How Likely is That?

# 2. And the Average is...

Suggested Learning Intentions

• To describe, interpret and compare data using mode, median, mean and range
• To explain the difference between a population and a sample

Sample Success Criteria

• I can describe data using words such as mode, median, mean and range
• I can explain the difference between a population and a sample
• I can use a range of manipulatives to model and explain my thinking
• graphs and data from Stage 1. 'What does our village look like?' (or created data)
• Our Average Student: docx PDF
• Our Average Student - Simplified: docx PDF
• Counters, coloured tiles or blocks

In groups of three, students solve the following problem:

Three siblings go fishing and catch an average of five fish each. No one catches exactly five fish. How many fish could each sibling have caught? (Sullivan, n.d.).

This question supports students to understand that it is possible to have a set of data in which no person has the average amount. Many students will miss the ‘no one catches exactly five fish’ and automatically calculate that there were fifteen fish in total and then automatically divide fifteen by three.

Enable students by engaging them in role play. Students split into groups of three and each student has five fish (use counters or blocks). Encourage students to find different ways to share the fish amongst three people. This question also uses fish as an example to demonstrate to students that we cannot always break numbers into fractions or decimals - you cannot catch a fraction or a decimal of a fish.

This activity can be used as a standalone single lesson, or it can follow on from Stage 1.

If you teach this lesson as a single lesson, prepare ‘What does our village look like?’ pie charts with a variety of categorical and numerical data. Below is a list of a few options you may like to consider.

 Categorical Data Numerical Data Eye colour Hair colour Religion practiced Favourite fruit/ drink/ cereal Gender (M/F/Other) Dominant hand (left or right or ambidextrous) Number of siblings Shoe size Height

To start, display the pie charts from the previous lesson around the room. Remind students that these pie charts represent selection of a population, not the entire population.

Encourage students to walk around the classroom and study the pie charts.

• What do you notice?
• What is the same/different?
• Did you expect these results?
• What surprised you the most?
• Do you think if we surveyed students in another class, we would have similar results? What do you think might be different?
• Before looking at these pie charts, what did you think the most popular drink was?

Ask students to select five different pie charts, summarise the information and record it on an Our Average Student poster (differentiated posters are available in the Materials and texts section above). Students calculate the mode, mean, median and range. Students then record the value they think describes the average student for that attribute in the My Answer section.

Once students fill in their poster, have them describe the average student in their class.

Enable students by having them complete a simpler Average Student poster (Poster 1). Support student tos calculate the mode and then record the value they consider describes the average student for that attribute in the My Answer section.

Extend students by asking them to estimate how their data may be affected by the addition of one or more students to their data. Tell students that there are two new students who are enrolling at the school and that you want to add their data to the pie charts. Provide students with two pre-prepared sets of data for the imaginary students.

• How might these additions change the data?
• Do you expect the mode to change?
• Will the range remain the same?

Have students add the additional sets of data and compare their estimates to the new outcomes.

Areas for further exploration

Once each student has completed an Our Average Student poster, have them compare their poster with another one or two students in the class. Do they each have the exact same average student? Is it possible to have two or more posters each displaying a different average student? Why do they think this is possible?

It is possible for students to have a different average student. This is because each of the pie charts from the previous task potentially represent a different sample of the population

To further assess students' understanding of mean, median, mode and range provide students with the following scenario. Have students work collaboratively and encourage students to justify their answer mathematically and using manipulatives.

Provide students with the following scenario:

Jenny and Marisa have played 8 games of European Handball together. Marisa is the top goal-scorer for the team and scored the following goals over the 8 games.

?,   7,  8,   3,  9,   11,   2,   4

The club has provided the following data of Marisa’s scores:

Mean = 6, Median = 5.5, Range = 9

Both Jenny and Marisa have forgotten what Marisa scored in the first game of the season. Can you help?

Enable students by providing them with the following scenario:

Jenny and Marisa have played 7 games of European Handball together. Marisa is the top goal-scorer for the team and scored the following number of goals over the 7 games.

 Game Marisa's goal score 1 8 2 3 3 5 4 9 5 3 6 10 7 4

Marisa and Jenny have tried to summarise the data by displaying some average statistics. However, they have managed to mix up the numbers. Can you unscramble the numbers, and provide the correct statistics?

 Mean 3 Median 7 Mode 5 Range 6

Support students to order the data before they analyse them.

Extend by providing students with the following scenario. Have students consider what it means to be a good goal-scorer. Do you need to be consistent in the number of goals you score each week  or do you need to just score a lot of goals occasionally?

Marisa has a younger sister Thalia who also plays European Handball. Each sister thinks they are the better player. Marisa has scored a total of 34 goals in 6 games while Thalia has played 7 games and scored a total of 36 goals. Thalia is considered the top goal scorer out of the sisters. Use your knowledge of data and statistics to settle the argument.

Freund, M. et al., 2019. The prevalence and correlates of gambling in secondary school students in Victoria, Australia, 2017. [Online]