Mathematics · 6th Grade · Data Displays and Cumulative Review · Weeks 28-36

Choosing Appropriate Measures

Students will choose appropriate measures of center and variability based on the shape of the data distribution.

• Common Core State Standards
• CCSS.Math.Content.6.SP.B.5d

About This Topic

When students choose between mean and median, they are making a decision about which number best represents a typical value in a data set. In the US Common Core framework (CCSS.Math.Content.6.SP.B.5d), students are expected to understand that the shape of a distribution provides key evidence for this choice. A symmetric distribution with no extreme values makes the mean appropriate, while a skewed distribution or one with outliers calls for the median.

Similarly, choosing between range and interquartile range depends on whether extreme values are part of the story or a distraction from it. Range is easy to calculate and includes all values, but a single outlier can inflate it dramatically. The IQR describes the spread of the middle 50% of the data, making it resistant to extreme values and a more reliable companion to the median.

Active learning benefits this topic because choosing appropriate measures is fundamentally a judgment call. Students who debate and defend their choices with peers develop the reasoning they need far more than those who simply follow a memorized rule.

Key Questions

1. Justify the selection of mean or median for a given data set.
2. Differentiate when to use range versus interquartile range.
3. Critique the use of inappropriate measures for a particular data distribution.

Learning Objectives

• Justify the selection of mean or median as the most appropriate measure of center for a given data set, referencing its distribution shape.
• Differentiate between the appropriate use of range and interquartile range to describe data variability based on the presence of outliers.
• Critique the choice of a specific measure of center or variability for a given data set, explaining why it is or is not appropriate.
• Calculate the mean, median, range, and interquartile range for small data sets to support measure selection.

Before You Start

Calculating Mean, Median, and Range

Why: Students need to be able to compute these basic statistical measures before they can evaluate their appropriateness.

Identifying Data Distribution Shapes

Why: Students must be able to recognize symmetry and skewness in data displays to make informed choices about measures of center.

Key Vocabulary

Mean	The average of a data set, calculated by summing all values and dividing by the number of values. It is sensitive to extreme values.
Median	The middle value in a data set when the values are ordered from least to greatest. It is not affected by extreme values.
Range	The difference between the highest and lowest values in a data set. It is easily affected by outliers.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) of a data set. It represents the spread of the middle 50% of the data and is resistant to outliers.
Symmetric Distribution	A data distribution where the left and right sides are mirror images of each other, often bell-shaped. The mean and median are typically close in value.
Skewed Distribution	A data distribution that is not symmetric. In a right-skewed distribution, the tail extends to the right, and the mean is usually greater than the median. In a left-skewed distribution, the tail extends to the left, and the mean is usually less than the median.

Watch Out for These Misconceptions

Common Misconception

The mean is always the best measure of center because it uses all the data.

What to Teach Instead

A single large outlier can pull the mean far above where most data falls, making it unrepresentative. Concrete examples like income data or house prices help students see this. Having students calculate both measures for a skewed data set and compare them in pairs is more persuasive than stating the rule.

Common Misconception

Range is always sufficient for describing variability.

What to Teach Instead

Range only uses the two most extreme values and ignores everything in between. Two data sets can have the same range but very different distributions. Working through a paired counterexample, where both sets have a range of 20 but very different IQRs, makes this gap concrete and visible.

Common Misconception

Skew and outliers are the same thing.

What to Teach Instead

Outliers are individual data points far from the rest. Skew is an asymmetry in the overall shape of the distribution. They often occur together but are distinct concepts. Box plots and dot plots displayed side-by-side help students distinguish between a long tail and a single extreme value.

Active Learning Ideas

Gallery Walk: Measures Match-Up

Set up four stations, each displaying a different data set (symmetric, right-skewed, left-skewed, and outlier-heavy). Students visit each station, record which measure of center and variability they would use, and leave a sticky note with one sentence of justification. Whole-class debrief focuses on stations where groups disagreed.

40 min·Small Groups

Think-Pair-Share: The Outlier Problem

Present a data set of seven home prices where one is dramatically higher than the rest. Pairs calculate both the mean and the median, then discuss which better represents the typical home price and why. Each pair shares their reasoning before the class settles on a choice together.

20 min·Pairs

Collaborative Debate: Which Measure Wins?

Provide three data sets (annual salaries, test scores, home run counts) alongside journalist-style claims that each use the wrong measure of center or spread. Small groups identify the mismatch, correct it, and present their case using the data as evidence.

35 min·Small Groups

Card Sort: Range or IQR?

Students sort scenario cards (e.g., 'salaries at a company' versus 'quiz scores in a class') into categories based on whether range or IQR is the more useful measure of spread. Each decision must be defended in writing before pairs compare their sorted results.

25 min·Pairs

Real-World Connections

• A real estate agent analyzing home prices in a neighborhood must decide whether to report the mean or median price. If there are a few very expensive mansions, the median might better represent a typical home price for most buyers.
• A meteorologist reporting on daily temperatures might use the mean temperature for the month, but if there was one extreme heatwave, they might also report the median temperature or the IQR to show the typical spread of temperatures without being skewed by the outlier.

Assessment Ideas

Quick Check

Present students with two data sets: one symmetric with no outliers, and one skewed with an outlier. Ask them to write down which measure of center (mean or median) is more appropriate for each set and briefly justify their choice.

Discussion Prompt

Pose this scenario: 'A company reports that the average salary is $75,000. However, the median salary is $50,000. What does this tell you about the salary distribution? Which number do you think is a better representation of a typical employee's salary, and why?'

Exit Ticket

Provide students with a small data set (e.g., test scores). Ask them to calculate the range and the IQR. Then, ask them to explain which measure of spread (range or IQR) is more useful for this specific data set and why.

Frequently Asked Questions

When should you use median instead of mean for 6th grade statistics?

Use the median when the data set contains outliers or is clearly skewed, because the median stays near where most data falls regardless of extreme values. Use the mean when the distribution is roughly symmetric and no values are unusually far from the rest. Household income is a classic example where the median is more representative than the mean.

What is interquartile range and when is it used in 6th grade math?

The IQR is the difference between the third quartile (Q3) and first quartile (Q1). It describes the spread of the middle 50% of a data set and is not affected by outliers. It pairs naturally with the median as a measure of center and is most useful when data is skewed or contains extreme values that would distort the range.

How does active learning help students choose appropriate statistical measures?

Choosing the right measure is a judgment call, not a procedure to follow. Active learning tasks like debates, card sorts, and group data analysis require students to articulate and defend their reasoning. When students hear a peer make a different choice and explain why, they build the critical thinking needed to apply these concepts flexibly in new contexts.

How do you tell if a data distribution is skewed?

Look at the overall shape of the display. If the tail extends to the right, the data is right-skewed. If the tail extends to the left, the data is left-skewed. In right-skewed distributions, the mean is typically higher than the median. Histograms and dot plots are useful displays for observing the direction of skew visually.