Working with Data (CIE A Level Maths: Probability & Statistics 1)

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 x (interquartile range) above the upper quartile or less than 1.5 x  (interquartile range) below the lower quartile.

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

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

The cumulative frequency diagram below shows completion times for 100 competitors at the 2019 Rubik’s cube championships. The quickest completion time was 9.8 seconds and the slowest time was 52.4 seconds.

The grid below shows a box plot of the 2020 championship data.  Draw a box plot on the grid to represent the 2019 championship data.

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:

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

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:

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

An outlier is an observation which lies more than  ±2  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. 

The cumulative frequency diagram below shows the distribution of income of 120 managers across a supermarket chain.

The income of a sample of 120 other employees across the supermarket chain are recorded in the table below.

On the grid above, draw a cumulative frequency graph to show the data for the other employees and compare the income of managers and other employees.

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.

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.

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. 

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.

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

An outlier is an observation which lies more than  ± 2  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:

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.

Sam, a zoologist, is a member of a group researching the masses of gentoo penguins.  The research group takes a sample of 100 male and 100 female penguins and records their masses.

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.

Given that values are outliers if they are less than 4.2 kg or more than 8.5 kg, calculate the upper and lower quartiles for the mass of the 200 gentoo penguins.

Casey is another member of Sam’s research group.  She believes that the masses of male and female gentoo penguins follow different distributions.  The cumulative frequency graphs below show the masses of the male and female gentoo penguins in the sample.

By calculating a measure of central tendency and a measure of variation, compare the two distributions.

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  2 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.
   (i)     Explain the effect this new value would have on the median.
   (ii)    Explain the effects this new value could have on the lower quartile.

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: 

(i)   the modal class 

(ii)   the class containing the median

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.

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.

 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 data set. Calculate the mean and standard deviation for the remaining values.

The ages of 27 children invited to a birthday party are summarised in the table below. 

Given that the eldest child invited to the party was 7 years old, estimate the mean and standard deviation of the ages of the children.

Briefly describe any problems you encountered in answering part (a) and how you overcame them.

State one disadvantage of grouping data before analysing it.

The cumulative frequency graph below shows the information about the lengths of time taken for 80 students to run a lap of the sports hall.

Complete the table below:

Estimate the mean and the interquartile range.

Any times under 32 seconds or over 90 seconds are classed as outliers.
Estimate the percentage of the students’ times that are outliers.

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 a sample of 150 patients from both surgeries and finds their waiting times during their last visit.  Tim illustrates the data for each surgery using box-and-whisker plots as shown below.

Give a reason as to why box-and-whisker plots are suitable for Tim’s purposes.

Briefly describe why there is an individual data value for FitFirst denoted with a cross (×) on its box-and-whisker plot.

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 (  ) 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,  , 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).