Representation of Data (CIE A Level Maths: Probability & Statistics 1)

Jeanette works for a conservation charity who rescue orphaned otters. Over many years she records the weight (g) of each otter when it first arrives.  The data is illustrated in the following box and whisker diagram:

Using the box plot above:

Otters are then weighed weekly to track their growth.  Summary data on the weights (g) of otters after one month is shown in the table below:

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

120 competitors enter an elimination race for charity.  Runners set off from the same start running as many laps of the course as possible.  Their total distance is tracked and the competitor who runs the furthest over a 6-hour period is the winner.  The distances runners achieved are recorded in the table below:

On the grid below, draw a cumulative frequency graph for the information in the table.

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

The total amount of time cleaners spent dealing with unplanned incidents in a supermarket was recorded each day.  Data collected over 49 days is summarised in the table below.

Time t (minutes)	Frequency
0 ≤ t < 90	9
90 ≤ t < 120	24
120 ≤ t < 200	12
200 ≤ t < 250	4

On the grid below, draw a histogram to represent this data.

Estimate how often cleaners spent longer than 3 hours dealing with incidents.

A taxi firm, JustDrive, records data on the amount of time, to the nearest minute, that customers had to wait before their taxis arrived.  A random sample of 20 times is given below:

Find the median and interquartile range of the waiting times.

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

Filmworld cinemas collected data on the ages of visitors to their cinemas during a 24-hour period.  The incomplete histogram and frequency table show some of the information they collected:

Age a (years)	Frequency
0 ≤ a < 5	15
5 ≤ a < 10
10 ≤ a < 20
20 ≤ a < 30	12
30 ≤ a < 50	18
50 ≤ a < 60	7

Use the information to complete the histogram and fill out the missing data in the frequency table.

Police check the speed of vehicles travelling along a stretch of highway.  The cumulative frequency curve below summarises the data for the speeds, in kmph, of 80 vehicles:

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

The speed limit for this section of road is 80 kmph.

Vehicles travelling above the speed limit are issued with a speeding ticket. Those travelling more than 10% over the speed limit are pulled over.  Use the graph to estimate the percentage of vehicles that the police pull over.

An advertisement for a charity is shown on TV at the same time every weekday for four weeks.  To assess the impact of the advert, the charity’s manager decides to record the number of donations the charity receives each day in the hour after the advert is broadcast.  The results are listed below: 

                                    21       27       24       31       17

                                    22       25       26       27       9

                                    32       29       25       24       40

                                    23       22       19       12       14 

Represent these data in a sorted stem-and-leaf diagram.

The manager decides that the advert is not cost effective unless the median number of donations per day in the hour after broadcast is at least 25. Determine whether the manager should continue to run the TV advert.

Give one advantage of using a stem-and-leaf diagram as opposed to grouping the data into a frequency table.

The amounts of time engineers spent dealing with individual faults in a power plant were recorded to the nearest minute.  Data on 30 different faults is summarised in the table below.

Time t (minutes)	Frequency
90 - 129	6
130 - 169	8
170 - 199	12
200 - 249	4

Give a reason to support the use of a histogram to represent these data.

On the grid below, draw a histogram to represent the data.

Estimate the proportion of individual faults on which engineers spent longer than three hours.

A teacher took 19 students on an international trip. The incomplete box plot below shows part of the summary of the weights, in kg, of the luggage brought by each student.  Each student’s luggage weighed a different value.

The median weight is more than the lower quartile.  The range of weights is three times the interquartile range of weights.

Use the information above to complete the box plot.

Calculate the proportion of luggage weights which were less than .

Students had to pay an additional fee if the weight of their luggage exceeded 23 kg.

Find the number of students who had to pay the additional fee.

Remy was studying how long it takes each of his 80 rats to find the exit to a maze.  Every two and a half minutes he records the number of rats which have found the exit which he then represents as a cumulative frequency curve.

Based on the graph, write down an inequality for the time, t, taken by the fastest rat.  

Remy’s assistant also recorded the actual times taken by the fastest and slowest rats.  She has used this information to begin constructing a box plot to represent the data.

Use the cumulative frequency curve to complete the box plot for the times.

An annual cheese-rolling contest involves participants chasing a 4 kg round of cheese down a steep 200 yard long hillside.  A group of 60 friends participated in the contest and the table below summarises the distances travelled by each before first falling over.

How many of the 60 friends made it to the bottom without falling over?

Draw a cumulative frequency graph for the information in the table.

The steepest part of the hill is between 100 and 140 yards away from the start. Using your graph, estimate how many people fell during this section of the hill.

The histogram below shows the masses, in grams, of 80 apples.

Find estimates for the mean and standard deviation.

Given that an outlier is classified as any data value falling more than two and a half standard deviations away from the mean, show that there are no outliers in the data above.

Wendy is investigating the running times of 199 movies and finds the following information: 

   The interquartile range is 63 minutes. 

   The median splits the interquartile range in the ratio 3:4. 

   The middle 80% of the data covers a period of 123 minutes. 

On the grid below draw a box plot for the information above.

Aggie runs a local bowls club and wants to attract younger members.  First Aggie needs to find out some information about the ages of current members of the club.  Selecting 35 male members and 35 female members at random, Aggie illustrates the ages of the 70 members on an ordered back-to-back stem-and-leaf diagram as shown below. 

                                                                               (35 Female, 35 Male)

Compare the central tendency and variation of the male and female members’ ages.

The cumulative frequency graph below shows the IQs of 160 employees of a company.

Using the graph, estimate the number of employees whose IQ is within 5 of the median IQ.

The employees who are within the top 10% of IQs are offered training for a management role. Estimate the lowest IQ of an employee who is offered the training.

The employees who are within the bottom 5% of IQs must attend a meeting with the director of the company. Estimate the highest IQ of an employee who must attend a meeting with the director.

There are 180 dogs in a rescue shelter. The histogram below shows the highest sound level reached by each individual dog’s bark, measured in decibels (dB).

Write down the underlying feature associated with each of the bars in a histogram.

Estimate how many dogs had a bark which ranged between 99 dB and 107 dB.

Mr Shapesphere, a history teacher, records the time, to the nearest minute, it takes him to mark each student’s essay. The times were summarised in a grouped frequency table and an extract is shown below:

Time t (minutes)	Frequency
0 - 10	7
11 – 30	16
31 – 35	4

A histogram was drawn to represent these data. The 11 – 30 group was represented by a bar of width 6 cm and height 4.5 cm.

Find the width and height of the 0 – 10 group.

The total area under the histogram was 60.75 cm².

Find the number of essays which Mr Shapesphere recorded as taking longer than 35 minutes, to the nearest minute.

The grouped frequency table below contains information about the lengths of time of Susie’s calls with her customers. The table was used to draw the cumulative frequency curve also shown below.

Use the graph to find the values of .

Use the graph to calculate the interquartile range of times for Susie’s calls.

Use the graph to estimate the percentage of customers whose calls lasted longer than 10 minutes.

Crystal is given an incomplete box plot showing the lengths of 99 unicorn horns. She also knows that the median length is the midpoint of the minimum and maximum lengths and that the range is 2.5 times as big as the interquartile range.

Complete the diagrams below to show that there are two possible distributions given the information above.

The box plot below shows the masses of the 99 unicorn horns.

Crystal discovers that two masses were recorded incorrectly; 11 kg should have been 8 kg and 9 kg should have been 10 kg. 

Explain why at most one value will need to be changed to fix the box plot.

Explain why it is possible that the box plot will remain unchanged when it is fixed.

Football’s Premier League was launched in 1992 and the champions at the end of each season (up to and including the 2020-21 season) had scored the following number of goals:

                        67       80       80       73       76       68      80

                        97       79       79       74       73       72      72

                        83       80       68      103      78       93      86

                        102     73       68       85       106      95      85       83

(i) Justify the use of a stem-and-leaf diagram for these data.
(ii)   Draw a stretched, ordered stem-and-leaf diagram for these data.