Estimating Population Measures of Center
5 min
Teacher Prep
Setup
Narrative
This Warm-up asks students to decide whether to use the mean or median based on the distribution of the data. As students compare groups in this section, the choice of measure of center will be important.
Launch
Arrange students in groups of 2. Give students 1 minute quiet work time, followed by 2 minutes to discuss their work with a partner. Then follow with a whole-class discussion.
Student Task
Would you use the median or mean to describe the center of each data set? Explain your reasoning.
heights of 50 basketball players
ages of 30 people at a family dinner party
backpack weights of sixth-grade students
number of books students read over summer break
Sample Response
- Basketball players: Mean. Sample reasoning: The distribution is symmetric.
- Ages at a party: Median. Sample reasoning: The distribution is not symmetric.
- Backpacks: Median. Sample reasoning: The point at 16 would affect the mean much more than the median.
- Books: The mean can be used if it is known because the data set is approximately symmetric, but if what's given on the box plot is the only thing known, then the median should be used.
Activity Synthesis (Teacher Notes)
Select students to share their chosen measure of center and reasoning for their choice. Ask students what measures of variability should be used with each measure of center. Ask students,
- “How does the symmetry of the distribution affect which measure of center we might choose to represent what is typical for a group?” (If the distribution is not symmetric, median is often a better choice for the measure of center because values that are very far from the center of the distribution do not affect the median as much as they do the mean. If the distribution is symmetric, the mean is better because it uses information from all of the data collected rather than ignoring some values to find the middle.)
- “What measures of variability should be used with each measure of center?” (Mean absolute deviation, or MAD, is used with mean and interquartile range, while IQR is used with median.)
Standards
- 6.SP.5.d·Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- 6.SP.B.5.d·Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- 7.SP.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <em>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</em>
- 7.SP.4·Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <em>For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.</em>
- 7.SP.A.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <span>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</span>
- 7.SP.B.4·Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <span>For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.</span>
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Estimating Population Measures of Center
5 min
Teacher Prep
Setup
Narrative
This Warm-up asks students to decide whether to use the mean or median based on the distribution of the data. As students compare groups in this section, the choice of measure of center will be important.
Launch
Arrange students in groups of 2. Give students 1 minute quiet work time, followed by 2 minutes to discuss their work with a partner. Then follow with a whole-class discussion.
Student Task
Would you use the median or mean to describe the center of each data set? Explain your reasoning.
heights of 50 basketball players
ages of 30 people at a family dinner party
backpack weights of sixth-grade students
number of books students read over summer break
Sample Response
- Basketball players: Mean. Sample reasoning: The distribution is symmetric.
- Ages at a party: Median. Sample reasoning: The distribution is not symmetric.
- Backpacks: Median. Sample reasoning: The point at 16 would affect the mean much more than the median.
- Books: The mean can be used if it is known because the data set is approximately symmetric, but if what's given on the box plot is the only thing known, then the median should be used.
Activity Synthesis (Teacher Notes)
Select students to share their chosen measure of center and reasoning for their choice. Ask students what measures of variability should be used with each measure of center. Ask students,
- “How does the symmetry of the distribution affect which measure of center we might choose to represent what is typical for a group?” (If the distribution is not symmetric, median is often a better choice for the measure of center because values that are very far from the center of the distribution do not affect the median as much as they do the mean. If the distribution is symmetric, the mean is better because it uses information from all of the data collected rather than ignoring some values to find the middle.)
- “What measures of variability should be used with each measure of center?” (Mean absolute deviation, or MAD, is used with mean and interquartile range, while IQR is used with median.)
Standards
- 6.SP.5.d·Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- 6.SP.B.5.d·Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.
- 7.SP.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <em>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</em>
- 7.SP.4·Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <em>For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.</em>
- 7.SP.A.2·Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. <span>For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.</span>
- 7.SP.B.4·Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. <span>For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.</span>