## Lesson

Let's use dot plots to describe distributions and answer questions.

Exercise (PageIndex{1}): Packs on Backs

This dot plot shows the weights of backpacks, in kilograms, of 50 sixth-grade students at a school in New Zealand.

- The dot plot shows several dots at 0 kilograms. What could a value of 0 mean in this context?
- Clare and Tyler studied the dot plot.
- Clare said, “I think we can use 3 kilograms to describe a typical backpack weight of the group because it represents 20%—or the largest portion—of the data.”
- Tyler disagreed and said, “I think 3 kilograms is too low to describe a typical weight. Half of the dots are for backpacks that are heavier than 3 kilograms, so I would use a larger value.”

Do you agree with either of them? Explain your reasoning.

Exercise (PageIndex{2}): On the Phone

Twenty-five sixth-grade students were asked to estimate how many hours a week they spend talking on the phone. This dot plot represents their reported number of hours of phone usage per week.

- How many of the students reported not talking on the phone during the week? Explain how you know.
- What percentage of the students reported not talking on the phone?

- What is the largest number of hours a student spent talking on the phone per week?
- What percentage of the group reported talking on the phone for this amount of time?

- How many hours would you say that these students typically spend talking on the phone?
- How many minutes per day would that be?

- How would you describe the
**spread**of the data? Would you consider these students’ amounts of time on the phone to be alike or different? Explain your reasoning. - Here is the dot plot from an earlier activity. It shows the number of hours per week the same group of 25 sixth-grade students reported spending on homework.

- How would you describe the

Overall, are these students more alike in the amount of time they spend talking on the phone or in the amount of time they spend on homework? Explain your reasoning.

- Suppose someone claimed that these sixth-grade students spend too much time on the phone. Do you agree? Use your analysis of the dot plot to support your answer.

Exercise (PageIndex{3}): Click-Clack

- A keyboarding teacher wondered: “Do typing speeds of students improve after taking a keyboarding course?” Explain why her question is a statistical question.
- The teacher recorded the number of words that her students could type per minute at the beginning of a course and again at the end. The two dot plots show the two data sets.

Based on the dot plots, do you agree with each of the following statements about this group of students? Be prepared to explain your reasoning.

- Overall, the students’ typing speed did not improve. They typed at the same speed at the end of the course as they did at the beginning.
- 20 words per minute is a good estimate for how fast, in general, the students typed at the beginning of the course.
- 20 words per minute is a good description of the
**center**of the data set at the end of the course. - There was more variability in the typing speeds at the beginning of the course than at the end, so the students’ typing speeds were more alike at the end.

- Overall, how fast would you say that the students typed after completing the course? What would you consider the center of the end-of-course data?

Are you ready for more?

Use one of these suggestions (or make up your own). Research to create a dot plot with at least 10 values. Then, describe the center and spread of the distribution.

- Points scored by your favorite sports team in its last 10 games
- Length of your 10 favorite movies (in minutes)
- Ages of your favorite 10 celebrities

### Summary

One way to describe what is typical or characteristic for a data set is by looking at the **center **and **spread **of its distribution.

Let’s compare the distribution of cat weights and dog weights shown on these dot plots.

The collection of points for the cat data is further to the left on the number line than the dog data. Based on the dot plots, we may describe the center of the distribution for cat weights to be between 4 and 5 kilograms and the center for dog weights to be between 7 and 8 kilograms.

We often say that values at or near the center of a distribution are typical for that group. This means that a weight of 4–5 kilograms is typical for a cat in the data set, and weight of 7–8 kilograms is typical for a dog.

We also see that the dog weights are more spread out than the cat weights. The difference between the heaviest and lightest cats is only 4 kilograms, but the difference between the heaviest and lightest dogs is 6 kilograms.

A distribution with greater spread tells us that the data have greater variability. In this case, we could say that the cats are more similar in their weights than the dogs.

In future lessons, we will discuss how to measure the center and spread of a distribution.

### Glossary Entries

Definition: Center

The center of a set of numerical data is a value in the middle of the distribution. It represents a typical value for the data set.

For example, the center of this distribution of cat weights is between 4.5 and 5 kilograms.

Definition: Distribution

The distribution tells how many times each value occurs in a data set. For example, in the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.

Here is a dot plot that shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35.

Definition: Frequency

The frequency of a data value is how many times it occurs in the data set.

For example, there were 20 dogs in a park. The table shows the frequency of each color.

color | frequency |
---|---|

white | (4) |

brown | (7) |

black | (3) |

multi-color | (6) |

Definition: Spread

The spread of a set of numerical data tells how far apart the values are.

For example, the dot plots show that the travel times for students in South Africa are more spread out than for New Zealand.

## Practice

Exercise (PageIndex{4})

Three sets of data about ten sixth-grade students were used to make three dot plots. The person who made these dot plots forgot to label them. Match each dot plot with the appropriate label.

- Dot Plot A
- Dot Plot B
- Dot Plot C

- Ages in years
- Number of hours of sleep on nights before school days
- Numbers of hours of sleep on nights before non-school days

Exercise (PageIndex{5})

The dot plots show the time it takes to get to school for ten sixth-grade students from the United States, Canada, Australia, New Zealand, and South Africa.

- List the countries in order of
*typical travel times*, from shortest to longest. - List the countries in order of
*variability in travel times*, from the least variability to the greatest.

Exercise (PageIndex{6})

Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution. A rating of 0 means “not at all important” and a rating of 10 means “very important.” Here is a dot plot of their responses.

Explain why a rating of 6 is *not* a good description of the center of this data set.

Exercise (PageIndex{7})

Tyler wants to buy some cherries at the farmer’s market. He has $10 and cherries cost $4 per pound.

- If (c) is the number of pounds of cherries that Tyler can buy, write one or more inequalities or equations describing (c).
- Can 2 be a value of (c)? Can 3 be a value of (c)? What about -1? Explain your reasoning.
- If (m) is the amount of money, in dollars, Tyler can spend, write one or more inequalities or equations describing (m).
- Can 8 be a value of (m)? Can 2 be a value of (m)? What about 10.5? Explain your reasoning.

(From Unit 7.2.3)

## Identifying Statistical Questions

Statistics is the study of variability. Students need to be able to identify and pose questions that can be answered by data that vary. The purpose of this task is to help students learn to distinguish between statistical questions and questions that are not statistical. A statistical question is one that can be answered by collecting data and where there will be variability in that data. This is different from a question that anticipates a deterministic answer. For example, "How many minutes do 6th grade students typically spend on homework each week?" is a statistical question. We would answer this question by collecting data from 6th graders, and we expect that not all 6th grade students spend the same amount of time on homework (meaning there will be variability in the data). On the other hand, "How much time did Juana spend on homework last night?" is not a statiscal question--it has a deterministic answer and is not answered by collecting data that vary.

## 6.8 Data Sets and Distributions

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

## Lesson 3

In this lesson, students represent distributions of numerical (and optionally categorical) data after organizing them into frequency tables. They construct dot plots for numerical data (and bar graphs for categorical data). Using graphical representations of distributions, they continue to develop a spatial understanding of distributions in preparation for understanding the concepts of “center” and “spread” in future lessons. Students make use of the structure of dot plots (MP7) to describe distributions and draw conclusions about the data.

### Learning Goals

Let’s represent data with dot plots and bar graphs.

### Required Materials

### Required Preparation

1 sticky note and 1 dot sticker for each student. Straightedges should be made available to create dot plots.

### Learning Targets

### CCSS Standards

### Glossary Entries

The distribution tells how many times each value occurs in a data set. For example, in the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.

Here is a dot plot that shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35.

Expand Image

The frequency of a data value is how many times it occurs in the data set.

For example, there were 20 dogs in a park. The table shows the frequency of each color.

### Print Formatted Materials

Teachers with a valid work email address can click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.

### Additional Resources

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

This site includes public domain images or openly licensed images that are copyrighted by their respective owners. Openly licensed images remain under the terms of their respective licenses. See the image attribution section for more information.

## Lesson 4

### Problem 1

Clare recorded the amounts of time spent doing homework, in hours per week, by students in sixth, eighth, and tenth grades. She made a dot plot of the data for each grade and provided the following summary.

- Students in sixth grade tend to spend less time on homework than students in eighth and tenth grades.

The homework times for the tenth-grade students are more alike than the homework times for the eighth-grade students.

Use Clare's summary to match each dot plot to the correct grade (sixth, eighth, or tenth).

### Problem 2

Mai played 10 basketball games. She recorded the number of points she scored and made a dot plot. Mai said that she scored between 8 and 14 points in most of the 10 games, but one game was exceptional. During that game she scored more than double her typical score of 9 points. Use the number line to make a dot plot that fits the description Mai gave.

### Problem 3

A movie theater is showing three different movies. The dot plots represent the ages of the people who were at the Saturday afternoon showing of each of these movies.

- One of these movies was an animated movie rated G for general audiences. Do you think it was Movie A, B, or C? Explain your reasoning.
- Which movie has a dot plot with ages that that center at about 30 years?
- What is a typical age for the people who were at Movie A?

### Problem 4 (from Unit 5, Lesson 13)

Find the value of each expression.

## Creating a Dot Plot

Video and solutions to help grade 6 students learn how to create a dot plot of a given data set and summarize a given data set using equal length intervals and construct a frequency table.

### New York State Common Core Math Module 6, Grade 6, Lesson 3

Lesson 3 Student Outcomes

&bull Students create a dot plot of a given data set.

&bull Students summarize a given data set using equal length intervals and construct a frequency table.

&bull Based on a frequency table, students describe the distribution.

This lesson described how to make a dot plot. This plot starts with a number line labeled from the smallest to the largest value. Then, a dot is placed above the number on the number line for each value in your data.

This lesson also described how to make a frequency table. A frequency table consists of three columns. The first column contains all the values of the data listed in order from smallest to largest. The second column is the tally column, and the third column is the number of tallies for each data value.

Example 1: Hours of Sleep

Robert, a 6th grader at Roosevelt Middle School, usually goes to bed around 10:00 p.m. and gets up around 6:00 a.m. to get ready for school. That means that he gets about 8 hours of sleep on a school night. He decided to investigate the statistical question: How many hours per night do 6th graders usually sleep when they have school the next day?

Robert took a survey of 6th graders and collected the following data to answer the question:

7 8 5 9 9 9 7 7 10 10 11 8 8 8 12 6 11 10 8 8 9 9 9 8 10 9 9 8

Robert decided to make a dot plot of the data to help him answer his statistical question. Robert first drew a number line and labeled it from 5 to 12 to match the lowest and highest number of hours slept.

He then placed a dot above for the first piece of data he collected. He continued to place dots above the numbers until each number was represented by a dot.

Exercises 1–9

1. Complete Robert’s dot plot by placing a dot above the number on the number line for each number of hours slept. If there is already a dot above a number, then add another dot above the dot already there.

2. What are the least and the most hours of sleep reported in the survey of 6th graders?

3. What is the most common number of hours slept?

4. How many hours of sleep describes the center of the data?

5. Think about how many hours of sleep you usually get on a school night. How does your number compare with the number of hours of sleep from the survey of 6th graders?

Here are the data for the number of hours 6th graders sleep when they don’t have school the next day:

6. Make a dot plot of the number of hours slept when there is no school the next day.

7. How many hours of sleep with no school the next day describe the center of the data?

8. What are the least and most hours slept with no school the next day reported in the survey?

9. Do students sleep longer when they don’t have school the next day than they do when they do have school the next day? Explain your answer using the data in both dot plots.

Example 2: Building and Interpreting a Frequency Table

A group of 6th graders investigated the statistical question: “How many hours per week do 6 sport or outdoor game?”

Here are the data the students collected from a sample of 26 6th graders showing the number of hours per week spent playing a sport or a game outdoors:

To help organize the data, the students placed the number of hours into a frequency table. A frequency table lists items and how often each item occurs.

To build a frequency table, first draw three columns. Label one column “Number of Hours Playing a Sport/Game,” label the second column “Tally,” and the third column “Frequency.” Since the least number of hours was 0, and the most was 12, list the numbers from 0 to 12 under the “Number of Hours” column.

As you read each number of hours from the survey, place a tally mark opposite that number. The table shows a tally mark for the first number 3.

Exercises 10–15

10. Complete the tally mark column.

11. For each number of hours, find the total number of tally marks and place this in the frequency column.

12. Make a dot plot of the number of hours playing a sport or playing outdoors.

13. What number of hours describes the center of the data?

14. How many 6th graders reported that they spend eight or more hours a week playing a sport or playing outdoors?

2. A sixth grader rolled two number cubes 21 times. The student found the sum of the two numbers that he rolled each time. The following are the sums for the 21 rolls of the two number cubes.

9, 2, 4, 6, 5, 7, 8, 11, 9, 4, 6, 5, 7, 7, 8, 8, 7, 5, 7, 6, 6

a. Complete the frequency table.

b. What sum describes the center of the data?

c. What sum occurred most often for these 21 rolls of the number cubes?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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## Planning a statistical investigation (Level 2)

In this unit students will identify how to plan and carry out a statistical investigation, looking at facts about their class as a context.

- Write investigative questions for statistical investigations and design a method of collection of data.
- Display collected data in an appropriate format.
- Make statements about implications or possible actions based on the results of an investigation.
- Make conclusions on the basis of statistical investigations.

It is vital when planning statistical investigations that the students understand the importance of the way that they plan, collect, record and present their information. If they are not consistent in the way they carry out any of these steps, they could alter their findings, therefore making their investigation invalid.

In this unit the students will first look at choosing investigative questions to explore, making sure that the topic lends itself to being investigated statistically. They will then collect their data using structured recording methods. Once they have collected and recorded their data, they will present their findings using appropriate displays and making descriptive statements about their displays to answer the investigative question.

Dot plots are used to display the distribution of a numerical variable in which each dot represents a value of the variable. If a value occurs more than once, the dots are placed one above the other so that the height of the column of dots represents the frequency for that value. Sometimes the dot plot is drawn using crosses instead of dots.

**Investigative questions**

At Level 2 students should be generating broad ideas to investigate and the teacher works with the students to refine their ideas into an investigative question that can be answered with data. Investigative summary questions are about the class or other whole group. The variables are categorical or whole numbers. Investigative questions are the questions **we ask of the data**.

The investigative question development is led by the teacher, and through questioning of the students identifies the variable of interest and the group the investigative question is about. The teacher still forms the investigative question but with student input.

**Survey questions**

Survey questions are the questions **we ask to collect the data** to answer the investigative question. For example, if our investigative question was “What ice cream flavours do the students in our class like?” a corresponding survey question might be “What is your favourite ice cream flavour?”

As with the investigative question, survey question development is led by the teacher, and through questioning of the students, suitable survey questions are developed.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

- directing students to collect category data or whole number data – whole number is harder
- giving students summarised data to graph rather than them having to collect it and collate it
- giving students a graph of the display and ask them to “notice” from the graph rather than having them draw the graph
- writing starter statements that students can fill in the blanks to describe a statistical graph e.g. I notice that the most common XXXX is ________, More students chose _______ than chose _______.

The context for this unit can be adapted to suit the interests and experiences of your students.

Although this unit is set out as five sessions, to cover the topic of statistical investigations in depth will likely take longer. Some of the sessions, especially sessions 4 and 5 dealing with data presentation and description, could easily be extended as a unit in themselves. Alternatively, this unit could follow on from a unit on data presentation to give students an appreciation of practical applications of data display.

#### Session 1

Session 1 provides an introduction to statistical investigations. The class will work together to answer the investigative question – *How many brothers and sisters do people in our class have?*

- Explain to the class that their job for maths this week will be to gather information or data on the class, summarise the information and then present as a report which will be sent home for parents.
- Ask students whether they can explain what the word statistics means.

Explain that statistics concerns the collection, organisation, analysis and presentation of data in a way that other people can understand what it shows. - Explain that the class will work in small groups, each of them with the job of finding information about the class.
- First we will work as a whole class to answer the investigative question:
*How many brothers and sisters do people in our class have?* - Ask the students what information we need to get from everyone in the class to answer our investigative question.

Students might suggest that we can ask how many siblings, or they might suggest we ask how many brothers, how many sisters and how many altogether.

The idea of asking about brothers and sisters separately allows for a deeper exploration of the data and a more in depth answer to our investigative question. - Agree as a class to ask about the three pieces of information. See if anyone can suggest how we could collect the data.
- Working with student ideas move towards a solution whereby each student records their information on a piece of paper.

Sticky notes could be a good way to collect this information from the students as it will allow rearrangements of the data quickly. - Suggest that the students divide their paper into three as shown in the diagram below to answer three survey questions.
- How many brothers do you have?
- How many sisters do you have?
- How many siblings do you have (or total number of brothers and sisters)?

- Get the students to fill in for their brothers and sisters. Check what a response of zero means – in one or all the sections (no brothers/no sisters, no siblings/only child)
- Work with a partner to check that the information is correct and in the correct place. A good way to do this is for the partner to take the piece of paper and describe to another student the number of brothers and sisters the student has.

For example:

Pip records the following information about her brothers and sisters. She gives it to her partner. Her partner, Kaycee shares this information with another student. Kaycee says that Pip has three brothers and one sister. Altogether Pip has four siblings. - Collect all the pieces of paper (sticky notes) and ask:
*How can we use the pieces of paper (sticky notes) to show someone else how many brothers and sisters people in the class have?**How can we show the information so that people can easily understand what it is showing?*

Hopefully, someone will suggest a more organised list, or counting the number of 0s, the number of 1s etc and writing sentences to explain how many there are of each. - Carry out these suggestions to show how much clearer they make the information.
- Ask for suggestions for other ways to show the information. If nobody suggests it introduce the idea of using a dot plot.
- Demonstrate how to draw a dot plot of the information, ensuring that you highlight important features of dot plots axes, scale and labels on the axes, title (use the investigative question), and accurately plotted points.
- Students could draw their own versions as a practice exercise. It may be useful to provide a template with an appropriate scale for students to use.
- Encourage students to draw separate graphs showing just brothers and just sisters as well.
- Now that we have made a display of the data, in this case a dot plot, we need to describe the dot plot.
- Ask the students what things they notice about the data, capture all the ideas on the board. Write the words “I notice…” on the board or chart paper and capture ideas under this. They might notice:
- What is the most common number of siblings/brothers/sisters?
- The largest number of siblings, the smallest number of siblings
- Where most of the data lies e.g. most of our class have 2-4 siblings

- Work with the students to tidy up their statements to ensure that they include the variable and reference the class. For example:
- The most common number of siblings for people in our class is 2.
- The largest number of siblings in our class is 7.
- Most of the people in our class have between 2 and 4 siblings.

- Explain that over the next few days students will be investigating some other ideas about the class, making their own graphs to show the information and describing what the information shows.

#### Session 2

This session is ultimately about choosing an appropriate topic to investigate about the class. There will be a real need for discussion about measurable data and realistic topics that can be investigated in the given time frame. It would be a good idea to provide the students with a list of topics if they get stuck, but they should be encouraged to try and come up with something original where possible.

- Recap the previous session’s work, discussing how the information was collected, how it was presented, and how it was discussed.

__PROBLEM: Generating ideas for statistical investigation and developing investigative questions__

- Explain that in this session students will work in small groups to come up with three topics to explore about the class. The topics need to be ones that they can collect information from the class about and therefore complete the investigation..
- Discuss criteria that the topics must meet.
- Is this a topic that the students in our class would be happy to share information with everyone?
- Would the topic apply to everyone in the class?
- Is the topic interesting or purposeful?

- Put students into small groups and give them a few minutes to come up with some ideas that they think they might use. Encourage them to think of topics that use categories and topics that use counts (e.g. number of siblings). Ideally they should have at least one of each across their three topics.

If groups are having trouble thinking of ideas, you could try writing a list of suggestions on the board but limiting groups to using one of your ideas only, to encourage them to think of their own. Some ideas could be:- Favourite flavour of ice cream/pizza/soft drink etc.
- Favourite pet
- Number of pets
- Colour of eyes
- Shoe size
- Favourite vegetable
- Favourite type of music
- Handedness
- Birthday month
- Number of skips (using a skipping rope) in 30 seconds
- Number of hops in 30 seconds
- How they travel to school
- Number of (whole) hours sleep the previous night
- Number of languages students speak
- Number of letters in their first name
- Number of letters in their first and last names
- Number of items in their school bag

- Once groups have decided on their topics work with them to pose
**investigative questions.**Model examples of these to help the students pose their own.- How many brothers and sisters do people in our class have?
- What are Room 30’s favourite pets?
- What eye colours do the people in our class have?
- How did Room 30 students get to school today?
- How long are our class’s first names?
- When are Room 30’s birthdays?

- Once they have posed their investigative questions, share them as a class, and ensure that they are all appropriate, checking in on the criteria specified in 3.
- If groups need to change any of their investigative questions, give them time to do so now.

__PLAN: Planning to collect data to answer our investigative questions__

- Explain to the students that they need to think about what question or questions they will ask to collect the information they need to answer their investigative question.
- Explain that these questions are called survey questions and they are the questions
**we ask to get the data**. Work with groups to generate survey questions. For example:- If the investigative question is: “What are Room 30’s favourite pets?”, ask the students how they could collect the data.
- A possible response is to ask the other students “What is your favourite pet?”
- Also, the students might want to ask, “What is your favourite pet out of cat, dog, fish, bird?” You could challenge them as to if this would really answer the investigative question and suggest that possibly they might change the survey question to allow for other answers.

- What is the colour of your eyes?
- How did you travel to school today?
- How many letters are in your first name?
- What month is your birthday?

**are Room 30’s**favourite pets? – overall about the class data) rather than a survey question (What**is your**favourite pet? – asking the individual). - Ensure that all groups record their investigative and survey questions for next session.

#### Session 3

Data collection is a vital part of the investigation process. In this session students will plan for their data collection, collect their data and record their data and summarise using a tally chart or similar for analysis in the following sessions.

__PLAN continued: Planning to collect data to answer our investigative questions__

- Get the students to think about how they will record the information they get. Options may include:
- Tally chart
- Writing down names and choices
- Using pre-determined options
- Using a class list to record responses

- Let them try any of the options they suggest. They are likely to encounter problems, but this will provide further learning opportunities as they reflect on the difficulties and how they can improve them.

__DATA: Collecting and organising data__

- Students collect data from the rest of the class using their planned method. Expect a bit of chaos. Possible issues aka teaching opportunities include:
- Pre-determined options
- What happens for students whose choice is not in the pre-determined options?
- What if nobody likes the options given and they end up with a whole lot of people choosing other, they only have tally marks so they cannot regroup to new categories?

- Using tally marks only
- The discussed issue about the “other” category
- Have less tally marks than the number of students in the class
- and they think they have surveyed everyone
- or they do not know who they have not surveyed yet

- Have more tally marks than the number of students in the class

- Possible solutions to the above issues could be (generated by the students please)
- Recording the name of the student and their response and then tallying from the list
- Giving everyone a piece of paper to write their response on, then collecting all the papers in and tallying from the papers

- Pre-determined options
- Regardless of the process of data collection we are aiming for a collated summary of the results.

#### Session 4

In this session the students will work on creating data displays of the data collected in the previous session.

__ANALYSIS: Making and describing displays__

*Numerical data – displaying count data e.g. “How many…” investigative questions*

- Show the dot plot created in Session 1 of numbers of siblings.
- Discuss how it was made and what needs to be included on it.
- Get students to identify which one (or more) of their investigative questions involves count data. Choose one of these to work on first if they have more than one.
- Give students time to work on their first graph, providing support as required. Providing pre-drawn axes may be useful, but students may still need help selecting an appropriate scale to use and placing the “dots”.
- After all students have completed one of their graphs bring the class together to share what students have done.
- Discuss and compare graphs between groups.

*Categorical data – displaying data that has categories e.g. “What…” investigative questions*

- Get students to identify which one (or more) of their investigative questions involves category data. Choose one of these to work on first if they have more than one.
- Ask students if they can remember how to graph categorical data (you may have already done some work on using pictographs or bar graphs e.g. Parties and favourites). Reference back to this previous work and discuss how it was made and what needs to be included.
- Give students time to work on their categorical data graph, either a pictograph or a bar graph. You may want to encourage a bar graph depending on how much statistics you have already done prior to this unit.
- After all students have completed one of their categorical data graphs bring the class together to share what the students have done.
- Discuss and compare graphs between groups.
- Send students to work on the last graph of their three.

#### Session 5

Session 5 is a finishing off session. Students should be given time to complete their graphs if they have not already, and to write statements about what the graphs show.

- Give groups time to finish graphs as required.
- Students should also write statements under each graph telling what the graph shows. Ideas for describing graphs were discussed in session 1. Refer to these ideas. In addition, starters for these statements could be given:
- The most popular…
- The most common…
- The least popular…
- The least common…
- Most students in our class…
- The largest number of…
- The smallest number of…

- Check their descriptive statements for the variable and the group. For example, favourite pet and our class travel to school and Room 30.
- Discuss with the students whether there is any action we should take as a result of any of the information we have found out in our investigations.
- Ask if there are any conclusions we can make from the investigations we have done.
- Students could compile their displays as a booklet to take home to their families entitled "About our class" or similar. Alternatively, create a class display of the findings, or share them with another class.

During the next week we will be working on statistical investigations in maths. Over this time, your child will be gathering data on the class and presenting it using data displays such as dot plots and bar graphs. If you know of any graphs or tables of information suitable to discuss with your child, either in the newspaper, or in a book, or perhaps on some advertising material, this week would be a good time to do so.

## 6.3 Population of States

Every ten years, the United States conducts a census, which is an effort to count the entire population. The dot plot shows the population data from the 2010 census for each of the fifty states and the District of Columbia (DC).

Here are some statistical questions about the population of the fifty states and DC. How difficult would it be to answer the questions using the *dot plot*?

In the middle column, rate each question with an E (easy to answer), H (hard to answer), or I (impossible to answer). Be prepared to explain your reasoning.

Here are the population data for all states and the District of Columbia from the 2010 census. Use the information to complete the table.

Use the grid and the information in your table to create a histogram.

Return to the statistical questions at the beginning of the activity. Which ones are now easier to answer?

In the last column of the table, rate each question with an E (easy), H (hard), and I (impossible) based on how difficult it is to answer. Be prepared to explain your reasoning.

### Are you ready for more?

## Step 1: Choose a scale and set it up.

We are going to make a horizontal scale and it needs to cover all values. For this data set, the smallest value is 16 and the largest is 40.

You will notice that I chose to count by 2 instead of 1. This isn’t required but just makes the plot a little more compact. You can count by 1, 5, or even 10 if you like!

## INTERPRETING A DOT PLOT WORKSHEET

The data values are spread out from 3 to 7 with no outliers.

The data has a cluster fromਃ to 7 with one peak at 5, which is the center of theistribution.

The distribution is symmetric. The data valuesਊre clustered around the center of the distribution.

Describe the spread, center, and shape of the dot plot given below.

The data values are spread out from 1 to 9. The data value 1ਊppears to be an out lier .

The data has a cluster from 6 to 9 with one peak at 9, which is the greatest value in the data set.

The distribution is not symmetric. The data values are clustered at one end of the distribution.

Describe the spread, center, and shape of the dot plot given below.

The data values are spread out from 0 to 11. The data value 11ਊppears to be an out lier .

The data has a cluster from 0 to 7 with one peak at 2, which is the greatest value in the data set.

The distribution is not symmetric. The data values are clustered at one end of the distribution.

Describe the spread, center, and shape of the dot plot given below.

The data values are spread out from 0 to 11 with no outliers.

The data has a cluster from 0 to 11 with one peak at 5, which is the center of theistribution.

The distribution is symmetric. The data valuesਊre clustered around the center of the distribution.

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