Mathematics · Year 6 · Data, Chance and Probability · Term 3

Calculating Mean, Median, and Mode

Calculating averages to summarize data sets and identify outliers.

• ACARA Content Descriptions
• AC9M6ST02

About This Topic

Mean, median, and mode are measures of 'central tendency' used to summarize and describe data sets. Students learn to calculate the average (mean), find the middle value (median), and identify the most frequent result (mode). This topic, aligned with AC9M6ST02, also introduces the concept of 'outliers', extreme values that can skew the results. Understanding which measure to use in different situations is a key part of statistical literacy.

In Australia, students might use these measures to analyze house prices, cricket scores, or daily temperatures. They explore why a 'median' might be a fairer representation of income than a 'mean' if there are a few billionaires in the mix. This topic comes alive when students can collect data about themselves and use these tools to find the 'typical' student in their classroom.

Key Questions

1. Which measure of center is most affected by an extreme outlier?
2. Why might a researcher choose to report the median instead of the mean?
3. How do averages help us compare two different groups of data?

Learning Objectives

• Calculate the mean, median, and mode for a given data set.
• Identify outliers within a data set and explain their potential impact on the mean.
• Compare the mean, median, and mode to determine the most appropriate measure of central tendency for different data sets.
• Explain why the median might be a more suitable measure than the mean when a data set contains extreme values.

Before You Start

Ordering Numbers

Why: Students need to be able to order numbers from least to greatest to find the median.

Basic Arithmetic Operations (Addition, Division)

Why: Calculating the mean requires addition and division skills.

Identifying Patterns and Frequency

Why: Finding the mode requires students to identify which number occurs most often.

Key Vocabulary

Mean	The average of a data set, calculated by summing all values and dividing by the number of values.
Median	The middle value in a data set when the values are arranged in order. If there is an even number of values, it is the average of the two middle values.
Mode	The value that appears most frequently in a data set. A data set can have one mode, more than one mode, or no mode.
Outlier	A value in a data set that is significantly different from other values. Outliers can greatly affect the mean.

Watch Out for These Misconceptions

Common Misconception

The mean is always the 'best' average.

What to Teach Instead

Students often default to the mean. Use data sets with extreme outliers to show how the mean can be misleading. Peer discussion about 'is this number really typical?' helps them see the value of the median.

Common Misconception

To find the median, you just pick the middle number in the list.

What to Teach Instead

Students often forget to put the numbers in order from smallest to largest first. A physical activity where students line up by height before finding the 'middle person' reinforces this essential step.

Active Learning Ideas

Inquiry Circle: The Typical Year 6 Student

Students collect data on height, arm span, or number of siblings. In groups, they calculate the mean, median, and mode for each category and create a profile of the 'average' student.

50 min·Small Groups

Whole Class: The Outlier Effect

Calculate the mean height of a small group of students. Then, 'add' a giant (like a 3-meter tall fictional character) to the data and recalculate. Discuss how the mean changes while the median stays almost the same.

25 min·Whole Class

Think-Pair-Share: Which Average is Best?

Students are given scenarios (e.g., shoe sizes in a shop, test scores with one zero, house prices). They must decide whether mean, median, or mode is the most useful 'average' for that specific case.

20 min·Pairs

Real-World Connections

• Sports statisticians use mean, median, and mode to analyze player performance, such as the average points scored per game (mean) or the most common score in a series (mode). This helps in player evaluation and team strategy.
• Economists and financial advisors use these measures to understand income and wealth distribution. For instance, the median income is often reported to give a more representative picture than the mean, which can be skewed by a few very high earners.
• Meteorologists calculate the mean daily temperature for a month or year to identify climate trends and predict future weather patterns. They might also report the median temperature to understand typical conditions.

Assessment Ideas

Quick Check

Provide students with a small data set (e.g., 5-7 numbers including one outlier). Ask them to calculate the mean, median, and mode. Then, ask: 'Which measure is most affected by the outlier and why?'

Discussion Prompt

Present two scenarios: 1) The average height of students in Year 6. 2) The median house price in a local suburb. Ask students: 'Which scenario is better described by its mean, and which by its median? Justify your answers, considering potential outliers in each case.'

Exit Ticket

Give each student a card with a different data set. Ask them to calculate the mean, median, and mode. On the back, they should write one sentence explaining which measure best represents the 'typical' value in their data set and why.

Frequently Asked Questions

How can active learning help students understand mean, median, and mode?

Active learning turns data into a story about the students themselves. When they use their own heights or reaction times, they are personally invested in the 'typical' result. Physically lining up to find the median or using blocks to 'level out' columns to find the mean (the 'fair share' method) provides a visual and tactile understanding that goes beyond just following a calculation formula.

What is an outlier?

An outlier is a data point that is much higher or much lower than the rest of the values. It can 'pull' the mean away from the center.

When is the mode most useful?

The mode is useful for categorical data, like finding the most popular flavor of ice cream or the most common shoe size sold in a store.

How do you find the median if there are two middle numbers?

If there is an even number of data points, you find the mean (average) of the two middle numbers to get the median.