Draw Scatter Plots and Best-Fitting Lines

Section 2.6

A scatter plot is a graph of a set of data pairs (x, y). If y tends to increase as x increases, then the data have a positive correlation.

If y tends to decrease as x increases, then the data have a negative correlation.

If the points show no obvious pattern, then the data have approximately no correlation.

Negative Correlation No Correlation

Describe the correlation shown by each scatter plot.

Example 1

Positive Correlation

Col

ors

in a

Rai

nbow

Shoe Size

No Correlation

Negative Correlation

A correlation coefficient, denoted by r, is a number from -1 to 1 that measures how well a line fits a set of data pairs (x, y).

If r is near 1, the points lie close to a line with a positive slope.

If r is near -1, the points lie close to a line with negative slope.

If r is near 0, the points do not lie close to any line.

This line is called the best fit line or line of best fit.

Correlation Coefficients

Approximating a Best-Fit Line

1. Draw a scatter plot of the data.

2. Sketch the line that appears to follow most closely the trend given by the data points. There should be about as many points above the line as below.

3. Choose two points on the line, and estimate the coordinates of each point. These points do not have to be original data point.

4. Write an equation of the line that passes through the two points. This is a model for the data.

Dogs are aged differently from humans. You may have heard someone say that a dog ages 1 year for every 7 human years. However, that is not the case. The table on the next slide shows the relationship between dog years and human years.

Example 2

DogYears

HumanYears

1 152 243 284 325 376 427 47

a. Plot each of these points on the graph.

b. What type of correlation does this data have?

Positive

Dog Years

Hum

an Years

c. Is the correlation coefficient closer to -1, 0, or 1?

Closer to 1.

Dog Years

Hum

an Years

d. Sketch the best-fit line for this data.

Dog Years

Hum

an Years

e. Write the slope-intercept form of equation of the best-fit line for this data.

Dog Years

Hum

an Years

(5, 37)

(6, 42)

42 375

6 5m

37 5 5y x

5 12y x

Graphing a scatterplot and best fit line on a graphing calculator.

1. Press 2nd + 4 to clear all lists.

2. Press STAT EDIT.

3. Type the x values in L1 and the y values in L2.

Press ENTER to move to the next x value or next y value

4. Press STAT CALC 4 and press ENTER.

Arrow down to Calculate and press ENTER.

y = 5.03x + 12 is the approximate equation of the best fit line.

5. To graph the scatterplot press 2nd STAT PLOT (above Y=). Press ENTER

6. Make an appropriate window for your data

Press WINDOW. Type the following and then press GRAPH.

7. Go to Y= and type in the best fit equation and then press GRAPH.