Working with Data (OCR A Level Maths: Statistics)

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?

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 (interquartile range) above the upper quartile or less than 1.5   (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  x= 5301  and  x²= 2 113 195,  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.

By removing any outliers identified in part (b), clean the data and recalculate the mean and standard deviation.

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.

An extract of data from the large data set on the number of people who work from home in 2011 in each region of England and Wales is given below.

Calculate the mean number of people per Local Authority who work from home.

Any value more than two standard deviations from the mean can be identified as an outlier. The mean number of people per region who work from home is 272 401 and the standard deviation is 111 488.6 to 1 decimal place.

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:

 = 286  and  ²= 4238

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 4.2kg or more than 8.5kg, 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:

‘-’ represents a job for which the length of time taken was not recorded.

An outlier is an observation which lies more than  ±2  standard deviations away from the mean.

By first cleaning the data, 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:

Doris is the owner ofa business in the northeast of the UK and she is considering the option for her employees of working from home if they wish. Doris uses the large data set to investigate the change in proportion of people in employment working from home between 2001 and 2011 in the northeast. Doris has the following information:

Outliers are values that are more than 2 standard deviations away from the mean percentages calculated in (a) (i).

Show that Northumberland is an outlier for both years.

Compare the distributions of the percentages of people in employment working from home in 2001 and 2011.

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.

For each scenario state, with a reason, whether the identified outlier should be included or excluded in the data set.

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:

Hence estimate the mean and the standard deviation of the times.

Given that the fastest time was 21 seconds and the slowest time was 100 seconds, show that these values are outliers using the definition that an outlier is more than 2 standard deviations away from the mean.

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.

Simon, an economist, is investigating the trends in employment rates in London and Wales. The large data set for 2001 does not show the number of people that are not in employment.

Below is an extract from the large data set from 2001:

Using your knowledge of the large data set, explain why there is not enough information in the table above to calculate the number of people that are not in employment.

Simon wants to compare unemployment between the two regions for each year.

Explain why Simon should use the proportions of unemployed people in each local authority instead of the number of unemployed people.

Simon calculates the median and interquartile range of the unemployment rates using all of the local authorities in London and Wales for the years 2001 and 2011.

Using the information in the table, compare the unemployment rates for London and Wales and discuss how they have changed between 2001 and 2011.