Mathematics · Grade 7 · Data Analysis and Statistics · Term 4

Measures of Center: Mean, Median, Mode

Calculating and interpreting mean, median, and mode for various data sets.

• Ontario Curriculum Expectations
• 7.SP.B.4

About This Topic

Visualizing data in Grade 7 focuses on creating and interpreting sophisticated graphical representations like box plots and histograms. In the Ontario curriculum, students move beyond simple bar graphs to tools that show the distribution and quartiles of a data set. This topic is vital because it allows students to see patterns, trends, and outliers that are invisible in a list of numbers. It is a key skill for data literacy in the 21st century.

Students learn how to identify the 'shape' of data, whether it is symmetrical, skewed, or has multiple peaks. They also explore how different graphs can be used to tell different stories from the same data set, which is a critical part of media literacy. This topic comes alive when students can use digital tools or physical 'human graphs' to model their own data. Students grasp this concept faster through structured discussion and peer explanation.

Key Questions

1. Differentiate between mean, median, and mode and their appropriate uses.
2. Analyze how outliers affect each measure of center.
3. Justify when the median is a better representation of a 'typical' value than the mean.

Learning Objectives

• Calculate the mean, median, and mode for given data sets.
• Compare the mean, median, and mode to determine the most appropriate measure of center for a specific data set.
• Analyze the impact of outliers on the mean, median, and mode of a data set.
• Explain why the median is sometimes a better representation of a typical value than the mean, using examples.
• Differentiate between the appropriate uses of mean, median, and mode in various contexts.

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 Frequency

Why: Finding the mode requires students to identify which numbers appear most often in a set.

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 data point that is significantly different from other observations in the data set. Outliers can skew the mean.

Watch Out for These Misconceptions

Common Misconception

A longer 'whisker' in a box plot means there are more people in that section.

What to Teach Instead

Students often confuse length with frequency. Using a 'human box plot' helps them see that each of the four sections always contains 25% of the data, regardless of how long the whisker is; a longer whisker just means the data is more spread out.

Common Misconception

Histograms and bar graphs are the same.

What to Teach Instead

Students often put spaces between the bars of a histogram. Peer teaching about 'continuous' vs 'categorical' data helps them understand that histograms show ranges of numbers, so the bars must touch to show the flow of the data.

Active Learning Ideas

Simulation Game: Human Box Plot

Students line up by height. The class identifies the median, the minimum, the maximum, and the quartiles. They use a long rope to create a physical 'box and whiskers' around the students to visualize the four sections of the data.

30 min·Whole Class

Inquiry Circle: Histogram vs Bar Graph

Groups are given a large data set (e.g., ages of people in a community centre). They must create both a bar graph and a histogram. They then discuss which graph better shows the 'age groups' and why the intervals in a histogram are useful.

40 min·Small Groups

Gallery Walk: Misleading Graphs

Post various graphs from advertisements or news sites that use 'tricks' (like non-zero axes). Students walk around in pairs to 'debunk' the graphs and explain how they could be redrawn to be more honest.

35 min·Pairs

Real-World Connections

• Sports statisticians use the mean to report average player performance, like a basketball player's average points per game, but may use the median to understand typical player earnings to avoid skew from superstar salaries.
• Retail managers analyze sales data using mean, median, and mode to understand customer purchasing habits. For example, the mode might show the most popular item size, while the median helps set typical price points for promotions.

Assessment Ideas

Exit Ticket

Provide students with a small data set (e.g., test scores: 75, 80, 85, 90, 100). Ask them to calculate the mean, median, and mode. Then, ask: 'Which measure best represents a typical score for this set and why?'

Quick Check

Present two data sets: one with an outlier (e.g., ages: 10, 12, 11, 10, 50) and one without (e.g., ages: 10, 12, 11, 10, 13). Ask students to calculate the mean and median for both sets and write one sentence comparing how the outlier affected the mean.

Discussion Prompt

Pose the question: 'Imagine you are reporting the average salary for a company. Would you use the mean or the median if a few executives earn millions of dollars while most employees earn much less? Explain your reasoning, referring to the definitions of mean and median.'

Frequently Asked Questions

What is a box plot (box and whisker plot)?

A box plot is a graph that shows the five-number summary of a data set: minimum, first quartile, median, third quartile, and maximum. It is a great way to see the spread and symmetry of data at a glance.

What is the difference between a histogram and a bar graph?

Bar graphs are for categorical data (like 'favourite fruit'), while histograms are for continuous numerical data (like 'height ranges'). In a histogram, the bars touch to show that the data is part of a continuous scale.

How can active learning help students visualize data?

Active learning, like creating 'human graphs,' turns abstract lines into physical reality. When students are the data points, they can see exactly where the 'middle' is and how much space is between them. This makes the concept of quartiles and distribution much more intuitive.

Why do we use quartiles in Grade 7 math?

Quartiles help us understand the 'inner' part of the data. By looking at the middle 50% (the box in a box plot), we can see where most of the population falls without being distracted by extreme outliers at the ends.