Rugby World Cup stats

Purpose

This unit requires students to use statistics about the top ranked teams in the 2019 Rugby World Cup to predict the winner of the World Cup, justifying their prediction using data.

Specific Learning Outcomes
  • Recognise variables and measures in an existing data set.
  • Establish criteria and use these to sort data.
  • Display data using dot plots and scatterplots for numeric data, and bar and pie charts for categoric data.
  • Use graphs to make comparisons between groups.
  • Justify conclusions based on data.
Description of Mathematics

This unit requires students to use a data set containing statistics about the top ranked teams in the 2019 Rugby World Cup. Students will examine the variables, and consider the attribute that each variable measures. They will compare groups (e.g. nations, positions) using digital technology to graph distributions and calculate measures. Finally, they will use their knowledge to predict the winner of the World Cup, justifying their prediction using data.

Students will carry out a statistical investigation using the PPDAC cycle (Pose, Plan, Data, Analysis, Conclusion). The data set of statistics about players in the rugby world cup, 2019, is provided. Students will need to work out which variables they will use to make their predictions about a likely winner.

In doing so they may use measures of central tendency and spread to compare groups. Since much of the data is numeric students might calculate means, medians or modes as measures of centrality.

Mean – the sum of all scores divided by the number of scores

Median – the middle score when all the data is ordered by size

Mode – the most common score

Measures of centrality can be used to represent whole groups to compare those groups, e.g. Wales might have a mean age of 26.5 compared to New Zealand’s mean age of 27.5.

Range might be considered as a measure of variability, to indicate the spread in ages, e.g. South Africa’s age range might be 15 years and New Zealand’s might be 12 years.

Creating graphs of data also allows for ‘eyeballing’ the data to look for similarities and differences.

Required Resource Materials
Activity

Lesson One

Discuss the upcoming 2019 rugby World Cup tournament, that is being held in Japan.

Why is the tournament of interest to New Zealanders?

The world rankings for rugby going into the tournament are:

  1. Ireland
  2. New Zealand
  3. England
  4. Wales
  5. South Africa
  6. Australia
  7. Scotland
  8. France
  9. Fiji
  10. Japan

Note: These rankings are correct as of September 9 2019. If you are using this unit at a later date they are likely to have changed.

How do you think the rankings are decided?

Students will likely suggest that the record of wins and losses is used to set the rankings. For example, New Zealand losing to Australia cost it the number one ranking as it was overtaken by Wales. In fact, the system is more complicated than it first appears. Teams lose points if they lose, and gain points if they win. How many points they gain or lose depends on the difference in ranking of the teams, which team is playing at home, and if the winning margin is greater than 15 points. If students are interested you can read more at https://www.world.rugby/rankings/explanation.

Does this mean that Ireland will win the World Cup this year?

Discuss the idea that the rankings are based on form, but the result of an individual game can be unpredictable. Demonstrate how luck can play a significant part even when one team has a higher rank than the other.

New Zealand and Australia played in the last world cup final. Since then the sides have played ten test matches and New Zealand have won 8 of those games.  Let’s put ten cards, Ace, one, two … ten, into a box. Australia won games six and nine. If a six or nine comes out, then Australia win, and any other card is an All Black win.

Imagine that New Zealand and Australia are in the final of the 2019 World Cup. Can Australia win?

Let your students discuss the chances of Australia winning. Look for ideas like:

  • They probably won’t win.
  • It’s possible they can win in a ‘one -off’ game.
  • Their chances are 1 in 5 or one fifth.

Discuss the uncertainty of predicting the result even though the odds favour New Zealand. They are ranked second and Australia are ranked sixth. You might simulate a single game of drawing a card. Surprises do happen.

Actually, Australia won two of the last six test matches. Let’s change the cards to match that. (Six cards, Australia win on 2 and 5)

Discuss how the chances of an upset change. Simulate taking out a single card again.

Slide Two of PowerPoint One shows how the draw for the World Cup tournament works. Discuss what is meant by pool play and how the winners play runners-up in the Quarter Finals.

Suppose you are organising the tournament. You have a table of the top eight ranked teams.

How would you allocate the teams to the pools?

You want (if the team win as expected):
Teams ranked 1-8 to probably be in the Quarter Finals
Teams ranked 1-4 to probably be in the Semi Finals
Teams ranked 1 and 2 to probably be in the Final

Copymaster One gives an organised space for students to work in. Let students work together in pairs to allocate the teams to pools. Do they:

  • Work backwards from the outcome of Ireland and New Zealand in the final Ireland, New Zealand, England and Wales in the semi-finals?
  • Look to the rankings to give the winner of each game?

After an appropriate time bring the class together to share the draws they have created. You might compare their answers to the actual draw on Slide Three of PowerPoint One.

Finish the lesson with this question:

What data about the players might be useful to predict the winner of the World Cup?

Let students brainstorm in small groups about the data they might want to know. Accept attributes (characteristics) they suggest at this stage but do ask how that attribute might be measured. For example:

I want to know how fast players can run.

How would that be measured?

Make a list of attributes as a class. Ask students for justification about why that attribute might be important, and how they think the attribute might be measured, e.g. Rugby players frequently measure a time to sprint 40 metres from a standing start. (Why is that more useful than 100 metres?).

Lesson Two

In this lesson students explore tools that will be important to justifying their prediction about which team they think will win the World Cup. Use the Spreadsheet One.

Bring the spreadsheet up on a screen or interactive whiteboard.

What data have we got about each player in the New Zealand squad?

Spend time ensuring that students know what each variable is referring to and having a sense of the measures. For example:

  • How tall is a player who is 2.04 metres tall? What about 1.80 metres?
    (You might measure those heights and mark them on a door frame. Students might measure their own heights in metres.)
  • How heavy is a player who is 126?
    (That means 126 kilograms. While being sensitive to students’ personal feelings you might find two or three students whose combined weights equal that of the player.)
  • A full two litre bottle of milk weighs about 2 kilograms.
    How many full bottles does this player weigh? 63 bottles are a reasonable approximation of the player’s volume as well.
  • What is meant by ‘caps’? (Appearances for New Zealand)

In the next three parts of the lesson, which may span two sessions, students look at answering three types of questions:

  1. Summary questions about one group, such as:
    How many players are each position? (category data)
    What are the weights of the All Black players? (numeric data)
     
  2. Comparison questions about two or more groups, such as:
    Which position has the tallest players?
     
  3. Relationship questions about the association of one variable with another, such as:
    How are players’ weights related to their heights?
    How are players’ total points related to their number of caps?
     

Question One: How many players are in each position?

Sort the spreadsheet data by position. Ask the students to scan the list of All Blacks and create a summary table like this:

Position

Number of players (frequency)

Hooker

3

Prop

5

Lock

etc.

Loose forward

 

Scrum half

 

Fly half

 

Centre

 

Outside back

 

You may need to explain the positions to some students, e.g. fly half is often referred to as ‘number ten’ or ‘first five eighth’ in New Zealand.

Create two different graphs of these data.

Provide the students with grid paper if needed to draw their graphs. Watch for students to:

  • Choose an appropriate type of graph, such as a bar graph, strip graph, or pie chart.
  • Label axes and keys in their graphs clearly. (Pay attention to frequency on the vertical axis of a bar graph)
  • Give the graph a sensible title.
  • Format the graph so it clearly displays patterns and differences.

After a suitable time gather the class to share the students’ graphs. Slides Two and Three of PowerPoint Two provide model bar and pie graphs.

Which type of graph best answers the question, “How many players are in each position?”

Both graphs provide answers to the question, but the bar graph highlights differences in frequency better.

Which type of graph best answers the question, “What fraction of the players are forwards and what fraction are backs?”

 

Question Two: What are the weights of the All Black players?

Sort the spreadsheet data by weight.

How might we display these data?

The most useful graph type at this Level is a dot plot. (See Slide Four) The displayed graph avoids students attending to considerations when constructing it.

What is the first thing someone drawing this graph by hand would do? (create an axis from lowest to highest weight)

Copymaster Two provides three copies of a suitable axis (on copy per three students). Ask the students to create their own dot plot of the All Blacks’ weights using the data on the spreadsheet.

What statements can you make about the weights?

What would the central weight be? How could you measure that? (Discuss mean and median as measures of centrality. Possibly use Excel to calculate the mean after students have estimated. The median is the 16th weight in order since there are 31 players in the squad.)

Are there any outliers? (Brad Weber, at 75 kg is light compared to the others)
Who do you think might be the heaviest player?
What is the range between the lightest and heaviest players?

Slide Five shows a box and whisker graph of the weight data. This is a more sophisticated graph as it uses the summary statistics, minimum, lower quartile, median, upper quartile, and maximum, to represent the data. Confident students at Level Four can understand the idea that the weights have been put in order of size then divided into quarters.

 

Question Three: Which position has the tallest players?

Students will have their own ideas about the answer to this question. Locks and loose forwards are the ball winners in the lineout so tend to be taller than players in other positions.

Slide Six shows a dot plot with the positions sorted.

What positions have the tallest players? (As expected, the locks are the tallest, but loose forwards, and props have similar median heights)
What positions have the shortest players? (As expected, the scrum halves followed by the fly halves are the shortest groups)
You might find the median heights for each group, reading the height of the central player in each group.

Position

Median height

Scrum half

1.72m

Fly half

1.81m

Hooker

1.84m

Centre

1.86m

Outside back

1.86m

Loose forward

1.89m

Prop

1.89m

Lock

2.00m

 

Question Four: How are players’ weights related to their heights?

Relationship questions require looking for an association between two or more variables, in this case height and weight.

How could we display these data to look for a pattern?

Students may not be aware of scatterplots as a representation so you might use Slide Seven to build up how such a graph might be created.

Ask your students to create a scatterplot and graph the data of the six players shown on Slide Eight. After students create their own graph you might show them how Excel can be used to create a scatterplot. Use the data for the All Black squad.

A complete scatterplot is provided on Slide Nine.

Is there any pattern in this graph?
If we were to draw a line through the middle of the points where would it go?
What does the line show?

A line of best fit is called a regression line. In this case it shows a pattern that heavier players tend to be taller, though there is a lot of variation.
 

Question Five: How are players’ total points related to their number of caps?

The point of this question is to illustrate that there is often no clear association between variables. It makes intuitive sense to think that the more tests a rugby player participates in, the more points they are likely to score.

Slide ten shows a scatterplot of the association between number of caps and number of points. The line of regression is not shown but it is obvious that the slope of the line is close to zero (horizontal).

Is there a pattern in the points for this graph?
Are there any outliers in these data? (Beauden Barrett scored 617 points in 77 tests)
Why does playing more tests not necessarily lead to more points?

Students who are familiar with rugby will know that most points are scored by the fly halves, who tend to be the goal kickers.

Lesson Three

In this lesson students are encouraged to come up with a prediction about which team will win the World Cup. It is important that they use evidence, related to the variables in the data set.

Begin with a discussion.

How might we use these data to predict a winning team?

You may need to discuss that the prediction will always be a ‘best guess’ based on form, as chance and form will play a big part in the outcome.

Show the students Spreadsheet Two which has statistics on the squads of the six highest ranked team. Historically, the winner usually comes from one of those teams. Discuss the meaning of each variable, and how it is measured.

PowerPoint Three has some considerations about making a prediction. Take care to present the ideas as opinions, in order to open students’ minds to possible ways to predict the outcome.

Take some time now to think about what measures you will use to decide which team/s have the best chance of winning. Write some ideas down.
When you present your ideas in the next lesson you will need to justify your prediction.
Data displays will be important to convincing others that your method was sound.

Provide students with access to Spreadsheet Two. You may wish to convert it to a CSV (Comma Separated Variable) version to upload into an online graphing tool. Look for tools that can graph two or three variables simultaneously.

Let students work in teams of two with the graphing tools, Excel included, to create a way to predict the winning team. You might provide Copymaster Three as template to help students structure their responses. Watch as students work:

  • Can they justify their choice of variables and measures?
  • Do they use appropriate displays to represent the data, that allow them to compare teams?
  • Do they ‘eyeball’ the distributions to get a sense of possible differences.
  • Do they use software to calculate statistics, such as measures of centrality (e.g. mean, median) and spread (e.g. range).

Provide students appropriate time to investigate the data and create their report. If Copymaster Three is provided in Word form students can type within it and insert graphs and tables from the online tools. Snipping graphs is useful way to do this, though encourage students to insert the graphs/tables in text boxes for ease of formatting.

Lesson Four

In this lesson students are the consumers of others’ reports and are expected to use statistical literacy to evaluate the validity of findings. From the outset it is important for students to understand that there is ‘no one correct answer’ and that different choices and combinations of variables may lead to different predictions.

PowerPoint Four has an example of a report. Focus on the questions for discussion in the speech bubbles. Pay attention to justification, and sensible use of technology to present the data.

Put three teams of two students together to present their reports. Each team presents their findings to two other teams. Watch to see:

  • Do students focus on justification, especially where another team chooses measures that are different to their own?
  • Are they critical (in a positive way) about the use of graphs and statistics?
  • Do they provide quality feedback about the work of the other teams, including other points of view?

After an appropriate time bring the class together to discuss what they learned from the feedback.

What did you learn that might improve your report?

You might give students a chance to make alterations to their reports prior to submitting them. As a final exercise collate the predictions for winning team in a tally chart.

Do you think this table is a better predictor of the winning team than each individual prediction? Explain.

Display the reports from each pair of students on the wall.


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