2.3 Working with Data

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.

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:

Bartholomew is a fan of BMW cars and is using the large data set to compare the masses of BMW cars in 2002 to 2016.

Using your knowledge of the large data set, explain why Bartholomew may not be able to include all the 2002 BMW cars from the large data set in his investigation.

Bartholomew cleans the data and calculates the following summary statistics:

	Min	Max	Number of cars
2002	1355	2180	126	199490	320295950
2016	1375	2350	410	668330	1109748800

Bartholomew uses the definition that an outlier is more than 2 standard deviations away from the mean.

Using this definition show that both years have outliers.

Compare the masses of BMW cars in 2002 and 2016.

Kristoff plays a game on his mobile phone where a player must match items together. Upon completion of the game, Kristoff receives a score out of 100. Kristoff plays the game 200 times and attempts to draw a box plot showing the distribution of his scores. However, Kristoff suspects that some of his scores may be outliers, so he defines an outlier as a score that is either more than 1.5 (interquartile range) above the upper quartile or more than 1.5 (interquartile range) below the lower quartile.

Kristoff's three worst scores were 12, 25 and 30 and his three best scores were 85, 93 and 96.

Complete the box plot of Kristoff s scores. Indicate any outliers by using a cross ().

Comment on the skewness of the distribution. Give a reason for your answer.

The game gives the player a gold medal if the player achieves a score of at least 70. After playing the game 200 times, Kristoff claims that he was awarded a gold medal 60 times.

Comment on the validity of Kristoff s claim.

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  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 is given below.

Calculate the mean and standard deviation of the engine sizes.

Any value more than two standard deviations from the mean can be identified as an outlier.

Using this definition of an outlier, show that one of the values from the sample of engine sizes is an outlier. Fully justify your answer.

Mr Wiltshire, a maths teacher, starts every Year 7 lesson by setting his 27 students the task of completing a randomised multiplication grid. Mr Wiltshire suspects that the time of day affects the speed at which his students complete the task. To test his suspicion, he records the times it takes the students to complete the multiplication grid in a morning lesson and then again in an afternoon lesson.

Below are the 27 times, in minutes, it took the students to complete the multiplication grid in the morning.

The grid below shows a box plot of the times in the afternoon lesson. Draw a box plot on the grid to represent the times in the morning lesson.

Explain what is represented by the cross () on the box plot for the afternoon lesson.

Comment on the skewness of the distribution of the afternoon times. Give a reason for your answer.

Compare the two distributions of times taken by the 27 students to complete a randomised multiplication grid in the morning and in the afternoon.

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.

Georgina is investigating the change in carbon monoxide emissions from cars between 2002 and 2016. She uses the large data set to select a random sample of 100 cars from each year.

Using your knowledge of the large data set, explain why Georgina must clean the data before taking a sample.

Georgina cleans the data and forms two samples and calculates the following summary statistics:

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 only one of the samples contains outliers.

Georgina removes the outliers from the relevant sample, describe what effect this has on the range and interquartile range.