Lesson 13

More Standard Deviation

13.1: Math Talk: Outlier Math (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings for computing values from expressions of the form \(a - 1.5\boldcdot b\). These understandings will be useful in a later lesson when students use expressions like \(\text{Q1} - 1.5 \boldcdot \text{IQR}\) to determine if values are outliers. 

In this activity, students have an opportunity to notice and make use of structure (MP7) because they are building up strategies that will enable them to both multiply by 1.5 and subtract that answer from another value.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Evaluate mentally.

\(0.5 \boldcdot 30\)

\(1.5 \boldcdot 30\)

\(100- 1.5 \boldcdot 30\)

\(100-1.5 \boldcdot 18\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation

13.2: Info Gap: African and Asian Elephants (30 minutes)

Activity

This is the first info gap activity in the course. See the launch for extended instructions for facilitating this activity successfully.

This info gap activity gives students an opportunity to determine and request the information needed to investigate and interpret measures of center and variability. 

The information gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

InfoGap problem and data cards.

Launch

This is the first time in this course that students do the information gap cards instructional routine, so it is important to demonstrate the routine in a whole-class discussion before they do the routine with each other.

Explain the info gap routine: students work with a partner. One partner gets a problem card with a math question that doesn’t have enough given information, and the other partner gets a data card with information relevant to the problem card. Students ask each other questions like “What information do you need?” and are expected to explain what they will do with the information.. Once the partner with the problem card has enough information to solve the problem, both partners can look at the problem card and solve the problem independently. This graphic illustrates a framework for the routine:

Info Gap Flowchart

Tell students that first, a demonstration will be conducted with the whole class. As a class, they are playing the role of the person with the problem card while you play the role of the person with the data card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question. 

Display this problem for all to see: 

100 captive Asian elephants and 100 wild Asian elephants are weighed. Is there a significant difference between the weights of the two groups of elephants? Explain your reasoning.

Explain that you will be playing the role of the person with the data card. Ask students, “What specific information do you need to find out if there is a significant difference between the weight of captive and wild Asian elephants?” Select students to ask their questions. Respond to each question with, “Why do you need that information?” Here is information from the data card. After this point, only answer the question if these data are relevant:

Data Card

  • The distribution of weights for each set of elephants is approximately symmetric.

Captive Asian elephants:

  • Mean weight of captive elephants: 3,073 kg
  • Standard deviation of captive elephant weights: 282 kg
  • Median weight of captive elephants: 3,055 kg
  • IQR of captive elephant weights: 399 kg

Wild Asian elephants:

  • Mean weight of wild elephants: 2,373 kg
  • Standard deviation of wild elephant weights: 121 kg
  • Median weight of wild elephants: 2,386 kg
  • IQR of wild elephant weights: 163 kg

Explain that if the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, “I don’t have that information.” Ask students to explain to their partner (you) how they used the information to solve the problem. (There is a significant difference in weights between captive and wild Asian elephants. Since the distributions are symmetric, it makes sense to use the mean and standard deviation weights for the two groups. The difference in means of 700 kilograms is very different, even in light of the standard deviations for the groups.)

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving measure of centers and variability. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation 
Representation: Provide Access for Perception. Display or provide students with a physical copy of the written directions and read them aloud. Check for understanding by inviting students to rephrase directions in their own words. Consider keeping the display of directions visible throughout the activity. 
Supports accessibility for: Language; Memory

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

  1. Silently read the information on your card.
  2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
  3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
  4. Read the problem card, and solve the problem independently.
  5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to answer the question.
  2. Ask your partner for the specific information that you need.
  3. Explain to your partner how you are using the information to solve the problem.
  4. When you have enough information, share the problem card with your partner, and solve the problem independently.
  5. Read the data card, and discuss your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

  • “What was interesting about Problem Card 1?” (It was interesting to use a real context to think about variability.)
  • “Was it more appropriate to use the mean or the median to compare the typical weights for Problem Card 1? Why?” (The mean because of the symmetry of the data.)
  • “Which measure of variability did you choose to compare for Problem Card 1? Why?” (The standard deviation, because I used the mean and the data has a symmetric distribution.)
  • “What was challenging about Problem Card 2?” (It was difficult to figure out how to make the dot plot without the actual data).
  • “How do you know your dot plot is correct?” (There is no way to know. It can only be approximated based on the mean, standard deviation, and the approximate shape.)

Highlight for students how they used the information about the center, shape and variability of the distribution to solve the problems.

13.3: Interpreting Measures of Center and Variability (10 minutes)

Optional activity

This task is provided for optional, additional practice for understanding variability in context.

The mathematical purpose of this activity is to compare data sets, and to interpret the measures of center and measures of variability in the context of the problem.

Launch

Keep students in the same groups. Give students 5 minutes to work on the questions and follow with a whole-class discussion.

Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames such as “In this situation, a larger/smaller value for the standard deviation means _____.” or “Both dot plots are different because….”
Supports accessibility for: Language; Organization

Student Facing

For each situation, you are given two graphs of data, a measure of center for each, and a measure of variability for each.

  • Interpret the measure of center in terms of the situation.
  • Interpret the measure of variability in terms of the situation.
  • Compare the two data sets.
  1. The heights of the 40 trees in each of two forests are collected.

    mean: 44.8 feet, standard deviation: 4.72 feet

    Dot plot.

    mean: 56.03 feet, standard deviation: 7.87 feet

    Dot plot.
  2. The number of minutes it takes Lin and Noah to finish their tests in German class is collected for the year.

    mean: 29.48 minutes, standard deviation: 5.44 minutes

    Dot Plot.

    mean: 28.44 minutes, standard deviation: 7.40 minutes

    Dot plot
  3. The number of raisins in a cereal with a name brand and the generic version of the same cereal are collected for several boxes.

    mean: 289.1 raisins, standard deviation: 19.8 raisins

    Histogram.

    mean: 249.17 raisins, standard deviation: 26.35 raisins

    Histogram.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

One use of standard deviation is it gives a natural scale as to how far above or below the mean a data point is. This is incredibly useful for comparing points from two different distributions.

For example, they say you cannot compare apples and oranges, but here is a way. The average weight of a granny smith weighs 128 grams with a standard deviation of about 10 grams. The average weight of a navel orange is 140 grams with a standard deviation of about 14 grams. If we have a 148 gram granny smith apple and a 161 gram navel orange, we might wonder which is larger for its species even though they are both about 20 grams above their respective mean. We could say that the apple, which is 2 standard deviations above its mean, is larger for its species than the orange, which is only 1.5 standard deviations above its mean.

  1. How many standard deviations above the mean height of a tree in forest A is its tallest tree?

  2. How many standard deviations above the mean height of a tree in forest B is its tallest tree?

  3. Which tree is taller in its forest?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Identify students who struggle making the connections between the summary statistics and the problem context. Ask them what information the standard deviation conveys. Tell them to focus less on the actual value of the standard deviation and more on which of the two distributions have a greater standard deviation and what that might mean.

Activity Synthesis

The purpose of this discussion is for students to explain the mean and the standard deviation in context.

For each set of graphs, select students read their answers for all three prompts. When necessary, prompt students to revise their language to include the terms shape, measure of center, and variability.

Speaking: MLR 8 Discussion Supports. Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language (shape, measure of center, and variability). Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking and to clarify, if necessary. Call students' attention to any words or phrases that helped clarify the original statement. This will provide more students with an opportunity to produce language as they explain the mean and standard deviation in context. Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

Here are some questions for discussion.

  • “What information does standard deviation tell you about a data set?” (It tells you how variable the data is relative to the mean.)
  • “Two professional race car drivers have the same average lap times after fifty laps. What does it mean to say that the first driver’s lap times have a greater standard deviation than the second driver’s lap times?” (It means that even though the drivers had the same average lap time, the first driver showed greater variability in the lap times of the 50 laps. The first driver has lap times that were slower and faster and less clustered near the mean than the second driver’s lap times.)
  • “In one class, students had an average height of 68 inches and a standard deviation of 5 inches. In a second class, the students had an average height of 67 inches and a standard deviation of 3.5 inches. What differences would you expect to see if you looked at the dot plots that represent the distributions of data for each class?” (I would expect that the data for the first class would be more spread apart and centered slightly to the right when compared to the dot plot for second class. I might expect the data values for the second class to be more tightly clustered around 67 inches than the first class’s points are clustered around 68 inches.)

13.4: Cool-down - Majors and Salaries (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

The more variation a distribution has, the greater the standard deviation. A more compact distribution will have a lesser standard deviation.

The first dot plot shows the number of points that a player on a basketball team made during each of 15 games. The second dot plot shows the number of points scored by another player during the same 15 games.

Dot plot.
Dot plot.

The data in the first plot has a mean of approximately 3.87 points and standard deviation of about 2.33 points. The data in the second plot has a mean of approximately 7.73 points and a standard deviation of approximately 4.67 points. The second distribution has greater variability than first distribution because the data is more spread out. This is shown in the standard deviation for the second distribution being greater than the standard deviation for the first distribution.

Standard deviation is calculated using the mean, so it makes sense to use it as a measure of variability when the mean is appropriate to use for the measure of center. In cases where the median is a more appropriate measure of center, the interquartile range is still a better measure of variability than standard deviation.