Working with Data (Edexcel International A Level Maths: 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.

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.

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:

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

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 lengths of unicorn horns are measured in cm.  For a group of adult unicorns, the lower quartile was 87 cm and the upper quartile was 123 cm.  For a group of adolescent unicorns, the lower quartile was 33 cm and the upper quartile was 55 cm.

An outlier is an observation that falls either more than 1.5 (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile.

Which of the following adult unicorn horn lengths would be considered outliers?

32 cm                96 cm                123 cm              188 cm

Which of the following adolescent unicorn horn lengths would be considered outliers?

12 cm                52 cm                86 cm                108 cm

Students at two Karate Schools, Miyagi Dojo and Cobra Kicks, measured the force of a particular style of hit. Summary statistics for the force, in newtons, with which the students could hit are shown in the table below:

Miyagi Dojo	12	21873	41532545
Cobra Kicks	17	29520	52330890

The heights, in metres, of a flock of 20 flamingos are recorded and shown below:

0.4        0.9        1.0        1.0        1.2        1.2        1.2        1.2        1.2         1.2

1.3        1.3        1.3        1.4        1.4        1.4        1.4        1.5        1.5         1.6

An outlier is an observation that falls either more than 1.5  (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile.

Find the values of and .

Using your answers to part (a), draw a box plot for the data.

The number of daily Covid-19 vaccinations reported by one vaccination centre over a 14-day period are given below:

237       264       308       313       319       352       378

378       405       421       428       450       465       583

Given that  and  ,  calculate the mean and standard deviation for the number of daily vaccinations.

An outlier is an observation which lies more than  standard deviations away from the mean.

Identify any outliers for this data.

Outliers are to be removed from the data, and the mean and standard deviation recalculated. Without making any further calculations, state the effect on the value of the mean and standard deviation removing outliers would have.

A disgruntled chocaholic is complaining that their favourite tub of chocolates seems to have very few of their favourite ‘toffee deluxe’ sweets in, compared to the other four available. In a bid to show they have a valid complaint, the chocaholic bought 10 tubs of the sweets and counted how many of each of the five sweets are contained in each tub.

The results are summarised in the box and whisker diagrams below.

What value is indicated by the () on the ‘toffee deluxe’ box plot and why has this been plotted individually?

A café owner is analysing data about the number of customers they serve on weekdays during the busy lunchtime period between 12pm and 2pm. The data below shows the number of customers who were served between 12pm and 2pm each weekday over a three-week period.

52         64         58         49         52

71         63         52         56         58

53         52         47         68         56

Given the summary statistics and  , find the mean and the standard deviation.

David buys and sells small antique items as a hobby, aiming to make a few hundred pounds profit each month.  Over the last two years David has kept a record of how much profit he has made each month  and wants to analyse the figures by calculating his mean monthly profit – and, to account for the variation in the antiques market – the standard deviation of his monthly profit.

David codes the data using the relationship .
For one of the months, the  value is . Work out the profit David made in this month.

Given that the mean of is  and the standard deviation of  is 0.8, find the mean amount of profit David made and the standard deviation for the two-year period.

The stem-and-leaf diagram below shows the number of attempts 27 gamers took to complete the last level of a computer game.

Find the median, the lower quartile and the upper quartile.

By comparing the median with the quartiles, comment on the skewness of the data.

The computer game developer decided that the last level was not difficult enough so added an extra, even harder, level. Briefly describe what you would expect to happen to the median if the number of attempts taken by the same 27 gamers to complete the new last level were recorded.

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.

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 amount.

The median weight is 4 kg 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 20 kg.

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.

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

Find estimates for the median, lower quartile and upper quartile.

Given that the lightest apple weighs 41 g and that the range of masses is 97 g, draw a box plot to show the distribution of the masses of the apples.

As part of an experiment, 15 maths teachers are asked to solve a riddle and their times, in minutes, are recorded:

8            12         19         20         20

21         22         23         23         23

25         26         27         37         39

An outlier is an observation which lies more than  standard deviations away from the mean.

Show that there is exactly one outlier.

State, with a reason, whether the mean or the median would be the most suitable measure of central tendency for these data.

15 history teachers also completed the riddle; their times are shown below in the box plot:

Explain what the cross (×) represents on the box plot above. Interpret this in context.

By comparing the distributions of times taken to complete the riddle, decide which set of teachers were faster at solving the riddle.

Hugo, a newly appointed HR administrator for a company, has been asked to investigate the number of absences within the IT department. The department contains 23 employees, and the box plot below summarises the data for the number of days that individual employees were absent during the previous quarter.

An outlier is an observation that falls either more than 1.5  (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile.

Show that these data have an outlier, and state its value.

For the 23 employees within the department, Hugo has the summary statistics:

and  

Hugo investigates the employee corresponding to the outlier value found in part (a) and discovers that this employee had a long-term illness.  Hugo decides not to include that value in the data for the department.

Assuming that there are no other outliers, calculate the mean and standard deviation of the number of days absent for the remaining employees.

Ms Chew is an accountant who is examining the length of time it takes her to complete jobs for her clients. Ms Chew looks at her spreadsheet and lists the number of hours it took her to complete her last 12 jobs:

An outlier is an observation which lies more than  standard deviations away from the mean.

Show that 21 is the only outlier.

Ms Chew looks at her handwritten records and finds that the value 21 was typed into the spreadsheet incorrectly. It should have been 12.

Without further calculations, explain the effect this would have on the:

Jan is concerned about the availability of toilet rolls following news of supply issues due to a lack of lorry drivers.  Jan visits 19 shops, starting at the local village store and visiting numerous shops in the local town centre, before travelling to some of the larger out of town supermarkets.  At each shop, Jan counts the number of standard 4 packs of toilet rolls available for customers to buy.

Jan records the results in a stem-and-leaf diagram, shown below.

Show that the median is 29 standard 4 packs and find the lower quartile.

Jan later visits another shop and again counts the number of standard 4 packs of toilet rolls available. Jan did not have anywhere to record the number and later forgets, only recalling that it was between 10 and 19.

By considering the range of the number of standard 4 packs of toilet rolls available at the shops Jan visited, suggest a problem with Jan’s method of collecting the data.

A road safety team are investigating how fast vehicles are travelling along a village road with a speed limit of 40 mph.  The team record the speeds of 120 vehicles travelling along the road one day during the busy morning rush hour period.

The histogram below shows the speeds of the 120 vehicles.

Determine:

The road safety team decide they will recommend anti-speeding measures if 30% or more of the recorded speeds are higher than 2.5 mph less than the speed limit.

Determine whether the team will be recommending anti-speeding measures.

Andrew is investigating the life expectancy at birth for countries in Europe and Asia.

He takes a random sample of 11 countries from each continent, and using data from a reliable online source he notes the life expectancies at birth from the year 2010 for each of the countries in his samples.

Andrew codes the data for Asia, using the formula ,  where  is the life expectancy and  and  are positive integers.

Given that the interquartile ranges for  and  are 5.23 and 523 respectively, write down the value of .

Given further that the medians for  and  are 74.15 and  respectively, find the value of .

Andrew uses the same coding formula for the 11 European countries and calculates the coded median and interquartile range to be  and 551 respectively.
Compare the central tendency and variation of the life expectancies at birth in 2010 for the European and Asian countries in Andrew’s samples.

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.

Compare the skewness of the two distributions.

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:

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.

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

Marya is consistently late for work. David, Marya’s boss, records the number of minutes that she is late during the next six days. David calculates the mean is 18 minutes and the variance is 210 minutes². On one of the six days, Marya was 50 minutes late.

Show that 50 is an outlier, using the definition that outliers are more than 2 standard deviations away from the mean.

Marya states that the 50 minutes should not be included as it is an outlier.

Marya tells David that she was 50 minutes late that day due to a road accident, she shows David the traffic report as evidence.

David agrees to remove the 50 from the dataset, calculate the new mean and standard deviation for the remaining values.

Tim has just moved to a new town and is trying to choose a doctor’s surgery to join, HealthHut or FitFirst. He wants to register with the one where patients get seen faster. He takes of sample of 150 patients from HealthHut and calculates the range of waiting times as 45 minutes and the variance as 121 minutes².

An outlier is defined as a value which is more than 2 standard deviations away from the mean.

Prove that the sample contains an outlier.

Tim finds out that the outlier is a valid piece of data and decides to keep the value in his sample.

Which pair of statistical measures would be more appropriate to use when using the sample to compare the doctor’s surgeries: the mean and standard deviation or the median and interquartile range? Give a reason for your answer.

The box plots below show the waiting times for the two surgeries.

Given that there is only one outlier for HealthHut, label it on the box plot with a cross ().

Compare the two distributions of waiting times in context.

The captain of a ferry that transports vehicles is investigating the masses of cars in the UK. They collect data on the masses ( kg) of 30 randomly selected cars.

The data can be summarised by  and  , where  is a constant.

Given that the mean mass of the 30 cars,  kg, find the value of the constant  and the standard deviation of the masses of the 30 cars.

Given that an outlier is defined as any value  satisfying  find the lowest and highest masses (to the nearest whole kilogram) that are not considered outliers.

In a safety test, firemen carry out an experiment where a sofa in a typical living room setup is set on fire and allowed to burn.  After 10 minutes the fire is extinguished and a fire damage expert assesses and records the percentage of the room destroyed.
The firemen repeat this experiment with identical equipment except that the sofa is treated with a fire-resistant chemical.

This process is repeated several times and the percentages recorded are listed below.

 Without fire-resistant chemical (%):           85         76         83         48              91

80         67         82         79              85

                                                                   76         72         84         76              69

With fire-resistant chemical (%):               40         51         52         48              60

                                                                  48         77         46         40              51

                                                                  54         51         46         45              49

Draw an ordered, back-to-back stem and leaf diagram to illustrate the data.

Find the median destruction percentages for both “without chemical” and “with chemical”.

It is later discovered that two of the values were muddled up.  The reading of 48% for “without chemical” should have been in the “with chemical” data whilst the 77% for “with chemical” should have been in the “without chemical” data.
Explain how correcting these errors will affect your answers to part (b).