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:

 = 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:

David and Bowey are planning a trip in June to Beijing, Jacksonville or Perth.  The temperature of the city and the atmospheric pressure will be deciding factors, so they investigate these three cities using all of the data for June 2015 from the large data set.

Using all of the days in June 2015, the following summary statistics for the daily mean air temperatures (t ℃) and the daily mean pressure (p hPa) are calculated:

	Daily mean air temperature		Daily Mean Pressure
Beijing			1004	3.81
Jacksonville	26.4	1.80	1017	1.88
Perth	14.8	2.37	1021	5.63

David also has the following information for Beijing in June:

∑t = 741.8  and  ∑t² = 18513.2

Calculate the values of   and  .

David suffers from headaches when the atmospheric pressure changes quickly so he would like to choose a city where the pressure does not vary a lot.  Additionally, Bowey does not like it when the temperature is higher than 30 ℃.  It is known that all the temperatures for Beijing in June 2015 were within 2 standard deviations of the mean, whereas in Jacksonville there were temperatures that were higher than the mean by more than 2 standard deviations. 

Suggest a city which both David and Bowey would be happy to visit.  Give reasons for your answer.

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.

Summary statistics from the large data set for the daily mean windspeed (knots) measured in Heathrow throughout October 1987 and October 2015 are given in the table below.

	Min	Max	Median
1987	2	16	5	185	1401
2015	3	10	6	197	1357

Calculate the mean of the daily mean windspeeds for each of the two years.

The standard deviation for 2015 was 1.84.

Calculate the standard deviation for 1987 and compare the daily mean windspeeds for each of the two years.

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.

Ororo, a meteorologist, is investigating the great storm of 1987 which devastated the south of England. Ororo would like to compare the daily maximum gust in Hurn during the months of October 1987 and October 2015.

Using your knowledge of the large data set

Ororo calculates the following summary statistics for the daily maximum gust in Hurn using the available data for October:

	Number of available days	Maximum value	x
1987	25	61	665	3462
2015	31	27	586	612.56

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

Compare the daily maximum gust in Hurn for October 1987 and October 2015.