are the heights, in cm, of 12 children.

Show this information in a stem and leaf diagram.

Here is a list of numbers. 7 14 6 6 8 5 6 10 11 14

Find the mode.

Work out the range.

Here are two lists of numbers.

mean of list A: mean of list B = 3: 5 Work out the value of x.

Here are two lists of numbers.

mean of list A: mean of list B = 3: 5 Work out the value of x.

12The table gives some information about the ages, in years, of 32 actors.

Draw a box plot to represent this information.

Work out an estimate for the number of these actors with an age between 31 years and 42 years.

Mary says, “At least one of the actors is 35 years old because the median is 35”

Is Mary correct? Give a reason for your answer.

Katya counted the number of people in each of 50 cars. The table gives information about her results.

Find the median number of people.

Work out the mean number of people.

The table shows information about the distances that 60 people travel to work.

Distance (d miles)	Frequency
0 d 10	12
10 d 20	25
20 d 30	10
30 d 40	8
40 d 50	5

Draw a frequency polygon for the information in the table.

The cumulative frequency graph gives information about the marks that 60 students got in a test.

For thes e 60 students

On the grid below, draw a box plot for the distribution of the marks.

The pass mark for the test was 40 Sian says, “30% of the 60 students passed the test.”

Is Sian correct? You must show how you get your answer.

Here are the ages, in years, of 8 children. 14 10 10 13 15 9 15 10

Work out the mean age.

Work out the range of the ages.

One of the children is chosen at random.

On the probability scale below, mark with a cross () the probability that this child has an age of 15

15The table gives information about the ages of the 41 children in Blackrod football club.

Work out the mean age. Give your answer correct to 1 decimal place.

Rohan is working out the modal age of the children in Blackrod football club. He says, “The highest frequency is 15, so the modal age is 15”

Is Rohan’s answer correct? Give a reason for your answer.

The table shows information about the times, in minutes, 100 people took to complete a bike race.

Time (t minutes)	Frequency
120 t 140	12
140 t 160	28
160 t 180	30
180 t 200	22
200 t 220	8

On the grid below, draw a frequency polygon for this information.

The stem and leaf diagram shows the test scores of 23 students from School A.

23 students from School B did the same test. Their median score was 56 The range of their scores was 47 Compare the distribution of the test scores of the students from School A with the distribution of the test scores of the students from School B.

The stem and leaf diagram shows the test scores of 23 students from School A.

23 students from School B did the same test. Their median score was 56 The range of their scores was 47 Compare the distribution of the test scores of the students from School A with the distribution of the test scores of the students from School B.

Tessa recorded the times that 15 adults took to complete a run. She showed her results in a stem and leaf diagram.

Find the median.

Find the range.

Tessa also recorded the times that 15 children took to complete the run. For the children, the median was 75 minutes.

Compare the times that the adults took with the times that the children took.

Len has 8 parcels. The mean weight of the 8 parcels is 2.5 kg. The mean weight of 3 of the parcels is 2 kg. Work out the mean weight of the other 5 parcels.

The table shows information about the weights of 120 oranges.

Weight (w grams)	Frequency
50 w 100	34
100 w 150	29
150 w 200	27
200 w 250	19
250 w 300	11

Find the class interval that contains the median.

Calculate an estimate for the mean weight of the 120 oranges. Give your answer correct to 3 significant figures.

The frequency table gives information about the weights of 60 parcels.

Weight (wkg)	Frequency
0 w 2	7
2 w 4	21
4 w 6	15
6 w 8	11
8 w 10	6

Complete the cumulative frequency table.

On the grid opposite, draw a cumulative frequency graph for your table.

Use your graph to find an estimate for the interquartile range.

Use your graph to find an estimate for the number of these parcels with a weight greater than 7.4 kg.

Julie asks some students how they travel to school. The chart shows her results.

Write down which method of travel is the mode.

More students walk to school than cycle to school.

How many more?

Mina records the speeds, in mph, of some cars on a road on Friday. She uses her results to work out the information in this table.

On the grid, draw a box plot to show the information in the table.

Mina also records the speeds of some cars on the same road on Sunday. She uses her results to draw this box plot.

Compare the distribution of the speeds on Friday with the distribution of the speeds on Sunday.

Here is a list of numbers. 2 4 4 7 8 Work out the range of these numbers.

Miss Bailey asked 24 students where they each wanted to go on a school trip. Here are the results. museum castle castle farm farm castle farm farm castle farm castle castle castle farm castle museum museum farm castle museum museum museum castle castle

Complete the frequency table.

Write down the place that is the mode.

Draw a bar chart to show the results.

The mean length of 5 sticks is 4.2 cm. Nawal measured the length of one of the sticks as 7 cm.

Work out the mean length of the other 4 sticks.

Nawal made a mistake. The stick was not 7 cm long. It was 17 cm long.

How does this affect your answer to part (a)?

The mean length of 5 sticks is 4.2 cm. Nawal measured the length of one of the sticks as 7 cm.

Work out the mean length of the other 4 sticks.

Nawal made a mistake. The stick was not 7 cm long. It was 17 cm long.

How does this affect your answer to part (a)?

Alice recorded the number of cars going into a village on each of 80 days. The incomplete table and the incomplete box plot give information about her results.

Use the information in the table to complete the box plot.

Use the information in the box plot to complete the table. On some of these 80 days Alice saw fewer than 1200 cars going into the village.

Work out an estimate for the number of days Alice saw fewer than 1200 cars going into the village.

Here are the heights, in cm, of 16 sunflowers.

Show this information in a stem and leaf diagram.

The incomplete table and the incomplete histogram give information about the times taken by some students to run a race.

None of these students had a time for the race such that t ≤ 10 or t 26

Use the histogram to complete the table.

Use the table to complete the histogram.

The histogram below gives information about the times taken by 43 students to run a different race.

Work out an estimate for the median of the times taken by these 43 students to run the race.

Seija works at a weather station. The table gives information about the temperature, T °C, at midday for each of 50 cities in the UK on Tuesday.

Temperature (T °C)	Frequency
10 T 15	2
15 T 20	8
20 T 25	13
25 T 30	21
30 T 35	6

Calculate an estimate for the mean temperature.

Seija says, “The median temperature is 22.5 °C because 22.5 is the middle number in the middle group.”

Is Seija correct? Give a reason for your answer.

Seija works at a weather station. The table gives information about the temperature, T °C, at midday for each of 50 cities in the UK on Tuesday.

Temperature (T °C)	Frequency
10 T 15	2
15 T 20	8
20 T 25	13
25 T 30	21
30 T 35	6

Calculate an estimate for the mean temperature.

Seija says, “The median temperature is 22.5 °C because 22.5 is the middle number in the middle group.”

Is Seija correct? Give a reason for your answer.

Alan grew 80 plants of the same type outside. The cumulative frequency graph shows information about the heights, in cm, of these plants.

One of the plants is chosen at random.

Find an estimate for the probability that this plant will have a height greater than 90 cm.

Use the graph to find an estimate for the median height.

Use the graph to find an estimate for the interquartile range of the heights.

Alan also grew plants of the same type inside. The interquartile range of the heights of these plants is 30 cm.

Give one comparison between the distribution of the heights of the plants grown inside with the distribution of the heights of the plants grown outside.

Here are 6 numbers. 13 5 4 9 3 8 Work out the mean.

The table shows information about the daily rainfall in a town for 60 days.

Rainfall (R mm)	Frequency
0 R 5	8
5 R 10	24
10 R 15	13
15 R 20	11
20 R 25	4

Draw a frequency polygon for this information.

There are 3 cinemas A, B and C. The mean number of seats per cinema is 380 There are 350 seats in cinema A. There are 250 seats in cinema B. Work out the number of seats in cinema C.

The cumulative frequency table gives information about the ages of 80 people working for a company.

Age (a years)	Cumulative frequency
20 a 30	20
20 a 40	48
20 a 50	64
20 a 60	75
20 a 70	80

On the grid opposite, draw a cumulative frequency graph for this information.

Use your graph to find an estimate for the median age.

Age (years)

The table shows the number of books read by four people in one month.

Work out the median number of books.

Find the range.

On the grid, draw a bar chart to show the information in the table.

Some students took a guitar exam. The pie chart shows information about the grades the students got.

Write down the modal grade.

7 students got distinction.

Work out the total number of students who took the guitar exam.

The table shows some information about the heights of a group of adults.

On the grid, draw a box plot for the information in the table.

The box plot below shows the distribution of the heights of a group of teenagers.

Compare the distribution of the heights of the adults with the distribution of the heights of the teenagers.

Here is a list of 8 letters. B C A A A A B A

Write down the mode.

One of the 8 letters is going to be picked at random.

On the probability scale, mark with a cross () the probability that this letter will be B.

Find the probability that this letter will be C.

Hetvi asked her friends how many stickers they each have in their collection. Here are her results.

Show this information in a stem and leaf diagram.

Find the median number of stickers.

The table shows information about the heights of 80 teenagers.

Height (h cm)	Frequency
150 h 160	8
160 h 170	14
170 h 180	24
180 h 190	30
190 h 200	4

Work out an estimate for the mean height of the teenagers.

In an experiment, 60 students each completed a puzzle. The cumulative frequency graph shows information about the times taken for the 60 students to complete the puzzle.

For these 60 students, the least time taken was 24 seconds the greatest time taken was 96 seconds. On the grid below, draw a box plot for the distribution of the times taken by the students.

Four students play a game. The table shows the number of points each student has.

Barbara has more points than Danesh.

How many more?

Work out the mean number of points.

There are 30 women and 20 men at a gym. The mean height of all 50 people is 167.6 cm The mean height of the 20 men is 182 cm Work out the mean height of the 30 women.

There are 30 women and 20 men at a gym. The mean height of all 50 people is 167.6 cm The mean height of the 20 men is 182 cm Work out the mean height of the 30 women.

Here are the ages, in years, of 15 people. 19 28 29 33 27 27 37 25 27 37 17 45 47 25 26

Show this information in a stem and leaf diagram.

The table shows the amount of snow, in cm, that fell each day for 30 days.

Amount of snow (s cm)	Frequency
0 s 10	8
10 s 20	10
20 s 30	7
30 s 40	2
40 s 50	3

Work out an estimate for the mean amount of snow per day.

The table shows some information about the profit made each day at a cricket club on 100 days.

Profit (£x)	Frequency
0 x 50	10
50 x 100	15
100 x 150	25
150 x 200	30
200 x 250	5
250 x 300	15

Complete the cumulative frequency table.

On the grid, draw a cumulative frequency graph for this information.

Use your graph to find an estimate for the number of days on which the profit was less than £125

Use your graph to find an estimate for the interquartile range.

Here is a list of numbers. 3 3 3 3 4 4 5 7 8 Write down the mode of the numbers.

The table shows information about the number of social media accounts used by each of 300 students.

Work out the total number of social media accounts used by these students.

Find the median number of social media accounts used by these students.

This sign was in a doctor’s waiting room.

Work out the mean length of time for each missed appointment. Give your answer in minutes.

The box plot shows information about the sales, in thousands of pounds (£000s), of an online store each month.

Andrew says, “Three quarters of the given data lies between 160 000 and 350 000 because these are the values of the lower quartile and the upper quartile.” Andrew is wrong.

Explain why.

The table shows information about the sales, in £000s, in a shop each month.

On the grid below, draw a box plot for this information.

Compare the distribution of the sales of the online store with the distribution of the sales in the shop.

Here is the shoe size of each of 12 boys in a class. 4 5 6 6 6 7 7 8 8 8 8 9

Find the median.

Work out the range.

For the shoe sizes of each of 12 girls in the class, the median size is 6 the range is 3

Compare the distribution of the shoe sizes of the boys with the distribution of the shoe sizes of the girls.

The table gives information about the lengths, in cm, of some pieces of string.

Length (t cm)	Frequency
0 t 10	15
10 t 20	20
20 t 30	50
30 t 40	25
40 t 50	5

Amos draws a frequency polygon for the information in the table.

Write down two mistakes that Amos has made.

The cumulative frequency table gives information about the heights, in cm, of 40 plants.

Height (h cm)	Cumulative Frequency
0 h 5	4
0 h 10	11
0 h 15	24
0 h 20	34
0 h 25	38
0 h 30	40

On the grid, draw a cumulative frequency graph for this information.

Use the graph to find an estimate for the median height of the plants.

The box plot shows information about the length of time, in minutes, some people waited to see a doctor at a hospital on Monday.

Work out the interquartile range of the information in the box plot.

Becky says, “50% of the people waited for at least 2 hours.”

Is Becky correct? Explain why.

The table gives information about the length of time, in minutes, some people waited to see a doctor at the same hospital on Tuesday.

Becky was asked to compare the distribution of the lengths of times people waited on Monday with the distribution of the lengths of times people waited on Tuesday. She wrote, “People had to wait longer on Tuesday than on Monday.”

Give one reason why Becky may be wrong.

Here are the heights, in centimetres, of 15 plants. 15 20 25 33 17 22 25 18 22 19 32 35 24 28 19 Draw a stem and leaf diagram for these heights.

The table gives information about the number of points scored by each of 16 students in

Tina worked out the median of the number of points scored to be 5

Explain why it is not possible for the median to be 5

Tina also worked out the total number of points scored by the 16 students in the game. Here is her working.

Tina made a mistake in her working to find the total number of points scored.

Describe the mistake that Tina made.

26The table shows information about the heights, in cm, of a group of Year 9 girls.

This stem and leaf diagram shows information about the heights, in cm, of a group of 15 Year 9 boys.

Compare the distribution of the heights of the girls with the distribution of the heights of the boys.

The table shows information about the heights, in cm, of a group of Year 9 girls.

This stem and leaf diagram shows information about the heights, in cm, of a group of 15 Year 9 boys.

Compare the distribution of the heights of the girls with the distribution of the heights of the boys.

Aisha recorded the heights, in centimetres, of some girls. She used her results to work out the information in this table.

Aisha drew this box plot for the information in the table. The box plot is not fully correct.

Write down the two things Aisha should do to make the box plot fully correct.

Akhtar, Ben and Carl each have some money. Akhtar has £65 Ben has £100 Carl has three £5 notes, one £20 note and some £10 notes. The mean amount of money per person is £80 How many £10 notes does Carl have?

There is a total of 45 boys and girls in a choir. The mean age of the 18 boys is 16.2 years. The mean age of the 27 girls is 16.7 years. Calculate the mean age of all 45 boys and girls.

The histogram gives information about the distances 80 competitors jumped in a long jump competition.

Calculate an estimate for the mean distance.

4 red bricks have a mean weight of 5 kg. 5 blue bricks have a mean weight of 9 kg. 1 green brick has a weight of 6 kg.

Donna says,

“The mean weight of the 10 bricks is less than 7 kg.”

Is Donna correct? You must show how you get your answer.

Megan grows potatoes.

The box plot below shows information about the weights of Megan’s potatoes.

Megan says that half of her potatoes weigh less than 50 grams each.

Is Megan correct? Give a reason for your answer.

Amy also grows potatoes.

The box plot below shows information about the weights of Amy’s potatoes.

Compare the distribution of the weights of Megan’s potatoes with the distribution of the weights of Amy’s potatoes.

The stem and leaf diagram below gives information about the ages of people in a social club.

Find the range of these ages.

The table gives information about the times taken, in seconds, by 18 students to run a race.

Work out an estimate for the mean time. Give your answer correct to 3 significant figures.

The table gives information about the times taken, in seconds, by 18 students to run a race.

Work out an estimate for the mean time. Give your answer correct to 3 significant figures.

The stem and leaf diagram shows information about the heights, in cm, of 23 sunflowers.

On the grid, draw a box plot for this information.

In Adam’s garden, the flowers are only red or white or yellow or blue. The chart shows the number of red flowers, the number of white flowers and the number of yellow flowers.

The total number of flowers is 30

Work out the number of blue flowers.

Write down the mode.

A bus company recorded the ages, in years, of the people on coach A and the people on coach B. Here are the ages of the 23 people on coach A.

Complete the table below to show information about the ages of the people on coach A.

Here is some information about the ages of the people on coach B.

Richard says that the people on coach A are younger than the people on coach B.

Is Richard correct? You must give a reason for your answer.

Richard says that the people on coach A vary more in age than the people on coach B.

Is Richard correct? You must give a reason for your answer.

The table shows information about the numbers of points scored by 30 students in a quiz.

Find the modal number of points.

Work out the total number of points scored.

The table shows information about the heights of 80 plants.

Height (h cm)	Frequency
10 h 20	7
20 h 30	13
30 h 40	14
40 h 50	12
50 h 60	16
60 h 70	18

Find the class interval that contains the median.

On the grid, draw a frequency polygon for the information in the table.

The table shows information about the heights of 80 plants.

Height (h cm)	Frequency
10 h 20	7
20 h 30	13
30 h 40	14
40 h 50	12
50 h 60	16
60 h 70	18

Find the class interval that contains the median.

On the grid, draw a frequency polygon for the information in the table.

The times that 48 trains left a station on Monday were recorded.

The cumulative frequency graph gives information about the numbers of minutes the trains were delayed, correct to the nearest minute.

The shortest delay was 0 minutes. The longest delay was 42 minutes.

On the grid below, draw a box plot for the information about the delays on Monday.

48 trains left the station on Tuesday. The box plot below gives information about the delays on Tuesday.

Compare the distribution of the delays on Monday with the distribution of the delays on Tuesday.

Mary says,

“ The longest delay on Tuesday was 33 minutes. This means that there must be some delays of between 25 minutes and 30 minutes.”

Is Mary right? You must give a reason for your answer.

Fran asks each of 40 students how many books they bought last year.

The chart below shows information about the number of books bought by each of the 40 students.

Number of books

Work out the percentage of these students who bought 20 or more books.

Show that an estimate for the mean number of books bought is 9.5 You must show all your working.

Fran asks each of 40 students how many books they bought last year.

The chart below shows information about the number of books bought by each of the 40 students.

Number of books

Work out the percentage of these students who bought 20 or more books.

Show that an estimate for the mean number of books bought is 9.5 You must show all your working.

The table shows information about the distances 570 students travelled to a university open day.

Distance (d miles)	Frequency
0 d 20	120
20 d 50	90
50 d 80	120
80 d 150	140
150 d 200	100

Draw a histogram for the information in the table.

Estimate the median distance.

The table gives some information about the heights of 80 girls.

Draw a box plot to represent this information.

Work out an estimate for the number of these girls with a height between 133 cm and 157 cm.

The histogram shows information about the times taken by some students to finish a puzzle.

Complete the frequency table for this information.

Find an estimate for the lower quartile of the times taken to finish the puzzle.

Jake and Sarah each played a computer game six times.

Their scores for each game are shown below.

Who had the most consistent scores, Jake or Sarah? You must give a reason for your answer.

Jake played a different game 20 times.

The stem and leaf diagram shows information about his scores.

Jake said his modal score was 6 points because 6 occurs most often in the diagram.

Is Jake correct? You must explain your answer.

The table shows information about the weekly earnings of 20 people who work in a shop.

Weekly earnings (£x)	Frequency
150 x 250	1
250 x 350	11
350 x 450	5
450 x 550	0
550 x 650	3

Work out an estimate for the mean of the weekly earnings.

Nadiya says,

“The mean may not be the best average to use to represent this information.”

Do you agree with Nadiya? You must justify your answer.

The table shows information about the weekly earnings of 20 people who work in a shop.

Weekly earnings (£x)	Frequency
150 x 250	1
250 x 350	11
350 x 450	5
450 x 550	0
550 x 650	3

Work out an estimate for the mean of the weekly earnings.

Nadiya says,

“The mean may not be the best average to use to represent this information.”

Do you agree with Nadiya? You must justify your answer.

The table shows information about the heights, in cm, of a group of Year 11 girls.

Draw a box plot for this information.

The box plot below shows information about the heights, in cm, of a group of Year 7 girls.

Compare the distribution of heights of the Year 7 girls with the distribution of heights of the Year 11 girls.

Hasmeet walks once round a circle with diameter 80 metres.

There are 8 points equally spaced on the circumference of the circle.

Find the distance Hasmeet walks between one point and the next point.

Four of the points are moved, as shown in the diagram below.

Hasmeet walks once round the circle again.

Has the mean distance that Hasmeet walks between one point and the next point changed? You must give a reason for your answer.

The table shows information about the heights of 80 children.

Height (h cm)	Frequency
130 h 140	4
140 h 150	11
150 h 160	24
160 h 170	22
170 h 180	19

Find the class interval that contains the median.

Draw a frequency polygon for the information in the table.

The table shows information about the heights of 80 children.

Height (h cm)	Frequency
130 h 140	4
140 h 150	11
150 h 160	24
160 h 170	22
170 h 180	19

Find the class interval that contains the median.

Draw a frequency polygon for the information in the table.

Here is a list of numbers.

12 15 14 17 22 19 13

Bridgit says,

“To work out the median you find the middle number, so the median of these numbers is 17”

Bridgit’s answer is not correct.

What is wrong with Bridgit’s method?

Work out the range of the numbers in the list.

Work out the mean of the numbers in the list.

The table shows some information about the dress sizes of 25 women.

Find the median dress size.

3 of the 25 women have a shoe size of 7

Zoe says that if you choose at random one of the 25 women, the probability that she has either a shoe size of 7 or a dress size of 14 is because

Is Zoe correct? You must give a reason for your answer.

The table shows some information about the dress sizes of 25 women.

Find the median dress size.

3 of the 25 women have a shoe size of 7

Zoe says that if you choose at random one of the 25 women, the probability that she has either a shoe size of 7 or a dress size of 14 is because

Is Zoe correct? You must give a reason for your answer.

The box plot shows information about the distribution of the amounts of money spent by some male students on their holidays.

Work out the interquartile range for the amounts of money spent by these male students.

The table below shows information about the distribution of the amounts of money spent by some female students on their holidays.

On the grid above, draw a box plot for the information in the table.

Chris says,

“The box plots show that the female students spent more money than the male students.”

Is Chris correct? Give a reason for your answer.