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This lesson explores the basics of statistical analysis, including graphing data, calculating residuals, and making predictions using a line of best fit. Students will also learn about outliers and correlation coefficients.
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CCSS – Statistics ThreadLouisville, KY 2012 Lisa Fisher-Comfort, MN Regional Coordinator, Michael Long, HI Regional Coordinator
Lesson 6.1.2 - Closure • The first step in any statistical analysis is to graph the data. Graphs do not necessarily start at the origin; indeed, frequently in statistical analyses they do not. • A residual is a measure of how far our prediction using the best-fit model is from what was actually observed. • A residual has the same units as the y-axis. • A residual can be graphed with a vertical segment. The length of this segment (in the units of the y-axis) is the residual. • A positive residual means the actual observed y-value of a piece of data is greater than the y‑value that was predicted by the LSRL. • A negative residual means the actual data is less than predicted. • Extrapolation of a statistical model can lead to nonsensical results.
The following table shows data for one season of the El Toro professional basketball team. El Toro team member Antonio Kusoc was inadvertently left off of the list. Antonio Kusoc played for 2103 minutes. We would like to predict how many points he scored in the season.
Your Task (6-30) a. Obtain a Lesson 6.1.4 Resource Page from your teacher. Draw a line of best fit for the data and then use it to write an equation that models the relationship between total points in the season and minutes played. b. Which data point is an outlier for this data? Whose data does that point represent? What is his residual? c. Would a player be more proud of a negative or positive residual? d. Predict how many points Antonio Kusoc made.
LSRL on a Calculator 6-33. A least squares regression line (LSRL) is a unique line that has the smallest possible value for the sum of the squares of the residuals. a. Your teacher will show you how to use your calculator to make a scatterplot. (Graphing calculator instructions can also be downloaded from www.cpm.org/technology.) Be sure to use the checksum at the bottom of the table in problem 6-30 to verify that you entered the data into your calculator accurately. b. Your teacher will show you how to find the LSRL and graph it on your calculator. Sketch your scatterplot and LSRL on your paper.
Lesson 6.1.4 - Closure • This is a two-day lesson. Problem 6-34 is a Least Squares Demo that can be teacher led to summarize their understanding of Least Square Regression Lines. • LeastSquaresDemo.html
Find the Correlation Coefficient for the El Toro Basketball team • Describe the form, direction, strength, and outliers of the association. Form could be linear but the residual plot indicates a another model might be better. Direction is negative with a slope of 0.59; an increase of one minutes played produced 0.59 points scored on the average. Strength is a fairly strong and positive linear association because r = 0.865. Outliers: Scottie Sordan (a.k.a. Michael Jordan).
Lesson 6.2.2 Closure Computer Exploration of Correlation Coefficients Problem 6-72 Students are asked to create scatterplots with the following associations and record r: Strong positive linear association Weak positive linear association Strong negative linear association No linear association (random scatter) http://illuminations.nctm.org/LessonDetail.aspx?ID=L456#first
Displaying Temperatures • Is the planet getting hotter? Experts look at the temperature of the air and the oceans, the kinds of molecules in the atmosphere, and many other kinds of data to try to determine how the earth is changing. However, sometimes the same data can lead to different conclusions because of how the data is represented. • Your teacher will provide you with temperature data from November 1, 1975, and from November 1, 2000. To make sense of this data, you will first need to organize it in a useful way. • Your teacher will assign you a city and give you two sticky notes. Label the appropriately colored sticky note with the name of the city and its temperature in 1975. Label the other sticky note with its city name and temperature in 2000. • Follow the directions of your teacher to place your sticky notes on the class histogram. Use the axis at the bottom of the graph to place your sticky note. • How many cities were measured for this study? • Describe the spread and shape of each of the histograms that you have created. Which measure of central tendency would you use to describe a typical temperature for each year? Justify your choice. • http://www.almanac.com/weather
Boxplots and Temperatures • The histograms your class made in problem 8-44 display data along the horizontal axis. Another way to display the data is to form a box plot, which divides the data into four equal parts, or quartiles. To create a box plot, follow the steps below with the class or in your team. With a sticky dot provided by your teacher, plot the 1975 temperature for your city on a number line in front of the class. What is the median temperature for 1975? Place a vertical line segment about one-half inch long marking this position above the number line on your resource page. How far does the data extend from the median? That is, what are the minimum and maximum temperatures in 1975? Place vertical line segments marking these positions above the number line. The median splits the data into two sets: those that come before it and those that come after it when the data is ordered from least to greatest, like it is on the number line. Find the median of the lower set (called the first quartile). Mark the first quartile with a vertical line segment above the number line. Look at the temperatures that come after the median on your number line. The median of this portion of data is called the third quartile. Mark the third quartile with a vertical line segment above the number line. Draw a box that contains all of the data points between the first and third quartiles. Your graph should be similar to a box with outer segments like the one shown below. What does the box plot tell you about the temperatures of the cities in 1975 that the dot plot did not?
Two-day lesson Students use center, shape, spread and outliers to compare two sets of numerical data.
8-30. Mrs. Ross is the school basketball coach. She wants to compare the scoring results for her team from two different games. The number of points scored by each player in each of the games are shown below. Game 1: 12, 10, 10, 8, 11, 4, 10, 14, 12, 9 Game 2: 7, 14, 11, 12, 8, 13, 9, 14, 4, 8 a. How many total players are on the team? b. What is the mean number of points per player for each game? c. What is the median number of points per player for each game? d. What is the range of points scored by each player for each game? e. With your team, discuss and find another method for comparing the data. f. Do you think the scoring in two games is equivalent?
Mean Absolute Deviation Data from Game 1: 12, 10, 10, 8, 11, 4, 10, 14, 12, 9
Standard Deviation • Data from 11-60 Sugar W: 10, 32, 32, 34, 34, 36, 37, 39, 39, 40, 41, 43, 43, 44, 45, 46, 46, 49, 70checksum760
