Graph of a scatter plot, origin O. Horizontal axis, distance (miles), scale 0 to 10, by 2’s. Vertical axis, cost (dollars), scale 0 to 22, by 2’s. The data has a visible pattern rising from near (point 5 comma 4) to (9 point 5 comma 21).
B
Graph of a scatter plot, origin O, with grid. Horizontal axis, height (millimeters), scale 0 to 10, by 2’s. Vertical axis, weight (milligrams), scale 0 to 22, by 2’s. The data has no visible pattern.
C
Scatterplot, origin O. Horizontal, from 0 to 10, by 2’s, labeled precipitation, centimeters. Vertical, from 0 to 22, by 2’s, labeled water used for irrigation, thousands of gallons. 25 points represent the data, trend linearly downward and to the right. First data point approximately begins at 0 comma 10 and last is approximately at 7 comma 3 point 5.
D
Scatterplot, origin O. Horizontal, from 0 to 10, by 2’s, labeled temperature, degrees Celsius. Vertical, from 0 to 22, by 2’s, labeled number of phytoplankton, tens of thousands. 25 points that represent the data, trend slightly upward and to the right first and then very steep upward curve and to the right. First data point approximately begins at 0 comma 2 point 5 and last is approximately at 9 point 5 comma 21.
7.2
Activity
Card Sort: Scatter Plot Fit
Your teacher will give you a set of cards that show scatter plots.
Sort the cards into categories of your choosing. Be prepared to describe your categories.
Pause for a whole-class discussion.
Sort the cards into new categories in a different way. Be prepared to describe your new categories.
7.3
Activity
Matching Correlation Coefficients
Take turns with your partner to match a scatter plot with a correlation coefficient.
For each match you find, explain to your partner how you know it’s a match.
For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.
A
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 10, by 2’s. Vertical axis scale 0 to 10, by 2’s. Best fit line from approximately (0 comma 2) to near (10 comma 8). The data fits a linear model with a positive slope.
B
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 90, by 10’s. Vertical axis scale 0 to 500, by 50’s. Best fit line from approximately (35 comma 75) to near (90 comma 500). The data is slightly scattered and trends upward with a positive slope.
C
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 60, by 10’s. Vertical axis scale 0 to 30, by 5’s. Best fit line from approximately (5 comma 25) to near (60 comma 12). The data fits a linear model with a negative slope.
D
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 10, by 2’s. Vertical axis scale 0 to 10, by 2’s. Best fit line from approximately (0 comma 8) to near (10 comma 4). The data is slightly scattered and trends downward with a negative slope.
E
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 10, by 2’s. Vertical axis scale 0 to 10, by 2’s. Best fit line from approximately (3 comma 0) to near (10 comma 8). The data is slightly scattered and trends upward with a positive slope.
F
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 10, by 2’s. Vertical axis scale 0 to 10, by 2’s. Best fit line from approximately (0 comma 8) to near (10 comma 3). The data fits a linear model with a negative slope.
G
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 24, by 4’s. Vertical axis scale 0 to 14, by 2’s. Best fit line from approximately (8 comma 7) to near (20 comma 6). The data is scattered and trends downward slightly with a negative slope.
H
Graph of a scatter plot, xy-plane, origin O. Horizontal axis scale 0 to 24, by 3’s. Vertical axis scale 0 to 275, by 25’s. Best fit line from approximately (2 comma 50) to near (24 point 5 comma 200). The data is slightly scattered and trends upward with a positive slope.
Student Lesson Summary
While residuals can help pick the best-line to fit the data among all lines, we still need a way to determine the strength of a linear relationship. Scatter plots of data that are close to the best-fit line are better modeled by the line than are scatter plots of data that are farther from the line.
The correlation coefficient is a convenient number that can be used to describe the strength and direction of a linear relationship. Usually represented by the letter , the correlation coefficient can take values from -1 to 1. The sign of the correlation coefficient is the same as the sign of the slope for the best-fit line. The closer the correlation coefficient is to 0, the weaker the linear relationship. The closer the correlation coefficient is to 1 or -1, the better a linear model fits the data.
While it is possible to try to fit a linear model to any data, we should always look at the scatter plot to see if there is a possible linear trend. The correlation coefficient and residuals can also help determine whether the linear model makes sense to use to estimate the situation. In some cases, another type of function might be a better fit for the data, or the two variables we are examining may be uncorrelated, and we should look for connections using other variables.
Glossary
correlation coefficient
A correlation coefficient is a number between -1 and 1 that describes the strength and direction of a linear relationship between two numerical variables.
The sign of the correlation coefficient is the same as the sign of the slope of the best-fit line.
The closer the correlation coefficient is to 0, the weaker the linear relationship.
When the correlation coefficient is closer to 1 or -1, the linear model fits the data better.
Correlation coefficient is close to 1.
Correlation coefficient is positive, and closer to 0.
Correlation coefficient is close to -1.
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