Researchers, such as anthropologists, are often interested in how two measurements are related. The statistical study of the relationship between variables is called regression.

A scatter plot is helpful in understanding the form, direction, and strength of the relationship between two variables. Correlation is the strength and direction of the linear relationship between the two variables.

Try to have about the same number of points above and below the line of best fit.

Helpful Hint

If there is a strong linear relationship between two variables, a line of best fit, or a line that best fits the data, can be used to make predictions.

Ex 1:

Albany and Sydney are about the same distance from the equator. Make a scatter plot with Albany’s temperature as the independent variable. Name the type of correlation. Then sketch a line of best fit and find its equation.

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Step 1 Plot the data points.

Step 2 Identify the correlation.

Notice that the data set is negatively correlated–as the temperature rises in Albany, it falls in Sydney.

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Step 3 Sketch a line of best fit.

Draw a line that splits the data evenly above and below.

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Step 4 Identify two points on the line.

For this data, you might select (35, 64) and (85, 41).

Step 5 Find the slope of the line that models the data.

Use the point-slope form.

An equation that models the data is y = –0.46x + 80.1.

y – y1= m(x – x1)

y – 64 = –0.46(x – 35)

y = –0.46x + 80.1

Point-slope form.

Substitute.

Simplify.

The correlation coefficient r is a measure of how well the data set is fit by a model.

To display the correlation coefficient r, you may have to turn on the diagnostic mode. To do this, press and choose the DiagnosticOn mode.

You can use a graphing calculator to perform a linear regression and find the correlation coefficient r.

Ex 2:

Anthropologists can use the femur, or thighbone, to estimate the height of a human being. The table shows the results of a randomly selected sample.

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a. Make a scatter plot of the data with femur length as the independent variable.

b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem.

Enter the data into lists L1 and L2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4:LinReg. The equation of the line of best fit is h ≈ 2.91l + 54.04.

The slope is about 2.91, so for each 1 cm increase in femur length, the predicted increase in a human being’s height is 2.91 cm.

The correlation coefficient is r ≈ 0.986 which indicates a strong positive correlation.

c. A man’s femur is 41 cm long. Predict the man’s height.

The height of a man with a 41-cm-long femur would be about 173 cm.

h ≈ 2.91(41) + 54.04

The equation of the line of best fit is h ≈ 2.91l + 54.04. Use the equation to predict the man’s height. For a 41-cm-long femur,

h ≈ 173.35