Outliers

Tsena is researching the ages of Oscar winners.

She has collected data from the internet about the ages of the winners of the Best Actor and Best Actress categories from 2000 to 2022

Data from the internet is an example of secondary data.

Explain what is meant by secondary data.

Here are the ages of the winners of the Best Actor category from 2000 to 2022

40 36 47 29 43 37 38 45 50 48 60 50

39 55 44 32 41 41 59 37 83 53 54 (Source: www.openintro.org and www.oscars.org)

The stem and leaf diagram gives the ages of the winners of the Best Actress category from 2000 to 2022

Complete the stem and leaf diagram with the ages of the winners of the Best Actor category. You should also complete the key.

(Source: www.openintro.org and www.oscars.org)

Work out the median age for the winners of the Best Actress category from 2000 to 2022

Michelle thinks it is more appropriate to use the interquartile range than the range as a measure of spread for these sets of data.

Give a reason why Michelle may think this.

Work out the interquartile range for the winners of the Best Actress category from 2000 to 2022

The table gives information about the ages of the winners of the Best Actor and Best Actress categories before 2000 It also gives information about the ages of the winners of the Best Actor category from 2000 to 2022

Compare the distribution of ages for the winners of the Best Actor category with the distribution of ages for the winners of the Best Actress category. You may use the blank column in the table to record your answers to parts (c) and (e) to help you. You should interpret your comparisons in context.

David collected data on the times, in hours, taken for men to complete the Ironman Canada 2022 race and the Ironman UK 2022 race.

These graphs and summary statistics were produced for this data using statistical software.

Mean
13.53 Median
13.49 Number of male competitors 992

Standard deviation 1.839 Skewness 0.0653

Mean
13.98 Median
14.12 Number of male competitors 1128

Standard deviation 1.489 Skewness –0.282

(Source: www.coachcox.co.uk)

Show that there are no outliers in the times for the Ironman Canada 2022 men’s results.

Compare, in context, the distribution of the Ironman Canada 2022 men’s results with the distribution of the Ironman UK 2022 men’s results.

Dan competed in the Ironman Canada 2022 race.

Here is information about the performance of all male competitors in the swim, bike and run stages of the race, and some of Dan’s standardised scores.

Dan took 5.64 hours to complete the run.

Use standardised scores to compare Dan’s performances relative to the other competitors in each stage of the race. Give reasons for your comparisons.

The scatter diagram shows the marks each of ten students got for a Maths test and for a Music test.

One of these points is an outlier.

On the diagram, draw a circle around the outlier.

Excluding the outlier, describe the correlation shown in the diagram.

The outlier is removed from the data. The mean of the Maths marks is 52 The mean of the Music marks is 63

Excluding the outlier, draw a line of best fit that passes through the double mean point on the scatter diagram.

30 students did both tests. The scatter diagram shows the marks of 10 of these students.

Suggest one adjustment that could be made to improve the reliability of any conclusions made using the scatter diagram.

David asked 15 of his friends about the number of pets they each have. Here is the data he collected.

0 0 0 0 0 1 1 2 2 2 4 4 4 4 8

Circle the word in the list below that describes this type of data.

continuous qualitative discrete grouped

Write down the modal number of pets.

Find the median number of pets.

State which average, the mode or the median, best represents these data. Give a reason for your answer.

Find the interquartile range of the number of pets.

Wanda asked some of her friends about the number of pets they each have. The table below is a summary of the data she collected.

Compare the distribution of the numbers of pets for David with the distribution of the numbers of pets for Wanda. Give two comparisons and interpret each of your comparisons.

Wanda recorded the highest number of pets as 15 She says that this must be an outlier and concludes that it should be removed from her data.

Give one reason why Wanda’s conclusion may be appropriate.

Give one reason why Wanda’s conclusion may not be appropriate.

Sam used the internet to collect the times, in minutes, it took for 50 cyclists to compete in a hill climb competition. He used a group frequency table to record the results he collected.

Give one advantage of using grouped data rather than raw data.

Give one disadvantage of using grouped data rather than raw data.

Sam used this grouped frequency table to show the results for the hill climb.

Before Sam collected the data he did not know what the longest time would be. The longest time in the hill climb was 28.3 minutes.

Explain why this table cannot be used to show the data for all 50 riders.

Sam drew this frequency polygon for the hill climb results.

Sam decided not to include the value of 28.3 minutes on his frequency polygon.

Suggest a reason why Sam’s decision might be appropriate.

Describe the skew of the distribution.

Interpret the skew of the distribution in context.

Gaby is researching the final position of a football team in the English Premier League and the mean value of all the players in the team.

Suggest a diagram that Gaby could draw to see if there is an association between the final position and the mean value.

The table gives information about the data for the 2018–2019 season that Gaby used.

(Source: www.transfermarket.com)

Calculate Spearman’s rank correlation coefficient for the information in the table. You may complete the d column and d² column of the table to help you.

Interpret your answer to part (b) in the context of Gaby’s research. You should refer to the effects of any anomalous data.

Amelia suggests that Pearson’s product moment correlation coefficient should be used instead of Spearman’s rank correlation coefficient to measure the correlation between the data Gaby is researching.

Discuss whether or not Amelia’s suggestion is appropriate.

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

Irina is investigating whether the percentage of the population of a country living in urban areas has an effect on the life expectancy in that country.

Suggest a hypothesis Irina could use for her investigation.

Irina collected the following information about 10 countries.

The Urban population (%) refers to the percentage of the population of the country who live in urban areas.

Irina used statistical software to draw a scatter diagram for the information in the table.

Give a reason why a scatter diagram is an appropriate diagram to use.

For this investigation, which variable is the explanatory variable? Give a reason for your answer.

The scatter diagram from the statistical software is shown below.

Explain, giving a statistical reason, whether or not this scatter diagram supports your hypothesis in part (a).

For these 10 countries, the double mean point of the data is (63.9, 77.8).

Using this information, draw a line of best fit on the scatter diagram.

Using statistical software, Irina finds that the gradient of the line of best fit should be 0.19

Interpret the gradient of the line of best fit.

Irina now finds that South Africa has Urban population 65% and Life expectancy 63 years.

Determine how this information for South Africa fits with the relationship shown in the scatter diagram for the other countries.

Bethany is the union representative at the large company where she works. She is investigating the salaries of the employees at the company.

Bethany takes a simple random sample of 8 employees at the company and asks each of them to tell her their salary.

Describe what you understand by the term ‘simple random sample’.

The salaries of the 8 employees are listed below.

£16 000 £3 000 £23 000 £31 000

£21 000 £24 000 £17 000 £25 000

Work out the mean.

Show that the range is £28 000

Bethany believes that one of the salaries is an outlier.

Write down the salary that is most likely to be an outlier.

Explain why you think this value is an outlier.

Bethany removes the outlier and calculates the mean of the remaining seven salaries.

Without carrying out any further calculations, state whether the mean of the remaining seven salaries is greater than, is equal to or is less than the mean of all eight salaries. Give a reason for your answer.

Bethany uses the mean of the remaining seven salaries to draw conclusions about the salaries of all of the employees at her company.

Describe two things that could affect the reliability of her conclusions.

Anne is investigating the effect of caffeine on reaction times.

Here is what Anne has written down for her hypothesis and how she plans to collect, process and present her data.

Discuss whether Anne’s hypothesis and her plans for collecting, processing and presenting data are appropriate.

Anne is investigating the effect of caffeine on reaction times.

Here is what Anne has written down for her hypothesis and how she plans to collect, process and present her data.

Discuss whether Anne’s hypothesis and her plans for collecting, processing and presenting data are appropriate.

The glucose level in blood is measured in milligrams per 100 millilitres (mg/100 ml).

A doctor is investigating the glucose levels in blood of some adult females at the end of 12 hours of not eating or drinking.

The incomplete histogram and the incomplete grouped frequency table give information about the glucose levels in the blood of these adult females.

Glucose level in blood (mg/100 m l)

Glucose level in blood x (mg/100 ml)	Frequency
40 x 60	5
60 x 70	11
70 x 75	11
75 x 80	13
80 x 85	16
85 x 90
90 x 110

(Source: college.cengage.com)

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

The doctor used statistical software to produce the following summary statistics for his data.

By calculating limits for outliers using the mean and the standard deviation, explain whether or not there could be any outliers in the doctor’s data.

A second doctor is investigating the glucose levels in the blood of some adult males at the end of 12 hours of not eating or drinking.

The following summary statistics were calculated.

Calculate the skew for the distribution of the adult male glucose levels in blood and interpret this value in context. Give your answer correct to 2 decimal places.

Joel is investigating the heights of buildings in Paris and in London. He collects data on the heights of the tallest buildings in Paris and in London.

Joel decides to remove the height of the tallest building from each of his two data sets before carrying out the investigation.

Suggest a reason why it may be appropriate for Joel to do this.

The cumulative frequency diagram shows information about the heights, in metres, of a sample of 50 of the tallest buildings in Paris.

The box plot shows information about the heights, in metres, of a sample of 50 of the tallest buildings in London.

­ Height (metres)

Compare the distribution of the heights of the tallest buildings in Paris with the distribution of the heights of the tallest buildings in London.

You should give the value of any statistic that you use in your comparisons.

The cumulative frequency graph shows information about the maximum daily temperature, in °C, in Bingley for the 30 days of June 2018

Find the 90th percentile.

Interpret your 90th percentile in context.

For Bingley in June 2018, the lowest maximum daily temperature was 13 °C and the greatest maximum daily temperature was 28 °C.

Show that the greatest maximum daily temperature in June is not an outlier.

The box plot below shows information about the maximum daily temperature, in °C, in Aultbea for each day of June 2018

On the grid above, draw a box plot for the maximum daily temperatures in June 2018 for Bingley.

Compare the distributions of maximum daily temperatures in June 2018 for Aultbea and for Bingley.

Chris is carrying out an experiment to see if left handed people have a better memory than right handed people. He has two groups of people with the same number of people in each group.

Group A contains left handed people only and group B contains right handed people only.

Each person is given 15 objects to memorise and then they are tested to see how many objects they remember.

The variables for the experiment are shown in the table.

For each variable, put a tick () in the correct column of the table to show the type of variable.

Chris is not sure whether to work out the mean or the median number of objects each group remembered.

Describe a situation for which it would be more appropriate to work out the median than the mean.

Chris worked out these summary statistics for the number of objects people in group A remembered.

Lower quartile = 5 Median = 7 Upper quartile = 8

Sanjit is a member of group A.

He remembered 14 objects.

Determine whether or not the number of objects Sanjit remembered is an outlier.