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Lessons

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    (Also available in CODAP)

    Students compute the “line of best fit” using the function for linear regression, and summarize linear relationships in a dataset.

    Lesson Goals

    Students will be able to…​

    • interpret linear regression in the context of the animals table

    • use linear regression to quantify patterns in a dataset

    Student-facing Lesson Goals

    • Let’s learn how to determine the strength of relationships between variables.

    Materials

    Supplemental Materials

    Preparation
    Glossary
    explanatory variable

    When modeling a possible relationship between an input and an output (e.g. - height and age), we are curious about how a change in the input (typically graphed on the x-axis of a scatter plot) might "explain" the output (y). When the behavior of the output may be explained by the input, we refer to the input as the "explanatory variable".

    line of best fit

    summarizes the relationship (if linear) between two quantitative variables in such a way as to minimize the errors overall when using explanatory values to predict responses

    linear regression

    a type of analysis that models the relationship between two quantitative variables. The result is known as a regression line, or line of best fit.

    predictor function

    a model of patterns in data. These models reduce the complexity of the data, predicting the value of the response variable as if completely dependent on the explanatory (independent) variable.

    r

    a number between −1 and 1 that measures the direction and strength of a linear relationship between two quantitative variables (also known as correlation value)

    response variable

    the variable in a relationship, generally plotted on the y-axis of a scatter plot, that is presumed to be affected by the explanatory variable; in some contexts the response variable is referred to as the "dependent variable" or the "output"

    slope

    the steepness of a straight line on a graph reported as a number which tells how much y changes for every unit increase in x

    y-intercept

    the point where a line or curve crosses the y-axis of a graph

    🔗Intro to Linear Regression 10 minutes

    Overview

    Students are introduced to the concept of linear regression, and learn how to interpret the slope and y-intercept of predictor functions.

    For teachers who have the need and the bandwidth to go deeper, this is a good opportunity to teach the algorithm behind linear regression. This algorithm is not part of Bootstrap:Data Science.

    Launch

    Vocabulary Note

    We prefer the words “explanatory” and “response” in our curriculum, because in other contexts the words “dependent” and “independent” refer to whether or not the variables are related at all, as opposed to what role each plays in the relationship.

    The straight line that best fits the points on a scatter plot has several names, depending on the context, subject, or grade level. All of the following terms refer to the same concept:

    • Regression Line

    • Line of Best Fit

    • Least Squares Line

    • Predictor

    • Trendline

    The line itself is computed through a process called linear regression, which also goes by the name least squares regression.

    Make two scatterplots from the animals-table, using age as the explanatory variable in one plot and pounds as the explanatory variable in the other.

    In both plots, use weeks as your response variable and name for the labels. We will refer to the explanatory column as “xs” and the response column as “ys.”

    weeks-v-age scatter plot with some points in the lower left and some in the upper right

    Quickly survey the class about whether they think we can use an animal’s size or age to predict how long it will take to be adopted.

    • A scatter plot of adoption time versus size does suggest that smaller animals get adopted faster than larger animals.

    • Similarly, younger animals tend to be adopted faster than older ones.

    Can we be more precise about this, and actually predict how long it will take an animal to be adopted, based on these factors? And which one would give us a better prediction?

    Just as the mean, median, and mode can summarize data from a single quantitative variable, the “center” of points in a two-dimensional cloud can be described by a line of best fit. Given a value for the explanatory variable, this line allows us to predict what the corresponding value for the response variable might be.

    Open the Interactive Regression Line (Geogebra), and use it to complete Introduction to Linear Regression.

    Have students talk about their answers to the first section.

    • Why do x-values that are far to the left or right of the point cloud have a larger impact on R?

      • Think of their distance from the "anchor" of the data as the length of a wrench. The longer the handle, the more "force" P applies on the regression line.

    • Why can’t we have a regression line with all points above or below the line?

      • Because the points exert a balanced force on the line. If all the points were on one side, there would be nothing to balance them out on the other side.

    Data scientists use a statistical method called linear regression to pinpoint linear relationships in a dataset. Linear regression is the math behind the line of best fit, which allows us to make predictions based on our data.

    Each point in our scatter plot “pulls” on the line, with points above the line yanking it up and points below the line dragging it down. Points that are really far away — especially influential observations that are far to the left or right of the point cloud — pull on the line with more force.

    This line can be graphed on top of the scatter plot as a function, called the predictor function.

    Investigate

    • Turn to Drawing Predictors.

    • Draw a line of best fit through each of the scatter plots.

    • Decide whether the direction of the correlation (which could also be referred to as the slope of the line) is positive or negative.

    • Then assess the strength of the correlation.

    Common Misconceptions

    • Don’t forget to look at sample size! A linear regression plot with an 𝑟-value of 0.999 is strong…​but that’s useless if it’s a sample of just three data points!

    Synthesize

    Give students some time to experiment, then share back observations.

    🔗Performing Linear Regression 20 minutes

    Overview

    Students are introduced to the lr-plot function in Pyret, which performs a linear regression and plots the result.

    Launch

    Pyret includes a powerful display called lr-plot, which (1) draws a scatter plot, (2) draws the line of best fit, and (3) even displays the equation for that line.

    Just like the scatter-plot function, lr-plot takes in a Table and the names of 3 Columns, the first of which will be used to label the points.

    • Open your saved Animals Starter File, or make a new copy.

    • Create an `lr-plot`for the Animals Table.

      • Use "name" for the labels.

      • Use "age" for the x-axis.

      • Use "weeks" for the y-axis.

    A screenshot of a linear-regression plot in pyret highlighting the information displayed at the top. The predictor function comes first, in this case y = 0.789x + 2.309, and is followed by the correlation, in this case R: +0.448.

    1. The resulting scatter plot looks like those we’ve seen before!

    2. The line of best fit is now drawn onto the plot.

    3. Above the display,we see the predictor function for that line.

      • The predictor function is written in slope-intercept form. (𝑦 = 𝑚𝑥 + 𝑏, where 𝑚 describes the slope or rate of change and 𝑏 identifies the y-intercept.)

      • In this plot, we can see that the slope of the line is 0.789, which means that on average, each extra year of age results in an extra 0.789 weeks of waiting to be adopted (about 5 or 6 extra days).

      • The y-intercept is 2.2309. This is where the best-fitting line crosses the y-axis. We want to be careful not to interpret this too literally, and say that a newborn animal would be adopted in 2.309weeks, because none of the animals in our dataset was that young. Still, the regression line (or line of best fit) suggests that a baby animal, whose age is close to 0, would take only about 3 weeks to be adopted.

      • By subsituting an animal’s age for x in the predictor function, we can make a prediction about how many weeks it will take to be adopted.

      • For example, we predict a 4-year-old animal to be adopted in 0.789(5) + 2.309 = 6.254weeks. That’s the y-value when 𝑥 = 5 for a point that falls exactly on the regression line.

    4. We also see the 𝑟-value is +0.448.

      • The sign ispositive, consistent with the fact thatthe scatter plot point cloud and line of best fit both slope upward.

      • The fact that the 𝑟-value is close to 0.5 tells us that the strength is moderate.

      • This makes sense: the scatter plot points are somewhere between being really tightly clustered and really loosely scattered.

    Going Deeper

    Students may notice another value in the lr-plot, called 𝑅2.𝑅2 describes the percentage of the variation in the y-variable that is explained by least-squares regression on the x variable. In other words, an 𝑅2 value of 0.20 could mean that “20% of the variation in adoption time is explained by regressing adoption time on the age of the animal”. Discussion of 𝑅2 may be appropriate for older students, or in an AP Statistics class.

    Investigate

    Have students share their answers and discuss

    Synthesize

    A predictor only makes sense within the range of the data that was used to generate it.

    Toddlers grow a lot faster than adults. A regression line predicting the height of toddlers based on age would predict that a 60-year-old is 10 feet tall!

    Statistical models are just proxies for the real world, drawn from a limited sample of data: they might make a useful prediction in the range of that data, but once we try to extrapolate beyond that data we may quickly get into trouble!

    Simpson’s Paradox

    A common misconception is that "more data is always better", and the age-v-height worksheet challenges that assumption. Two sub-groups (girls and boys) can each have a strong correlation between age and height, but when they are combined the correlation is weaker. This phenomenon is called Simpson’s Paradox. Statistics (especially AP!) teachers will want to dive deeper on this topic.

    🔗Interpreting Linear Regression Lines 20 minutes

    Overview

    Students learn how to write about the results of a linear regression, using proper statistical terminology and thinking through the many ways this language can be misused.

    Launch

    How well can you interpret the results of a linear regression analysis? How would you explain it to someone else?

    • What does it mean when a data point is above the line of best fit?

      • It means the y-value is higher than the sample would have predicted for that x-value.

    • What does it mean when a data point is below the line of best fit?

      • It means the y-value is lower than the sample would have predicted for that x-value.

    • Turn to Interpreting Regression Lines & r-Values, and fill in the blanks for each scenario on the left using information from the predictor function and r-value on the right.

    Let’s take a look at how the Data Cycle can be used with Linear Regression, and how the result can be used to form our Data Story.

    • Read Data Cycle: Regression Analysis (Example).

    • What do you Notice? What do you Wonder?

    • Do the Ask Questions and Consider Data steps match each other? Why or why not?

      • Yes. We’re thinking about how age of cats impact time to adoption by looking a the cat rows and comparing age and weeks.

    • At the bottom of the page we have the Data Story for this question, which includes the results of the analysis and a responsible way to write about them.

    Investigate

    Common Misconceptions

    • Don’t call it "accuracy"! One of the most common misconceptions about Linear Regression is that the 𝑟 or 𝑅2 value is a measure of accuracy. For example, a student who sees a very high 𝑟-value when plotting age vs. weeks might say "this prediction is 95% accurate." But these values only speak to how much variation in the y-axis can be explained by variation in the x-axis, so the statement should be "95% of the variation in weeks can be explained by variation in the age."

    • X and Y matter! The correlation coefficient will be the same, even if you swap the x- and y-axes. However, the interpretation of the display is different! The column used for the x-axis will always be interpreted as "the explanation" for the "result" seen in the y-axis. It’s fine to say that being older tends to make an animal take longer to be adopted, but it is not true that taking longer to be adopted makes an animal older!

    Synthesize

    Have students read their data stories aloud, to get comfortable with the phrasing.

    A scatter plot with a dense cluster of points moving up along the y-axis from 40 to 73 and then curving out to the right between 73 and 84 as the x increases from 0 to 8000.

    The word “linear” in “linear regression” is important here. In the image on the right, there’s clearly a pattern, but it doesn’t look like a straight line!

    There are many other kinds of statistical models out there, but all of them work the same way: use a particular kind of mathematical function (linear or otherwise), to figure out how to get the “best fit” for a cloud of data.

    Project Option: Olympic Records

    In this project, students analyze Olympic data in running, swimming, or speed skating. They analyze change over time using scatter plots and linear regression. Optional Project: Olympic Records [rubric] can be used as a mid-term or formative assessment, or as a capstone for a limited implementation of Bootstrap:Data Science.

    (Project designed by Joy Straub)

    🔗Data Exploration Project (Linear Regression) flexible

    Overview

    Students apply what they have learned about linear regression to their chosen dataset. They will add at least one linear regression display to their Data Exploration Project Slide Template, along with their interpretation of what this display tells them. To learn more about the sequence and scope of the Exploration Project, visit Project: Dataset Exploration. For teachers with time and interest, Project: Create a Research Project is an extension of the Dataset Exploration, where students select a single question to investigate via data analysis.

    Launch

    Let’s review what we have learned about linear regression.

    Linear Regression is a way to calculate the line-of-best-fit (or "predictor function") for the relationship between two quantitative columns.

    • Will `lr-plot`still find a line of best fit, even if there’s no correlation?

      • Yes! Linear regression will always find a line of best fit — it just might not fit very well!

    • What does the slope of the line-of-best-fit tell us about a correlation?

      • If the slope is positive, the correlation is positive. If it’s negative, so is the correlation._

    • If the 𝑟-value is close to 1, does this mean the predictor function will always give us a good prediction of the y-value, based on any x-value? Why or why not?

      • No! Even with an exremely high 𝑟-value, the predictor function should not be used to make predictions far outside the range of the dataset.

    Investigate

    Let’s connect what we know about linear regression to your chosen dataset.

    Confirm that all students have created and understand how to interpret their LR plots. Once you are confident that all students have made adequate progress, invite them to access their Data Exploration Project Slide Template from Google Drive.

    • It’s time to add to your Data Exploration Project Slide Template.

    • Find the "Correlations I want to look into" section of the slide deck.

    • Choose one correlation you explored, and duplicate the slide.

    • On the new slide, replace your scatter plot with the linear regression plot and add your interpretation of that plot.

    Synthesize

    Have students share their findings.

    Did you discover anything surprising or interesting about their dataset?

    Did the results from `lr-plot`confirm your suspicions about the correlation? Were any of them surprising?

    🔗Additional Exercises

    These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.