12.1: First Name versus Last Name (5 minutes)
The purpose of this warm-up is for students to begin to see the need for samples of data when the population is too large. In this activity, students are asked to think about a question involving all the students at their school and compare the question to an earlier lesson in which the population was small and it was easy to obtain data for the entire population.
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Consider the question: In general, do the students at this school have more letters in their first name or last name? How many more letters?
What are some ways you might get some data to answer the question?
- The other day, we compared the heights of people on different teams and the lengths of songs on different albums. What makes this question about first and last names harder to answer than those questions?
The purpose of the discussion is to highlight the methods of getting data for the school more than it is the method of computing the answer.
Select some students to share their responses.
Students who have elected to sum all the letters in the first names in the school and all the letters in the last names in the school may note that it is a simple comparison to tell whether there are more in first or last names, since you get one single large number for each group. (Comparing data sets.)
Students who have elected to calculate the mean for each group and use MAD as a method of comparison may note that while the calculations may take more time, they give you more precise information, such as knowing about how long first names and last names are, as well as a way to compare the two sets. (Using the general rule from the previous lesson.)
Students who suggest surveying a small group of students may point out that it would be easier to do the calculation with a smaller group. The information would not be as accurate, but it would take a lot less time and might give a good general idea. It would depend on how accurate you needed your answer to be. (Introduction to sampling.)
12.2: John Jacobjingleheimerschmidt (10 minutes)
In this activity, students are asked to compare two groups (length of preferred names and last names) by collecting data from the class. They are asked if the data from the class gives enough information to draw a conclusion about a larger group (MP3). In the following activities, students will be introduced to the idea of sampling. This activity gives students the first chance to experience why sampling might be needed.
Compute the mean and MAD for the number of letters in each student’s preferred name (if students do not go by their first name, you may use their nickname, middle name, etc.). Do the same for their last names.
Give students 1 minute of quiet work time for the first 2 questions followed by a quick display of information then 5 more minutes of quiet work time and a whole-class discussion.
If a digital solution is available, input the data for the class to find the mean and mean absolute deviation for each data set. If a digital solution is not available, this information should be calculated based on the class roster prior to this activity. After students have had a minute to work on answering the first two questions, provide students with the mean and MAD for the names in the class.
Tell students that if they have a preferred name other than their official first name (nickname, middle name, etc.) they may use this in place of the first name.
Continue to consider the question from the warm-up: In general, do the students at this school have more letters in their first name or last name? How many more letters?
How many letters are in your first name? In your last name?
- Do the number of letters in your own first and last names give you enough information to make conclusions about students' names in your entire school? Explain your reasoning.
Your teacher will provide you with data from the class. Record the mean number of letters as well as the mean absolute deviation for each data set.
The first names of the students in your class.
The last names of the students in your class.
- Which mean is larger? By how much? What does this difference tell you about the situation?
- Do the mean numbers of letters in the first and last names for everyone in your class give you enough information to make conclusions about students’ names in your entire school? Explain your reasoning.
The purpose of the discussion is for students to see how the data they have might relate to a larger group. In particular, that a sample might give some estimate of a larger population, but the estimate should not be assumed to be exact.
Consider asking these questions for discussion:
- “Do you expect the mean length of first names for the school to be exactly the same as the mean length for the class?” (Probably not exactly the same. It may be close, though.)
- “Do you expect the mean length of first names for the school to be much larger or smaller or about the same as the mean length for the class? Explain your reasoning.” (Unless there are a few outliers in the class, it should be fairly close to the mean from the class.)
Design Principle(s): Support sense-making; Cultivate conversation
12.3: Siblings and Pets (10 minutes)
In this activity, students think a little more deeply about the data we would like to know and how that compares to the data we can collect easily and quickly (MP1). They are presented with a statistical question that does not have an obvious answer. Students are then asked to consider ways they might begin gathering data to answer the question, but are asked to realize that the data they could reasonably collect is not everyone addressed by the question. Following the activity, the discussion defines the terms population and sample.
Arrange students in groups of 2.
Set up the context by asking students, “Do people who are the only child have more pets?” then to provide a possible explanation for their answer. For example, maybe only children do have more pets because the family can better afford to take care of an animal with only 1 child. Maybe they do not because smaller families may live in smaller places and not have room for a lot of pets.
Give students 5 minutes of partner work time followed by 5 minutes of whole-class discussion.
Consider the question: Do people who are the only child have more pets?
- Earlier, we used information about the people in your class to answer a question about the entire school. Would surveying only the people in your class give you enough information to answer this new question? Explain your reasoning.
- If you had to have an answer to this question by the end of class today, how would you gather data to answer the question?
- If you could come back tomorrow with your answer to this question, how would you gather data to answer the question?
- If someone else in the class came back tomorrow with an answer that was different than yours, what would that mean? How would you determine which answer was better?
The purpose of the discussion is to show the difference between the data we would like to have to answer the question and the data we have available.
Some questions for discussion:
- “If we had all the time and money in the world and wanted to answer this question, who would we need to collect data from?” (Everyone in the world.)
- “What would you do with the data collected from everyone to answer the questions?” (Find the mean and MAD of the data from the two sets and compare them like we did in previous lessons.)
- “Why is it unreasonable to actually collect all the necessary data to answer the question?” (There are too many people to collect data from. There is not enough time to get to everyone in the world, and I cannot travel everywhere.)
- “Since it may be difficult to guess an answer without doing any research, but we cannot get all of the data we want, what data could you get that would help estimate an answer?” (It would be good to ask a few people in different parts of the world and try to get different groups represented.)
Define population and sample. A population is the entire pool from which data is taken. Examples include (depending on the question) “all humans in the world,” “all 7th graders at our school,” or “oak trees in North America.” In this usage, it does not have to refer only to groups of people or animals. A sample is the part of the population from which data is actually collected. Examples (related to the population examples) include “5 people from each country,” “the first 30 seventh graders to arrive at our school,” or “8 oak trees from the forest near our school.”
Ask students, "What is the population for the question about only children and their pets?" (Everyone in the world.) Note that we would need data from everyone, including those who don't have pets or do have siblings.
Ask students, "What might be a sample we could use to answer the question?" (The students in our class, my neighbors, a few people from different countries.) After getting several responses, ask, "What might be the benefits and drawbacks of each of these samples?" (Some may be more convenient, but would not represent the population as well or vice-versa.)
Explain: While it is best to have data for the entire population, there are many reasons to use a sample.
- More manageable. With very large populations, the amount of data can be hard to collect and work with, so a smaller subset may still be informative and easier to work with. Example: Find the average size of a grain of sand.
- Necessary. Sometimes it is impossible to reach the entire population, so a sample is all that is available. Example: Find the average lifespan of tuna.
- Speed. Sometimes a rough estimate is all that is needed and a sample of data is enough to estimate the population characteristic. Example: Out of curiosity, what is the median number of apps on smartphones.
- Cost. Sometimes it is very costly to obtain the data needed, so a sample can reduce the cost. Example: Find the average amount of hydrogen in moon rocks.
Supports accessibility for: Conceptual processing; Language
Design Principle(s): Support sense-making (for representation); Maximize meta-awareness
12.4: Sampling the Population (10 minutes)
This activity gives students the opportunity to practice the new vocabulary of population and sample by identifying the population from a set of questions and describing a possible sample that could be used to get some information to begin answering the question. Since these words have a very specific meaning in the context of statistics that is different from the colloquial use of the words, it is important for students to work with the vocabulary in specific situations to understand their meaning (MP6).
Arrange students in groups of 2. Allow students 3 minutes of quiet work time followed by 3 minutes of partner discussion then a whole-class discussion.
While in partner discussion, suggest students compare their answers and discuss any advantages or disadvantages for the samples they proposed.
For each question, identify the population and a possible sample.
- What is the mean number of pages for novels that were on the best seller list in the 1990s?
- What fraction of new cars sold between August 2010 and October 2016 were built in the United States?
- What is the median income for teachers in North America?
- What is the average lifespan of Tasmanian devils?
Are you ready for more?
Political parties often use samples to poll people about important issues. One common method is to call people and ask their opinions. In most places, though, they are not allowed to call cell phones. Explain how this restriction might lead to inaccurate samples of the population.
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The purpose of the discussion is to further solidify the meaning of the terms population and sample for students.
Consider asking these questions for discussion:
- “For each question, could there be another population than the one you gave?” (No. The population refers to all of the individuals that pertain to the question.)
- “For each question, could there be another sample than the one you gave?” (Yes. A sample refers to a few of the individuals from whom data will be collected and does not specify the number or how the individuals are selected.)
- “What are some of the advantages and disadvantages you determined for the samples you chose?” (Some are easy to work with, but might miss large sections of the population.)
- “What is a question you could ask for which the population would be all of the books in your house?” (For example, “What is the average number of pages in books in my house?”)
- “What is a question you could ask for which the sample could be all of the books in your house?” (For example, “What is the average number of pages in all the books ever written?”)
Explain that a well-phrased question should only have 1 population (a question that is not well-phrased should be reconsidered so that the purpose of the question is clear), but there are usually many ways to find samples within that population. In future lessons, we will explore some important aspects to consider while selecting a sample.
Consider asking these questions to reinforce the ideas from this lesson:
- “When the groups become too large, how can we obtain some data to begin answering a question about the group?”
- “What are some drawbacks of using samples instead of the entire population?” (The value for the measure of center will not be exact and some variability may be lost. Some groups may not have been included in the sample, so their input is lost.)
- “What are some reasons samples are necessary?” (More manageable, impossible to reach the entire population, speed, cost.)
- “Someone wants to know what breed of dog is most popular as a pet in the state. What is a sample that could be used?” (A few dog owners from each of the major cities in the state and a few dog owners from the rural areas.)
- “The principal of a school has access to the grades for students at the school. If we use these grades as a sample, what is a population that the data could be applied to?” (The entire school district, the state, the United States, or all students around the world.)
12.5: Cool-down - How Many Games? (5 minutes)
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Student Lesson Summary
A population is a set of people or things that we want to study. Here are some examples of populations:
- All people in the world
- All seventh graders at a school
- All apples grown in the U.S.
A sample is a subset of a population. Here are some examples of samples from the listed populations:
- The leaders of each country
- The seventh graders who are in band
- The apples in the school cafeteria
When we want to know more about a population but it is not feasible to collect data from everyone in the population, we often collect data from a sample. In the lessons that follow, we will learn more about how to pick a sample that can help answer questions about the entire population.