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John C. D. Diamantopoulos (Northeastern State University) and Cynthia J. Woodburn (Pittsburg State University)

Making a Right Angle the Maya Way

According to his doctoral dissertation, Christopher Powell of the Maya Exploration Center observed modern Maya using the geometric fact that a rhombus with equal diagonals must be a square to make sure the corners were square when laying out a building. This fact allows one to easily check whether or not a rhombus is a square by checking if the diagonals are equal with a cord or piece of rope. More generally, a parallelogram with equal diagonals must be a rectangle.

In a lecture during the 2011 MAA Study Tour, Powell explained another clever way that the Maya used a cord to form right angles in laying out a square that he learned from a master builder who had learned it while a shaman apprentice. The method uses a knotted cord and properties of equilateral triangles. The cord has eight knots on it, dividing the cord into seven equal segments with a knot at each end. There are loops at each of the knots for staking to the ground. Since the knots are evenly spaced, when knots 1 and 4 are held together and the cord pulled taut, an equilateral triangle with interior angles of 60° is formed. Then knot 6 is joined with knot 3 and the cord pulled tight resulting in another equilateral triangle formed by knots 4, 5 and 6. Finally knot 8 is joined with knot 5 forming a third equilateral triangle (and all together, half of a hexagon). If one runs a ray (or rope) from knot 1 through knot 2 and another ray from knot 1 through knot 7, the resulting angle is a right (60° + 30°) angle. The animation below (created by the first author) illustrates how the right angle is formed.

The first author also created the following video using student volunteers to demonstrate the Maya way of forming a right angle. See Activity 3 for detailed instructions for recreating this activity with your students.

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Maya Geometry in the Classroom - Making a Right Angle the Maya Way

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John C. D. Diamantopoulos (Northeastern State University) and Cynthia J. Woodburn (Pittsburg State University) According to his doctoral dissertation, Christopher Powell of the Maya Exploration Center observed modern Maya using the geometric fact that a rhombus with equal diagonals must be a square to make sure the corners were square when laying out a building.

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Measurement using your body - ethnomathematics

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Measurement using your body - ethnomathematics

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7th grade probability unit

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7.6(H) solve problems using qualitative and quantitative predictions and comparisons from simple
experiments

Have students describe their favorite food or meal using qualitative and quantitative descriptions:

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7.12(c)

topic: take random sample from wnba and nba, compare pay differences

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7.6(C): make predictions and determine solutions using experimental data for simple and compound events

Ideas:
- Predicting when their favorite YouTuber would reach a million subscribers by looking at previous trends.
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7.6(I) determine experimental and theoretical probabilities related to simple and compound
events using data and sample spaces

Have students actual perform hands on activities to record, compare, and contract theoretical and experimental probability and their relationship to one another

Ex.
Record basketball shots
Record soccer goals made
Record flip of coins
Record rolling dice

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Grade 5 Curriculum, Resources, Activities

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Grade 5 Curriculum, Resources, Activities

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Grade 4 Curriculum, Lesson Plans, and Resources

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Grade 4 Curriculum, Lesson Plans, and Resources

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4th grade Gingerbread houseProject

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The weeks leading up to winter break can be a challenging time to keep kids motivated at school. It can feel like you lose precious teaching time because they're distracted and anxious for the holidays to begin. A weeklong hands-on, cumulative math project may be just what you need to keep them engaged and thinking right up to the break.

One project I love to do with fourth graders is the Great Gingerbread House Project, a mix of review and new content. It keeps kids' minds active until the last day of the school term. The balance of old and new math works because the project is so hands on that students can see connections between previous understanding and new challenges.

Kids need a lot of motivation to do this much math during a time when they're thinking of snow, hot chocolate, and holidays. Before we begin, we look at pictures of gingerbread houses. The students learn how they're going to spend the next week or so in math class: designing and building their own "gingerbread" houses (for simplicity, we use graham crackers-they're cheap and easy to get, and they have a very consistent shape and size). The kids always become really excited, and they stay excited even after they find out it's going to take a lot of work and a lot of math to complete the project.

How the Great Gingerbread House Project Works

For this project, you'll need to provide 1x1 cm grid paper, base 10 blocks, and graham crackers as tools for designing a floor plan that doesn't exceed the area of six graham crackers. That's a realistic size for the kids to work with, and helps limit the number of graham crackers you need to supply. The students can use whole, half, or quarter graham crackers. They need to find the area and perimeter of the floor plan in centimeter and graham cracker units. This gets kids working with multiplication, addition, and fractions. Some students will use the base 10 blocks to create the floor plan, but they don't have to.

At this point in the year, some students may still be shaky when multiplying two 2-digit numbers, and often fourth graders haven't done much work with fractions yet. The floor plan provides an authentic scenario for applying math skills they've been introduced to in the first few months of school. Finding area and perimeter with visuals and hands-on tools helps them grapple with new concepts, like adding fractions.

This project is not just about numbers and sweet treats-there's also a heavy focus on craftsmanship. Students create drafts and neat, accurate final copies every step along the way. They work in pairs, but each partner is individually responsible for a draft and final copy of each step: a floor plan with labeled dimensions, the math work that shows how they found the total area and perimeter, and front- and side-view drawings of the gingerbread house they intend to build.

This project requires creativity, precise calculation, and spatial reasoning. Collaboration and compromise are necessary for partners to create a design and work accurately with numbers in the tens and hundreds-the numbers can get quite high when working in centimeters. During the project, partners help each other with math strategies and use each other to check for accuracy. They work together to record their predictions for how many graham crackers they'll need to build the floor, walls, and roof of their house.

Once the partners complete these steps, they create a poster with four quadrants: predictions, floor plan, front view, and side view. They staple all their work in these sections with their final drafts on top so that someone can flip through and see the work they've done along the way. The posters are one more way to emphasize the importance of the process, not just the final product.

Finally, when partners have shown accurate math and neat craftsmanship on every part of the planning, they get to build. You'll need to have royal icing-which is easy to make-on hand to serve as the glue. (It hardens completely, unlike regular frosting.) I ask parents to send in decorating materials-chocolate chips, mini marshmallows, Skittles, mini candy canes, etc. What comes in depends on the parents, and on student allergies. Partners carefully follow their plans to build delicious smelling graham cracker "gingerbread" houses. As they build, they keep track of how many graham crackers they actually use for the floor, walls, and roof. They record their "actuals" next to their predictions.

When the posters are complete and the houses are built, it's fun to invite parents and classes from other grade levels to see the project. Students are proud to show off the hard work they've done. They like to explain the process, flipping through each of their many drafts. It's also a good idea to provide some reflection time. Students can think back through the steps of the project, consider the challenges they faced, and reflect on how their partners helped them be successful.

For me, the Great Gingerbread House Project changed what it means to teach in December. It makes it possible for students to be motivated, engaged, and collaborating while doing lots of deep thinking and authentic math every day until winter break.

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The weeks leading up to winter break can be a challenging time to keep kids motivated at school. It can feel like you lose precious teaching time because they're distracted and anxious for the holidays to begin. A weeklong hands-on, cumulative math project may be just what you need to keep them engaged and thinking right up to the break.

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Financial Literacy Unit for Kindergarten, 1st, and 2nd grade

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I was a Chapter 220 student. This meant I was bussed from my neighborhood to suburban schools that were a different world from my home in Milwaukee's 53206 zip code, the most incarcerated zip code in the world-in 2016, 62 percent of Black men there were or had been incarcerated. The racial and financial inequities I experienced in my school years have influenced me both personally and professionally.

Today I'm the math supervisor for St. Paul Public Schools, overseeing curriculum and instruction for over 37,000 students and 3,000 teachers. We've tried many things in St. Paul to improve student learning: tracking, 5 Easy Steps, mastering math fluency, professional learning communities, and purchasing new curriculum. We were still not seeing high student growth and proficiency.

In 2017, I was told to find and purchase a new curriculum, at a projected cost of roughly $8 million. But there is no single off-the-shelf curriculum that addresses the varied needs and interests of a student population that is 31 percent Asian, 26 percent Black, 21 percent White, 14 percent Latino, 1 percent Indigenous, and 7 percent ethnically mixed.

If You Can't Find a Curriculum, Build One

Concluding that a packaged curriculum would not meet our needs, I had to come up with a better plan. Who knows these students better than their teachers? No one. I determined that St. Paul teachers would write curriculum for St. Paul students. My team was enthusiastic, and in 2018 we extended an invitation to teachers to write curriculum during the last three weeks of June.

The response was positive. We had 30 teachers working together to build the vertical alignment of our math curriculum. I knew we were on to something special when I heard teachers saying, "I can't wait to teach this!"

Throughout the school year, our curriculum writers saw significant improvement in student participation and achievement. More teachers wanted to join, including specialists. We adopted a task, question, and evidence format and added culturally relevant topics with themes of social justice. We immersed ourselves in the literature of Gloria Ladson-Billings and Zaretta Hammond. Ladson-Billings gave us the "why" we needed to do this work; Hammond gave us the "how" to build independent learners.

Some teachers readily embraced these new units; however, some were reluctant to introduce social justice topics in math. I was frequently told, "This could be a traumatic trigger." But we pushed on, and in 2020, two years into this work, I became inspired by Dr. Claud Anderson's PowerNomics. Anderson points out economic disparities steeped in historically racist policies of the United States, and provides a financial plan to help Black people overcome the initial and continual setbacks we face in this country.

Financial literacy is now our compass in St. Paul. Every lesson we create comes back to making students financially literate, and every financial literacy unit promotes independent learning through culturally relevant topics and themes of social justice.

The focus of our work is keeping money in the community. Anderson writes that Asian, Latino, and Indigenous communities have areas where members live, work, eat, educate their children, and spend their money, but it's hard to find that for Black people-we are more likely than other groups to spend our money outside of our neighborhoods. There are a lot of Black neighborhoods, he writes, but not many Black communities.

Financial Literacy as a Blueprint

My team created a financial literacy breakdown by grade level. Understanding that learning deepens when connections are made, we have made sure the primary units are interdisciplinary. During reading periods, for example, students are exposed to literature with financial literacy themes, like Cinders McLeod's Earn It! and her other picture books about money in grade 1, and Uncle Jed's Barbershop in grade 2. Students write about their hopes and dreams as they design entrepreneurial business plans, and they prepare persuasive sales pitches that reflect speaking and listening standards-all while learning about wants and needs, goods and services, producers and consumers.

Primary units approach traditional math skills with a focus on the value of money, spending and saving, and entrepreneurship. Students see themselves as producers, not just consumers. Our kindergarten unit focuses on saving, spending, and sharing money; entrepreneurship; and keeping money in the community. The first-grade unit builds on that, as children learn about scarcity, bartering, and trading. Using kid entrepreneurs as examples, students work to develop their own business plans.

Second graders learn about opportunity costs, startup costs, and marketing their businesses. Math team member Stacy Waskosky helped write this unit and taught it to 30 second graders in the virtual classroom during the pandemic. An introductory letter home increased family engagement. Students learned about entrepreneurship from a local barbershop owner, and 100 percent of the students completed the task of creating a business plan and sales pitch-many of them seriously considered taking their ideas all the way to Shark Tank. Examples of all of this can be found in our second-grade unit.

Our fourth-grade teachers now use data on housing and homelessness to teach division. One class was so inspired by this unit that they held a sock drive, collecting and donating over 400 pairs of socks to a local women's shelter.

I carry mixed feelings about my early educational experiences. I'm grateful for the opportunities the suburban schools provided me, but I wish I could have had them in my neighborhood school, with peers and teachers who looked like me and knew me at a deeper level. Those mixed feelings fuel my commitment to providing urban students with opportunities for rigorous, relevant, and real educational experiences.

I'd like to express my gratitude to my colleagues Peggy Nayar and Stacy Waskosky for their contributions to this article and, more importantly, their contributions to our ongoing work of improving math outcomes for every student in St. Paul Public Schools.

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Creating a Culturally Responsive Math Curriculum for the Elementary Grades

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I was a Chapter 220 student. This meant I was bussed from my neighborhood to suburban schools that were a different world from my home in Milwaukee's 53206 zip code, the most incarcerated zip code in the world-in 2016, 62 percent of Black men there were or had been incarcerated.

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3.MD.3 - About the Math, Learning Targets, and Rigor: Grade 3 Mathematics

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Ten sixth-grade students at a school were each asked five survey questions. Their answers to each question are shown here.

The five survey questions are shown below.

a. Match each question to a data set that could represent the students' answers. Explain your reasoning.

Question 1: Flip a coin 10 times. How many heads did you get?

Question 2: How many books did you read in the last year?

Question 3: What grade are you in?

Question 4: How many dogs and cats do you have?

Question 5: How many inches are in 1 foot?

b. Discuss with a partner: How are Question 3 and Question 5 different from the other questions?

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Fishtank Learning - 6th Grade - Unit 8: Statistics - Lesson 1

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Ten sixth-grade students at a school were each asked five survey questions. Their answers to each question are shown here. The five survey questions are shown below. a. Match each question to a data set that could represent the students' answers. Explain your reasoning. Question 1: Flip a coin 10 times.

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COVID-19 Cases in the USA by the Thousands since Feb 15, 2020

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COVID-19 Cases in the USA by the Thousands since Feb 15, 2020

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Power point for class

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Data is a collection of facts, such as numbers, words, measurements, observations or just descriptions of things.

Qualitative vs Quantitative

Data can be qualitative or quantitative.

Qualitative data is descriptive information (it describes something)

Quantitative data is numerical information (numbers)

Quantitative data can be Discrete or Continuous:

Discrete data can only take certain values (like whole numbers)

Continuous data can take any value (within a range)

Put simply: Discrete data is counted, Continuous data is measured

Example: What do we know about Arrow the Dog?

Qualitative:

He is brown and black

He has long hair

He has lots of energy

Quantitative:

To help you remember think " Quantitative is Quantity"

More Examples

Qualitative:

Your friends' favorite holiday destination

The most common given names in your town

How people describe the smell of a new perfume

Quantitative:

Height (Continuous)

Weight (Continuous)

Petals on a flower (Discrete)

Customers in a shop (Discrete)

Collecting

Data can be collected in many ways. The simplest way is direct observation.

Example: Counting Cars

You want to find how many cars pass by a certain point on a road in a 10-minute interval.

So: stand near that road, and count the cars that pass by in 10 minutes.

You might want to count many 10-minute intervals at different times during the day, and on different days too!

We collect data by doing a Survey.

Census or Sample

A Census is when we collect data for every member of the group (the whole "population").

A Sample is when we collect data just for selected members of the group.

Example: 120 people in your local football club

You can ask everyone (all 120) what their age is. That is a census.

Or you could just choose the people that are there this afternoon. That is a sample.

A census is accurate, but hard to do. A sample is not as accurate, but may be good enough, and is a lot easier.

Language

Data or Datum?

The singular form is "datum", so we say "that datum is very high".

"Data" is the plural so we say "the data are available", but data is also a collection of facts, so "the data is available" is fine too.

1893, 1894, 1895, 1896, 3762, 3763, 3764

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Data is a collection of facts, such as numbers, words, measurements, ...

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Alignments to Content Standards: 6.SP.A.1

Task

Last night, Jennifer and her family went out for dinner. The questions below came up on their way to the restaurant or during the meal. Decide whether or not each question is a statistical question, and justify your decision.

1. How far are we from the restaurant?

2. How long will it be until we get there?

3. Would Jennifer rather have burgers or pizza?

4. How much should we leave for the tip?

5. What was the first dish ordered in the restaurant this evening?

6. Do customers at the restaurant like pizza?

7. What is a typical bill for tables at this restaurant?

8. On average, how many people were sitting at each table this evening?

IM Commentary

The goal of this task is to promote a discussion of what makes a statistical question. This is an ideal place for a classroom discussion because answers are not always clear cut and there is a continuum going from questions that are clearly not statistical (Who was the King of France in 1716?) to questions that are definitely statistical (What is the average lifespan in the United States?). There is a lot of ground in between because a question can be interpreted and answered in different ways. A statistical question is one that can be answered by collecting data and for which there will be variability in that data. Questions that are answered with a single data point are not statistical questions because there is no variability in the data used to answer the question.

Solution

1. This is not a statistical question if the route to the restaurant is determined or if the question is intended ''as the crow flies''. In these cases, it would be a fixed numerical answer, which is likely what is intended by the question. It could be viewed as a statistical question if it referred to time rather than distance (e.g. On average, how long does it take to get home from the restaurant?).

2. Unlike the first question, this is a statistical question. Variables such as traffic signals and the amount of traffic rule out an exact answer. The answer will be an approximation based on repeated experience of similar conditions and drives. That is, the answer would be based on data that vary.

3. This question refers to a single person and so it would be answered based on a single response and not on data that vary.

4. This could be a non-statistical question if, for example, the family always leaves a fixed percent of the bill for a tip. On the other hand, there may be a discussion and consideration of how much everyone enjoyed the meal and service with each person suggesting a tip amount. This would make it a statistical question.

5. This is not a statistical question. The first dish ordered is a fixed item and this question would not be answered by collecting data that vary.

6. This is a statistical question (unless we know in advance that everyone is going to say yes or no).

7. This is a statistical question. It would be answered by collecting data on the amount of the bill for different tables, and this amount would vary from table to table.

8. Like the previous question, this is a statistical question. It would be answered by collecting data on the number of people at a table form many tables and there would be variability in the data collected.

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Alignments to Content Standards: 6.SP.A.1 Last night, Jennifer and her family went out for dinner. The questions below came up on their way to the restaurant or during the meal. Decide whether or not each question is a statistical question, and justify your decision. How far are we from the restaurant?

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See all the wonderful things Scott Walters has made

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Kyla & Jasmine Submission

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Alignments to Content Standards: 6.SP.A.1

Task

Last night, Jennifer and her family went out for dinner. The questions below came up on their way to the restaurant or during the meal. Decide whether or not each question is a statistical question, and justify your decision.

1. How far are we from the restaurant?

2. How long will it be until we get there?

3. Would Jennifer rather have burgers or pizza?

4. How much should we leave for the tip?

5. What was the first dish ordered in the restaurant this evening?

6. Do customers at the restaurant like pizza?

7. What is a typical bill for tables at this restaurant?

8. On average, how many people were sitting at each table this evening?

IM Commentary

The goal of this task is to promote a discussion of what makes a statistical question. This is an ideal place for a classroom discussion because answers are not always clear cut and there is a continuum going from questions that are clearly not statistical (Who was the King of France in 1716?) to questions that are definitely statistical (What is the average lifespan in the United States?). There is a lot of ground in between because a question can be interpreted and answered in different ways. A statistical question is one that can be answered by collecting data and for which there will be variability in that data. Questions that are answered with a single data point are not statistical questions because there is no variability in the data used to answer the question.

Solution

1. This is not a statistical question if the route to the restaurant is determined or if the question is intended ''as the crow flies''. In these cases, it would be a fixed numerical answer, which is likely what is intended by the question. It could be viewed as a statistical question if it referred to time rather than distance (e.g. On average, how long does it take to get home from the restaurant?).

2. Unlike the first question, this is a statistical question. Variables such as traffic signals and the amount of traffic rule out an exact answer. The answer will be an approximation based on repeated experience of similar conditions and drives. That is, the answer would be based on data that vary.

3. This question refers to a single person and so it would be answered based on a single response and not on data that vary.

4. This could be a non-statistical question if, for example, the family always leaves a fixed percent of the bill for a tip. On the other hand, there may be a discussion and consideration of how much everyone enjoyed the meal and service with each person suggesting a tip amount. This would make it a statistical question.

5. This is not a statistical question. The first dish ordered is a fixed item and this question would not be answered by collecting data that vary.

6. This is a statistical question (unless we know in advance that everyone is going to say yes or no).

7. This is a statistical question. It would be answered by collecting data on the amount of the bill for different tables, and this amount would vary from table to table.

8. Like the previous question, this is a statistical question. It would be answered by collecting data on the number of people at a table form many tables and there would be variability in the data collected.

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Alignments to Content Standards: 6.SP.A.1 Last night, Jennifer and her family went out for dinner. The questions below came up on their way to the restaurant or during the meal. Decide whether or not each question is a statistical question, and justify your decision. How far are we from the restaurant?

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Asked students for the number of siblings they have and built on a previous lesson of creating dot plots. Students then moved the boxes to find the "fain share" mean.

CCSS 6.SP.B.5.C

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The version of the browser you are using is no longer supported. Please upgrade to a supported browser. Dismiss

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https://docs.google.com/document/d/1kbiJcEYozDOPsomJvoX_VbCH9ETm2-dl4Z_hu9bO6tE/edit?usp=sharing

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Tiana Hamilton

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Measures of Central Tendency Lesson Plan

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Measures of Central Tendency Lesson Plan

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Tiana Hamilton

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Engaging texts to ask statistical questions about

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Engaging texts to ask statistical questions about

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Students view the texts individually, in groups, or as a whole class. Make a list of questions, then determine which questions are statistical and which questions aren't.

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Determining if Questions are Statistical

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SWBAT find the median, mode, range, & outlier of a given data set and recognize them as measures of central tendency or measures of variation.

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Today, the students will continue their exploration of mean and will also be introduced to median, mode, range, and outlier. They will focus on being able to find/calculate these measurements when given a set of data. I will demonstrate to my student how to calculate these different data measures by conducting a survey of how many total siblings each student has. The data gathered by this survey will be used for the purpose of modeling how to use the data items to calculate the different measures of data highlighted in this lesson.

The procedure by which we will conduct this survey is as follows:

1. I will ask each student to write down the number of siblings that they have on a sticky note.

2. Each student will then bring their sticky note to the board and place the notes in numerical order from least to greatest creating a list of data.

3. Once we have the data from each student, I will show the students how to calculate the mean, median, mode, range, and check for any outliers.

The students will follow along and take notes as I demonstrate how to find the mean, median, mode, range, and check for outliers. Then, I will provide my students with the following data set so that they can practice finding the mean, median, mode, range, and outlier.

1) 15, 19, 12, 15, 2, 18, 17, 15, 2, 14

After the students practice finding these measurements of data, then we will discuss what these measurements tell you about the data set as well as what these measurements cannot tell you about the data set

Some examples as to questions that I might ask during this time is:

Does this measurement give you an indication as to how random your data is?

Between what numbers on the number line would this data be located? How do you know?

Does the median tell me how spread out the data is?

***Note: These are just a sampling of possible questions. There are many more questions that can be asked during this time that will provide wonderful amount of information about what the students know and understand.

Things that I want students to pull form this discussion are:

That the median tells you the exact number in the middle of the data set which let's you know that half of the numbers in your data set are above this measurement and half are below this measurement.

The mode gives us an indication as to the randomness of a data set. If one or more numbers show up several times, chances are that the data is pretty consistent.

The range also gives us an indication of how random the data is. If the range is a small number that lets us know that the data in the data set are not very far apart from each other. If the range is a larger number than that lets us know that the data is spread further apart and therefore is a bit more random.

An outlier will help us to recognize whether or not the measurements of the range and the mean is reliable. Having an outlier can change the mean and range dramatically.

Students should remember the significance behind the data measure called mean from the previous lesson. They should remember that his is a measure of central tendency that takes all the items in a data set into consideration in its calculation of center.

***Note: Students should know that the most reliable data set should have measurements for mean, median, and mode that are very close if not the same. And, it should also have a small range, and no outliers.

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Median, Mode, Range, and the Outlaw called the Outlier

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Today, the students will continue their exploration of mean and will also be introduced to median, mode, range, and outlier. They will focus on being able to find/calculate these measurements when given a set of data. I will demonstrate to my student how to calculate these different data measures by conducting a survey of how many total siblings each student has.

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Investigate patterns of association in bivariate data.

CCSS.MATH.CONTENT.8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

CCSS.MATH.CONTENT.8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

CCSS.MATH.CONTENT.8.SP.A.3
Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

CCSS.MATH.CONTENT.8.SP.A.4
Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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Use random sampling to draw inferences about a population.

CCSS.MATH.CONTENT.7.SP.A.1
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

CCSS.MATH.CONTENT.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. 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.

Draw informal comparative inferences about two populations.

CCSS.MATH.CONTENT.7.SP.B.3
Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

CCSS.MATH.CONTENT.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. 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.

Investigate chance processes and develop, use, and evaluate probability models.

CCSS.MATH.CONTENT.7.SP.C.5
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

CCSS.MATH.CONTENT.7.SP.C.6
Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

CCSS.MATH.CONTENT.7.SP.C.7
Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

CCSS.MATH.CONTENT.7.SP.C.7.A
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

CCSS.MATH.CONTENT.7.SP.C.7.B
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

CCSS.MATH.CONTENT.7.SP.C.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

CCSS.MATH.CONTENT.7.SP.C.8.A
Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

CCSS.MATH.CONTENT.7.SP.C.8.B
Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

CCSS.MATH.CONTENT.7.SP.C.8.C
Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

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Develop understanding of statistical variability.

CCSS.MATH.CONTENT.6.SP.A.1
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.

CCSS.MATH.CONTENT.6.SP.A.2
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.

CCSS.MATH.CONTENT.6.SP.A.3
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.

Summarize and describe distributions.

CCSS.MATH.CONTENT.6.SP.B.4
Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

CCSS.MATH.CONTENT.6.SP.B.5
Summarize numerical data sets in relation to their context, such as by:

CCSS.MATH.CONTENT.6.SP.B.5.A
Reporting the number of observations.

CCSS.MATH.CONTENT.6.SP.B.5.B
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

CCSS.MATH.CONTENT.6.SP.B.5.C
Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

CCSS.MATH.CONTENT.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.

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Use probability to evaluate outcomes of decisions

CCSS.MATH.CONTENT.HSS.MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.MATH.CONTENT.HSS.MD.B.5.A
Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

CCSS.MATH.CONTENT.HSS.MD.B.5.B
Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

CCSS.MATH.CONTENT.HSS.MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

CCSS.MATH.CONTENT.HSS.MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

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Calculate expected values and use them to solve problems

CCSS.MATH.CONTENT.HSS.MD.A.1
(+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

CCSS.MATH.CONTENT.HSS.MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

CCSS.MATH.CONTENT.HSS.MD.A.3
(+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

CCSS.MATH.CONTENT.HSS.MD.A.4
(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

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Convert like measurement units within a given measurement system.

CCSS.MATH.CONTENT.5.MD.A.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Represent and interpret data.

CCSS.MATH.CONTENT.5.MD.B.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Geometric measurement: understand concepts of volume.

CCSS.MATH.CONTENT.5.MD.C.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

CCSS.MATH.CONTENT.5.MD.C.3.A
A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

CCSS.MATH.CONTENT.5.MD.C.3.B
A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

CCSS.MATH.CONTENT.5.MD.C.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

CCSS.MATH.CONTENT.5.MD.C.5
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

CCSS.MATH.CONTENT.5.MD.C.5.A
Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

CCSS.MATH.CONTENT.5.MD.C.5.B
Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

CCSS.MATH.CONTENT.5.MD.C.5.C
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

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Solve problems involving measurement and conversion of measurements.

CCSS.MATH.CONTENT.4.MD.A.1
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

CCSS.MATH.CONTENT.4.MD.A.2
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

CCSS.MATH.CONTENT.4.MD.A.3
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

Represent and interpret data.

CCSS.MATH.CONTENT.4.MD.B.4
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Geometric measurement: understand concepts of angle and measure angles.

CCSS.MATH.CONTENT.4.MD.C.5
Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:

CCSS.MATH.CONTENT.4.MD.C.5.A
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.

CCSS.MATH.CONTENT.4.MD.C.5.B
An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

CCSS.MATH.CONTENT.4.MD.C.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.

CCSS.MATH.CONTENT.4.MD.C.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

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Kindergarten

Describe and compare measurable attributes.

CCSS.MATH.CONTENT.K.MD.A.1
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

CCSS.MATH.CONTENT.K.MD.A.2
Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Classify objects and count the number of objects in each category.

CCSS.MATH.CONTENT.K.MD.B.3
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.¹

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Grade 1

Measure lengths indirectly and by iterating length units.

CCSS.MATH.CONTENT.1.MD.A.1
Order three objects by length; compare the lengths of two objects indirectly by using a third object.

CCSS.MATH.CONTENT.1.MD.A.2
Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

Tell and write time.

CCSS.MATH.CONTENT.1.MD.B.3
Tell and write time in hours and half-hours using analog and digital clocks.

Represent and interpret data.

CCSS.MATH.CONTENT.1.MD.C.4
Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

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Grade 2

Measure and estimate lengths in standard units.

CCSS.MATH.CONTENT.2.MD.A.1
Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

CCSS.MATH.CONTENT.2.MD.A.2
Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

CCSS.MATH.CONTENT.2.MD.A.3
Estimate lengths using units of inches, feet, centimeters, and meters.

CCSS.MATH.CONTENT.2.MD.A.4
Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

Relate addition and subtraction to length.

CCSS.MATH.CONTENT.2.MD.B.5
Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

CCSS.MATH.CONTENT.2.MD.B.6
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

Work with time and money.

CCSS.MATH.CONTENT.2.MD.C.7
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

CCSS.MATH.CONTENT.2.MD.C.8
Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

Represent and interpret data.

CCSS.MATH.CONTENT.2.MD.D.9
Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.

CCSS.MATH.CONTENT.2.MD.D.10
Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems¹ using information presented in a bar graph.

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Solve problems involving measurement and estimation.

CCSS.MATH.CONTENT.3.MD.A.1
Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

CCSS.MATH.CONTENT.3.MD.A.2
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).¹ Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.²

Represent and interpret data.

CCSS.MATH.CONTENT.3.MD.B.3
Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

CCSS.MATH.CONTENT.3.MD.B.4
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

CCSS.MATH.CONTENT.3.MD.C.5
Recognize area as an attribute of plane figures and understand concepts of area measurement.

CCSS.MATH.CONTENT.3.MD.C.5.A
A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.

CCSS.MATH.CONTENT.3.MD.C.5.B
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.

CCSS.MATH.CONTENT.3.MD.C.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

CCSS.MATH.CONTENT.3.MD.C.7
Relate area to the operations of multiplication and addition.

CCSS.MATH.CONTENT.3.MD.C.7.A
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

CCSS.MATH.CONTENT.3.MD.C.7.B
Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.

CCSS.MATH.CONTENT.3.MD.C.7.C
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

CCSS.MATH.CONTENT.3.MD.C.7.D
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

Geometric measurement: recognize perimeter.

CCSS.MATH.CONTENT.3.MD.D.8
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

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Math Activity - Determining if a Question is Statistical

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Math Activity - Determining if a Question is Statistical

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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Geogebra Measures of Central Tendency Tool

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Geogebra Measures of Central Tendency Tool

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Katie + Young

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Geogebra Measures of Central Tendency Tool

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Katie + Young

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Lesson plan about comparing data

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math methods lesson plan template -2.docx

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Also has data available for other cities

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The median household income in Grand Prairie, TX in 2019 was $66,097, which was 3.1% greater than the median annual income of $64,034 across the entire state of Texas. Compared to the median income of $46,816 in 2000 this represents an increase of 29.2%. The per capita income in 2019 was $28,086, which means an increase of 32.4% compared to 2000 when it was $18,978.

Whole page based on the data from 2000 / 2019 when the population of Grand Prairie, Texas was 127,427 / 194,522

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Grand Prairie, Texas (TX) income map, earnings map, and wages data

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The median household income in Grand Prairie, TX in 2019 was $66,097, which was 3.1% greater than the median annual income of $64,034 across the entire state of Texas. Compared to the median income of $46,816 in 2000 this represents an increase of 29.2%.

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Lesson Plan

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Video transcript

let's say that you have a cherry pie store and you've noticed that there is variability in the number of cherries on each pie that you sell some pies might have over 100 cherries while other pies might have fewer than 50 cherries so what you're curious about is what is the distribution how many of the different types of pies do you have how many pies you have that have a lot of cherries how many pies you have that have very few cherries how many pies are in between and so to do that you set up a histogram what you do is you take each pie in your store let's see if I can draw you draw a pie of some kind it's a cherry pie I don't know if this is an adequate drawing of a pie but you take each of the pies in your in your store and you count the number you count the number of cherries on it so this pie right over here is 1 2 3 4 5 6 7 8 9 10 so you keep counting and let's say it has 32 cherries and you do that for every buy and then you created buckets because you don't want to create just a graph of how many have exactly 32 you just want to get a general sense of things so you create buckets of 30 you say how many pies have between 0 and 29 cherries how many pies have between 30 and 59 including 30 and 59 how many pies have at least 60 and at most 89 cherries how many pies have at least 90 and at most 119 and then how many pies have at least 120 and at most 149 and you know that you don't have any pies that have more than 149 cherries so this should account for everything and then you count them so for example you say ok 5 5 30 to 59 cherries and so we create a histogram or you create a histogram and you make this magenta bar go up to 5 so that's how you would construct this histogram that's what this pie is at different cherry levels histogram is telling us so now that we know how to construct it let's see if we can interpret it based on the information given in the histogram so the first question is based on just this information can you figure out the total number of pies in your store zooming that they're all accounted for by this histogram and I encourage you to pause the video and try to figure it out on your own well what's the total number Five's well let's see there's 5 pies there's five pies that have between that have more than 30 or more and have at least 30 cherries but no more than 59 you have eight pies in this blue bucket you have four pies in this green bucket and then you have what is this five no this is three pies that have at least 120 but no more than 149 cherries and this accounts for all of the pies so the total number of pies you have at the store or five plus eight plus four plus three which is what five plus eight is thirteen plus four seventeen plus three is twenty so there are 20 pies in this store then you can ask more nuanced question what if you wanted to know the number of pies with more than 60 cherries the number of pies with more than 60 so number of pies with I'll say let's say 60 or more 60 or more 60 or more cherries so let's think about it well this magenta bar doesn't apply because these all have less than 60 but all of these other bars are counting pies that have 60 or more cherries this is 60 289 this is 19 119 this is 120 149 so it's going to be these eight cherries that are eight these eight pies that are in this bucket plus these four pies plus these three pies so it is going to be essentially everything but this first bucket everything but all the pies except for these five pies have 60 or more cherries so it should be five less than 20 and so let's see 8 plus 4 is 12 plus 3 is 15 which is five less than 20 so using this histogram we can answer a really interesting we can say well how many more pies do we have that have 60 289 cherries than 120 to 149 cherries well we say well we have eight pies that have 60 8960 289 cherries three that have 120 249 so we have five more pies in the sixty to eighty nine category than we do in the 120 to 149 category so a lot of questions that we can start to answer and hopefully this gives you a sense of how you can interpret histograms

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Interpreting a histogram (video) | Khan Academy

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(for Grand Prairie)

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A measure of variability is a summary statistic that represents the amount of dispersion in a dataset. How spread out are the values? While a measure of central tendency describes the typical value, measures of variability define how far away the data points tend to fall from the center. We talk about variability in the context of a distribution of values. A low dispersion indicates that the data points tend to be clustered tightly around the center. High dispersion signifies that they tend to fall further away.

In statistics, variability, dispersion, and spread are synonyms that denote the width of the distribution. Just as there are multiple measures of central tendency, there are several measures of variability. In this blog post, you'll learn why understanding the variability of your data is critical. Then, I explore the most common measures of variability-the range, interquartile range, variance, and standard deviation. I'll help you determine which one is best for your data.

The two plots below show the difference graphically for distributions with the same mean but more and less dispersion. The panel on the left shows a distribution that is tightly clustered around the average, while the distribution in the right panel is more spread out.

Related post: Measures of Central Tendency: Mean, Median, and Mode

Why Understanding Variability is Important

Let's take a step back and first get a handle on why understanding variability is so essential. Analysts frequently use the mean to summarize the center of a population or a process. While the mean is relevant, people often react to variability even more. When a distribution has lower variability, the values in a dataset are more consistent. However, when the variability is higher, the data points are more dissimilar and extreme values become more likely. Consequently, understanding variability helps you grasp the likelihood of unusual events.

In some situations, extreme values can cause problems! Have you seen a weather report where the meteorologist shows extreme heat and drought in one area and flooding in another? It would be nice to average those together! Frequently, we feel discomfort at the extremes more than the mean. Understanding that variability around the mean provides critical information.

Variability is everywhere. Your commute time to work varies a bit every day. When you order a favorite dish at a restaurant repeatedly, it isn't exactly the same each time. The parts that come off an assembly line might appear to be identical, but they have subtly different lengths and widths.

These are all examples of real-life variability. Some degree of variation is unavoidable. However, too much inconsistency can cause problems. If your morning commute takes much longer than the mean travel time, you will be late for work. If the restaurant dish is much different than how it is usually, you might not like it at all. And, if a manufactured part is too much out of spec, it won't function as intended.

Some variation is inevitable, but problems occur at the extremes. Distributions with greater variability produce observations with unusually large and small values more frequently than distributions with less variability.

Example of Different Amounts of Variability

Let's take a look at two hypothetical pizza restaurants. They both advertise a mean delivery time of 20 minutes. When we're ravenous, they both sound equally good! However, this equivalence can be deceptive! To determine the restaurant that you should order from when you're hungry, we need to analyze their variability.

Suppose we study their delivery times, calculate the variability for each place, and determine that their variabilities are different. We've computed the standard deviations for both restaurants-which is a measure that we'll come back to later in this post. How significant is this difference in getting pizza to their customers promptly?

The graphs below display the distribution of delivery times and provide the answer. The restaurant with more variable delivery times has the broader distribution curve. I've used the same scales in both graphs so you can visually compare the two distributions.

In these graphs, we consider a 30-minute wait or longer to be unacceptable. We're hungry after all! The shaded area in each chart represents the proportion of delivery times that surpass 30 minutes. Nearly 16% of the deliveries for the high variability restaurant exceed 30 minutes. On the other hand, only 2% of the deliveries take too long with the low variability restaurant. They both have an average delivery time of 20 minutes, but I know where I'd place my order when I'm hungry!

As this example shows, the central tendency doesn't provide complete information. We also need to understand the variability around the middle of the distribution to get the full picture. Now, let's move on to the different ways of measuring variability!

Range

Let's start with the range because it is the most straightforward measure of variability to calculate and the simplest to understand. The range of a dataset is the difference between the largest and smallest values in that dataset. For example, in the two datasets below, dataset 1 has a range of 20 - 38 = 18 while dataset 2 has a range of 11 - 52 = 41. Dataset 2 has a broader range and, hence, more variability than dataset 1.

While the range is easy to understand, it is based on only the two most extreme values in the dataset, which makes it very susceptible to outliers. If one of those numbers is unusually high or low, it affects the entire range even if it is atypical.

Additionally, the size of the dataset affects the range. In general, you are less likely to observe extreme values. However, as you increase the sample size, you have more opportunities to obtain these extreme values. Consequently, when you draw random samples from the same population, the range tends to increase as the sample size increases. Consequently, use the range to compare variability only when the sample sizes are similar.

The Interquartile Range (IQR) . . . and other Percentiles

The interquartile range is the middle half of the data. To visualize it, think about the median value that splits the dataset in half. Similarly, you can divide the data into quarters. Statisticians refer to these quarters as quartiles and denote them from low to high as Q1, Q2, and Q3. The lowest quartile (Q1) contains the quarter of the dataset with the smallest values. The upper quartile (Q4) contains the quarter of the dataset with the highest values. The interquartile range is the middle half of the data that is in between the upper and lower quartiles. In other words, the interquartile range includes the 50% of data points that fall between Q1 and Q3. The IQR is the red area in the graph below.

The interquartile range is a robust measure of variability in a similar manner that the median is a robust measure of central tendency. Neither measure is influenced dramatically by outliers because they don't depend on every value. Additionally, the interquartile range is excellent for skewed distributions, just like the median. As you'll learn, when you have a normal distribution, the standard deviation tells you the percentage of observations that fall specific distances from the mean. However, this doesn't work for skewed distributions, and the IQR is a great alternative.

I've divided the dataset below into quartiles. The interquartile range (IQR) extends from the low end of Q2 to the upper limit of Q3. For this dataset, the range is 21 - 39.

Using other percentiles

When you have a skewed distribution, I find that reporting the median with the interquartile range is a particularly good combination. The interquartile range is equivalent to the region between the 75th and 25th percentile (75 - 25 = 50% of the data). You can also use other percentiles to determine the spread of different proportions. For example, the range between the 97.5th percentile and the 2.5th percentile covers 95% of the data. The broader these ranges, the higher the variability in your dataset.

Related post: Percentiles: Interpretations and Calculations

Variance

Variance is the average squared difference of the values from the mean. Unlike the previous measures of variability, the variance includes all values in the calculation by comparing each value to the mean. To calculate this statistic, you calculate a set of squared differences between the data points and the mean, sum them, and then divide by the number of observations. Hence, it's the average squared difference.

There are two formulas for the variance depending on whether you are calculating the variance for an entire population or using a sample to estimate the population variance. The equations are below, and then I work through an example in a table to help bring it to life.

Population variance

The formula for the variance of an entire population is the following:

In the equation, σ ² is the population parameter for the variance, μ is the parameter for the population mean, and N is the number of data points, which should include the entire population.

Sample variance

To use a sample to estimate the variance for a population, use the following formula. Using the previous equation with sample data tends to underestimate the variability. Because it's usually impossible to measure an entire population, statisticians use the equation for sample variances much more frequently.

In the equation, s ² is the sample variance, and M is the sample mean. N-1 in the denominator corrects for the tendency of a sample to underestimate the population variance.

Example of calculating the sample variance

I'll work through an example using the formula for a sample on a dataset with 17 observations in the table below. The numbers in parentheses represent the corresponding table column number. The procedure involves taking each observation (1), subtracting the sample mean (2) to calculate the difference (3), and squaring that difference (4). Then, I sum the squared differences at the bottom of the table. Finally, I take the sum and divide by 16 because I'm using the sample variance equation with 17 observations (17 - 1 = 16). The variance for this dataset is 201.

Because the calculations use the squared differences, the variance is in squared units rather the original units of the data. While higher values of the variance indicate greater variability, there is no intuitive interpretation for specific values. Despite this limitation, various statistical tests use the variance in their calculations. For an example, read my post about the F-test and ANOVA.

While it is difficult to interpret the variance itself, the standard deviation resolves this problem!

Standard Deviation

The standard deviation is the standard or typical difference between each data point and the mean. When the values in a dataset are grouped closer together, you have a smaller standard deviation. On the other hand, when the values are spread out more, the standard deviation is larger because the standard distance is greater.

Conveniently, the standard deviation uses the original units of the data, which makes interpretation easier. Consequently, the standard deviation is the most widely used measure of variability. For example, in the pizza delivery example, a standard deviation of 5 indicates that the typical delivery time is plus or minus 5 minutes from the mean. It's often reported along with the mean: 20 minutes (s.d. 5).

The standard deviation is just the square root of the variance. Recall that the variance is in squared units. Hence, the square root returns the value to the natural units. The symbol for the standard deviation as a population parameter is σ while s represents it as a sample estimate. To calculate the standard deviation, calculate the variance as shown above, and then take the square root of it. Voila! You have the standard deviation!

In the variance section, we calculated a variance of 201 in the table.

Therefore, the standard deviation for that dataset is 14.177.

People often confuse the standard deviation with the standard error of the mean. Both measures assess variability, but they have extremely different purposes. To learn more, read my post The Standard Error of the Mean.

The Empirical Rule for the Standard Deviation of a Normal Distribution

When you have normally distributed data, or approximately so, the standard deviation becomes particularly valuable. You can use it to determine the proportion of the values that fall within a specified number of standard deviations from the mean. For example, in a normal distribution, 68% of the values will fall within +/- 1 standard deviation from the mean. This property is part of the Empirical Rule. This rule describes the percentage of the data that fall within specific numbers of standard deviations from the mean for bell-shaped curves.

Let's take another look at the pizza delivery example where we have a mean delivery time of 20 minutes and a standard deviation of 5 minutes. Using the Empirical Rule, we can use the mean and standard deviation to determine that 68% of the delivery times will fall between 15-25 minutes (20 +/- 5) and 95% will fall between 10-30 minutes (20 +/- 2*5).

Related post: The Normal Distribution

Which is Best-the Range, Interquartile Range, or Standard Deviation?

First off, you probably notice that I didn't include the variance as one of the options in the heading above. That's because the variance is in squared units and doesn't provide an intuitive interpretation. So, I've crossed that off the list. Let's go over the other three measures of variability.

When you are comparing samples that are the same size, consider using the range as the measure of variability. It's a reasonably intuitive statistic. Just be aware that a single outlier can throw the range off. The range is particularly suitable for small samples when you don't have enough data to calculate the other measures reliably, and the likelihood of obtaining an outlier is also lower.

When you have a skewed distribution, the median is a better measure of central tendency, and it makes sense to pair it with either the interquartile range or other percentile-based ranges because all of these statistics divide the dataset into groups with specific proportions.

For normally distributed data, or even data that aren't terribly skewed, using the tried and true combination reporting the mean and the standard deviation is the way to go. This combination is by far the most common. You can still supplement this approach with percentile-base ranges as you need.

Except for variances, the statistics in this post are absolute measures of variability because they use the original variable's measurement units. Read my post about the coefficient of variation to learn about a relative measure of variability that can be advantageous in some circumstances.

Analysts frequently use measures of variability to describe their datasets. Learn how to Analyze Descriptive Statistics in Excel.

If you're learning about statistics and like the approach I use in my blog, check out my Introduction to Statistics eBook!

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Measures of Variability: Range, Interquartile Range, Variance, and Standard Deviation - Statistics By Jim

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A measure of variability is a summary statistic that represents the amount of dispersion in a dataset. How spread out are the values? While a measure of central tendency describes the typical value, measures of variability define how far away the data points tend to fall from the center.

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One of the resources for our lesson

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Measures of Variability (Variance, Standard Deviation, Range, Mean Absolute Deviation)

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Suzie, Dave, Jeff - measures of spread DP MATH AI IB SL 4.3

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Suzie, Dave, Jeff - measures of spread DP MATH AI IB SL 4.3

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Measures of central tendency lesson plan

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Measures of central tendency lesson plan

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Explanation of the real-life, in class data collection for Scatterplots

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Explanation of the real-life, in class data collection for Scatterplots

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Explanation of the real-life, in class data collection for Scatterplots

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The gist of the lesson plan will explain more about collecting the data.

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8.5CD/8.11A Resources - Released STAAR Questions, Vocabulary, Example handout

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8.5CD/8.11A Resources - Released STAAR Questions, Vocabulary, Example handout

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Cohort 2019 is collecting data.  Here you will see them playing Trash-ket Ball.  Each square is a foot and the amount of baskets made versus distance is what is being collected.

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Cohort 2019 is collecting data.  Here you will see them playing Trash-ket Ball.  Each square is a foot and the amount of baskets made versus distance is what is being collected.

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Readiness
8.5(D) use a trend line that approximates the linear relationship between bivariate sets of data to
make predictions
Supporting
8.5(C) contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data
that do not suggest a linear relationship from a graphical representation
8.11(A) construct a scatterplot and describe the observed data to address questions of association
such as linear, nonlinear, and no association between bivariate data
8.11(B) determine the mean absolute deviation and use this quantity as a measure of the average
distance data are from the mean using a data set of no more than 10 data points
Not Assessed
8.11(C) simulate generating random samples of the same size from a population with known characteristics to develop the notion of a random sample being representative of the population from which it was selected

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Readiness
7.6(G) solve problems using data represented in bar graphs, dot plots, and circle graphs, including
part‐to‐whole and part‐to‐part comparisons and equivalents
7.12(A) compare two groups of numeric data using comparative dot plots or box plots by comparing
their shapes, centers, and spreads
Supporting
7.12(B) use data from a random sample to make inferences about a population
7.12(C) compare two populations based on data in random samples from these populations,
including informal comparative inferences about differences between the two populations

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Readiness Standards
6.12(C) summarize numeric data with numerical summaries, including the mean and median
(measures of center) and the range and interquartile range (IQR) (measures of spread), and
use these summaries to describe the center, spread, and shape of the data distribution
6.12(D) summarize categorical data with numerical and graphical summaries, including the mode, the percent of values in each category (relative frequency table), and the percent bar graph, and use these summaries to describe the data distribution
6.13(A) interpret numeric data summarized in dot plots, stem‐and‐leaf plots, histograms, and box plots
Supporting Standards
6.12(A) represent numeric data graphically, including dot plots, stem‐and‐leaf plots, histograms, and box plots
6.12(B) use the graphical representation of numeric data to describe the center, spread, and shape of the data distribution
6.13(B) distinguish between situations that yield data with and without variability