options: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
problem_activity: Activity
The goal of this activity is for students to be introduced to the idea that variability is important when analyzing and comparing data sets. Students will compare data sets, and make a decision as to which data set represents a better scenario. This activity prepares students to calculate variability in order to compare data sets in a later lesson. Monitor for students who calculate any
statistics
to inform their decisions. Make note of the students who only calculate a measure of center vs. students who calculate both a measure of center and a measure of variability.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Students will work individually. Tell them to use statistics to inform their decision.
Student Facing
In order from least to greatest, here are the prices per gallon of gas at two different gas stations over the past 7 days.
Station A:
2.38, 2.68, 2.82, 2.86, 2.99, 3.26, 3.59
Station B:
2.84, 2.85, 2.88, 2.95, 2.98, 3.03, 3.05
Suppose that these gas stations were the closest to your house, but not near each other. Which gas station would you go to for gas? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is to understand that the variability is worth paying attention to. If any students chose to compute the mean, median, IQR, MAD, or range for the two stations, ask them to share their results and how those values help them decide. Interestingly, the mean is the same for the gas stations. But any measure of variability shows that the prices at station B are more stable and predictable than the prices at station A. If no students suggest computing measures of center and variability and comparing them, be prepared to demonstrate.
Here are the summary statistics for each set of data:
For Station A:
Mean: 2.94
Five-number summary: 2.38, 2.68, 2.86, 3.26, 3.59
Range: 1.21
IQR: 0.58
MAD: 0.29
For Station B:
Mean: 2.94
Five-number summary: 2.84, 2.85, 2.95, 3.03, 3.05
Range: 0.21
IQR: 0.18
MAD: 0.07
If time permits, discuss which measures are helpful. Here are sample questions:
“Was the mean helpful in deciding the gas station from which you would purchase gas?”(No, because both means were the same.)
“How did the range (or IQR, MAD, five-number summary) help you decide which gas station you would go to?” (Whichever gas station had the highest range [or IQR, MAD] is the gas station I chose.)
“In the real world, how would you use the gas prices at each gas station to make a decision?” (I would think about which one generally had lower prices and choose that one, or I would see which one had more predictably cheaper prices and choose that one.)


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