# Is It Center or Is It Variability?

## Task

Statistical questions are questions that can be answered by collecting data and where we anticipate that there will be variability in that data. The data collected can be summarized in a distribution that can then be described in terms of center and in terms of spread. For some statistical questions, to answer the question you need to consider center. For other questions you might need to consider spread.

For each of the five statistical questions below, decide if you would answer the question by considering center or considering variability in the data distribution.

**Example 1**: The records office at an elementary school keeps daily attendance records.

**Question 1**: For students at this school, what is a typical number of school days missed in the month of April?

**Example 2: **Suppose that third graders at your school took both a math test and a social studies test. Scores on both tests could be any number between 0 and 100.

**Question 2**: On average, did the students score better on the math test or the social studies test?

**Question 3**: Were the students’ scores more consistent (more similar to one another) on the math test or on the social studies test?

**Example 3**: Bags of M&Ms don’t all have exactly the same number of candies in each bag. Suppose you count the number of candies in each of 25 bags of plain M&Ms and in each of 25 bags of peanut M&Ms, and make two dot plots—one for the number of candies in the plain M&M bags and one for the number of candies in the peanut M&M bags.

**Question 4**: If you wanted to give each student in your class a bag of M&Ms and you wanted to try to make sure that each student got the same number of candies, should you give them bags of plain M&Ms or bags of peanut M&Ms?

**Question 5**: If you wanted to give each student in your class a bag of M&Ms and you wanted to try to give students bags with the greatest number of candies, should you give them bags of plain M&Ms or bags of peanut M&Ms?

## IM Commentary

Standards 6.SP.A.2 and 6.SP.A.3 specify that students should understand that the set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape and that students should recognize that for numerical data, measures of center describe all of the values in a data set with a single number while measures of variability describe how the values in a data set vary. The purpose of this task is to challenge students to think about whether they should be most interested in the center of the data distribution or in the spread of a data distribution in order to answer a given statistical question.

Many students get the impression that center is the most important characteristic of a data distribution and always focus on center when presented with a data distribution. This task could be used as the basis of a classroom discussion in order to help students see that some questions are answered by considering variability rather than center.

This task may be difficult for students if it is the first time they have thought about statistical questions that focus on variability, so it probably works best as a small group activity or as the basis of a whole class discussion.

## Solution

Question 1: Because this question asks about a typical value, it would be answered by considering the center of the data distribution.

Question 2: Because this question is asking which of two groups has a higher score on average, it would be answered by comparing the centers of the two data distributions.

Question 3: Because the focus of this question is on consistency of scores, it would be answered by comparing the spreads of the two data distributions. The test with scores that are less variable would be the one that is the most consistent.

Question 4: In order to try to have each student get the same number of candies, the choice would be the one that has the smallest spread.

Question 5: To give students bags with the greatest number of candies, the choice would be the one that has the greatest center.

## Is It Center or Is It Variability?

Statistical questions are questions that can be answered by collecting data and where we anticipate that there will be variability in that data. The data collected can be summarized in a distribution that can then be described in terms of center and in terms of spread. For some statistical questions, to answer the question you need to consider center. For other questions you might need to consider spread.

For each of the five statistical questions below, decide if you would answer the question by considering center or considering variability in the data distribution.

**Example 1**: The records office at an elementary school keeps daily attendance records.

**Question 1**: For students at this school, what is a typical number of school days missed in the month of April?

**Example 2: **Suppose that third graders at your school took both a math test and a social studies test. Scores on both tests could be any number between 0 and 100.

**Question 2**: On average, did the students score better on the math test or the social studies test?

**Question 3**: Were the students’ scores more consistent (more similar to one another) on the math test or on the social studies test?

**Example 3**: Bags of M&Ms don’t all have exactly the same number of candies in each bag. Suppose you count the number of candies in each of 25 bags of plain M&Ms and in each of 25 bags of peanut M&Ms, and make two dot plots—one for the number of candies in the plain M&M bags and one for the number of candies in the peanut M&M bags.

**Question 4**: If you wanted to give each student in your class a bag of M&Ms and you wanted to try to make sure that each student got the same number of candies, should you give them bags of plain M&Ms or bags of peanut M&Ms?

**Question 5**: If you wanted to give each student in your class a bag of M&Ms and you wanted to try to give students bags with the greatest number of candies, should you give them bags of plain M&Ms or bags of peanut M&Ms?

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