Skewness

Benjamin collected data on the results for the male competitors in the Ironman Canada competition in 2022

The incomplete histogram and incomplete grouped frequency table give information about the times taken, in hours, for these competitors to complete the competition.

Time (t hours)	Frequency
8 t 9	2
9 t 10	23
10 t 11	58
11 t 12	134
12 t 13	188
13 t 14	183
14 t 15	..................
15 t 16	..................
16 t 17	102

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Benjamin also collected data about the results of the male competitors in the Ironman UK competition in 2022

The histogram below gives information about the times taken, in hours, for these competitors to complete the Ironman UK competition.

Identify and interpret the type of skew shown in this histogram of the times for these competitors in the Ironman UK competition.

The time taken, in hours, for each competitor to complete the competition was recorded when the digital tag worn by the competitor passed the finish line.

Discuss the reliability of any conclusions made based on this method of data collection. Give a reason for your answer.

Benjamin wants to find out how competitors prepare for Ironman competitions. He plans to give a questionnaire to the finishers of his local competition. There are 1200 people in the competition. Benjamin intends to use systematic sampling and ask every 10th competitor who crosses the finish line.

Comment on the appropriateness of this plan for data collection.

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 table gives information about the median house price in a town called Bulkington, in England, between the years 2015 and 2020

The table also gives, correct to one decimal place, some of the chain base index numbers for these prices.

Calculate the geometric mean of the chain base index numbers from 2016 to 2020 and interpret your result. You must show your working.

Here are some summary statistics for house prices in 2020 for all of England and Wales.

Calculate the skew of this distribution and interpret your result.

The time taken, in minutes, for some runners to complete a 5 km run was recorded. The incomplete histogram and incomplete grouped frequency table give information about the times taken, in minutes, for these runners to complete the 5 km run.

Time taken to run 5 km (t minutes)

Use the information in the histogram to complete the table.

Estimate the number of runners that took less than or equal to 23 minutes to complete the race.

Identify and interpret the skew shown on the histogram.

Kyle is investigating the heights and the weights of professional basketball players.

He found the weight, in kilograms, of some professional basketball players from 1950 to 1959

Circle the word in the list below that describes weight, in kilograms, as a type of data.

discrete continuous ordinal categorical

The incomplete histogram and incomplete grouped frequency table give information about the weights, in kilograms, of the professional basketball players from 1950 to 1959

Weight (w kilograms)
Frequency

65 w 70	5
70 w 75	15
75 w 80	61
80 w 85	81
85 w 90
90 w 95
95 w 100	35
100 w 105	14
105 w 110	9
110 w 115	1

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

Kyle also drew a histogram for the weights of professional basketball players from 2000 to 2009 This histogram was negatively skewed.

Interpret the negative skew of the weights of professional basketball players from 2000 to 2009

Kyle also collected data about the heights of professional basketball players from 1950 to 1959 and the heights of professional basketball players from 2000 to 2009

The grouped frequency table below gives information about the heights of professional

The estimate of the mean height for professional basketball players from 1950 to 1959 is calculated to be 190.9 cm to one decimal place.

Calculate an estimate of the mean height of basketball players from 2000 to 2009

Comment on how the mean height of professional basketball players has changed between the two sets of data.

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.

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

Weronika works for a road traffic organisation. One day she is investigating the speeds of cars and the speeds of motorcycles along a motorway.

Here is part of the spreadsheet that Weronika used to record her results.

Speed (s miles per hour)	Percentage of cars	Percentage of motorcycles
30 s 40	3	0
40 s 50	18	two
50 s 60	42	7
60 s 70	27	124
70 s 80	6	56
80 s 90		11
Total	100	100

Give a reason why Weronika will need to clean the data.

Weronika concludes that the value of 124 in the spreadsheet must be wrong.

Explain why.

Here is the information about motorcycles from the spreadsheet with the data cleaned.

Speed (s miles per hour)	Percentage of motorcycles
30 s 40	0
40 s 50	2
50 s 60	7
60 s 70	24
70 s 80	56
80 s 90	11
Total	100

Use linear interpolation to work out an estimate of the median speed of the motorcycles.

The frequency polygon for the speeds of cars has been drawn on the grid.

On the same grid, draw the frequency polygon for the speeds of the motorcycles.

Using the two frequency polygons, compare the skew of the distribution of the speeds of the cars with the skew of the distribution of the speeds of the motorcycles. Explain what your comparison means in context.

Viktoria collects information about the times taken, in seconds, by 50 boys to run 400 metres. She records her results in the grouped frequency table, shown below, and gives the grouped frequency table to her friend Emeka.

Time taken (t seconds)	Frequency (f)
50 t 60	1
60 t 65	5
65 t 70	15
70 t 75	17
75 t 80	12

Viktoria asks Emeka to use only the information in the grouped frequency table to work out an estimate of the standard deviation of the times taken.

Using Emeka’s estimates of the summary statistics, calculate an estimate of the skew of the distribution of the times taken by the 50 boys. Give your answer to 2 decimal places.

Interpret your answer to part (b) in the context of the data in the table.

Aminah collected information about the mean daily temperature, in C, in August in Scotland for the years 1910 to 2018

The table shows a summary of the data produced by statistical software.

(Source: https://www.metoffice.gov.uk/climate/uk/summaries/datasets)

On the grid below, draw a box plot for the information in the table.

Calculate the interquartile range for the data in the table. You must show your working.

Describe the skew of the distribution represented by the box plot. Give a reason for your answer.

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 Forestry Commission planted Field Maple trees and Silverleaf Maple trees in region A. They measured the heights of the trees after 14 years.

The incomplete histogram and incomplete grouped frequency table give information about the heights, in metres, of the Field Maple trees in region A.

Height (h metres)	Frequency
0.8 h 1.2	2
1.2 h 1.6	4
1.6 h 2.0	13
2.0 h 2.4	25
2.4 h 2.8	10
2.8 h 3.2
3.2 h 3.6

(Source: data.gov.uk)

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

The histogram below gives information about the heights of the Silverleaf Maple trees after 14 years.

Identify and interpret the type of skew shown in the histogram for Silverleaf Maple trees.

The Forestry Commission also planted Field Maple trees and Silverleaf Maple trees in region B. The grouped frequency table below gives information about the heights of these trees.

		Frequency
Height (h centimetres)	Field Maple	Silverleaf Maple
0 h 80	1	1
80 h 160	14	4
160 h 240	27	32
240 h 320	21	13
Total	63	50

The estimate of the mean for Field Maple trees is calculated to be 206.3 cm to 1 decimal place.

David thinks that the estimate of the means for Field Maple trees and for Silverleaf Maple trees suggests that Field Maple trees are taller than Silverleaf Maple trees.

Is David correct? You must show your working. Give one limitation of your conclusion.

The Forestry Commission planted Field Maple trees and Silverleaf Maple trees in region A. They measured the heights of the trees after 14 years.

The incomplete histogram and incomplete grouped frequency table give information about the heights, in metres, of the Field Maple trees in region A.

Height (h metres)	Frequency
0.8 h 1.2	2
1.2 h 1.6	4
1.6 h 2.0	13
2.0 h 2.4	25
2.4 h 2.8	10
2.8 h 3.2
3.2 h 3.6

(Source: data.gov.uk)

Use the information in the histogram to complete the table.

Use the information in the table to complete the histogram.

The histogram below gives information about the heights of the Silverleaf Maple trees after 14 years.

Identify and interpret the type of skew shown in the histogram for Silverleaf Maple trees.

The Forestry Commission also planted Field Maple trees and Silverleaf Maple trees in region B. The grouped frequency table below gives information about the heights of these trees.

		Frequency
Height (h centimetres)	Field Maple	Silverleaf Maple
0 h 80	1	1
80 h 160	14	4
160 h 240	27	32
240 h 320	21	13
Total	63	50

The estimate of the mean for Field Maple trees is calculated to be 206.3 cm to 1 decimal place.

David thinks that the estimate of the means for Field Maple trees and for Silverleaf Maple trees suggests that Field Maple trees are taller than Silverleaf Maple trees.

Is David correct? You must show your working. Give one limitation of your conclusion.

Bella collected the finish times, x minutes, of the men’s elite wheelchair race at the 2018 London Marathon.

She uses statistical software to calculate the following summary statistics.

Number of athletes = 39
Median = 97 minutes

(Source: http://results-2018.virginmoneylondonmarathon.com)

Calculate the skew for the distribution of the finish times and interpret this value in the context of Bella’s data.

The table shows information about the amount of time that each member of a group of 46 teenagers spent on social media during one day.

Time spent on social media (t minutes)	Frequency
0 t 50	1
50 t 100	4
100 t 150	8
150 t 200	17
200 t 300	16

Use linear interpolation to find an estimate of the median time spent on social media by the 46 teenagers.

The average person will spend approximately 116 minutes on social media each day.

(Source: www.socialmediatoday.com)

Compare the amount of time spent on social media by the 46 teenagers with the amount of time spent on social media by the average person.

The table below gives the mean, the standard deviation and the median for the times spent on social media during one day by a sample of sixty year olds.

Calculate the skew of the times spent on social media by the sample of sixty year olds.

Interpret your answer to part (b).