Cambridge International Mathematics

The following table is a guide for describing the strength of linear association using the coefficient of

determination:

Value Strength of association

r^2 =0 no correlation

0 <r^2 < 0 : 25 very weak correlation

0 : 256 r^2 < 0 : 50 weak correlation

0 : 506 r^2 < 0 : 75 moderate correlation

0 : 756 r^2 < 0 : 90 strong correlation

0 : 906 r^2 < 1 very strong correlation

r^2 =1 perfect correlation

For example, for the daily temperature data, as r^2 ¼ 0 : 631 and r> 0 , the two variablesNandtshow

moderate positive correlation only.

LINEAR REGRESSION BY COMPUTER

Click on the icon to obtain a computer statistics package. This will enable you to find the

equation of the linear regression line, as well asrandr^2.

INTERPOLATION AND EXTRAPOLATION

Suppose we have gathered data to investigate the association between two variables. We

obtain the scatter diagram shown below. The data values with the lowest and highest values

ofxare called thepoles.

We use least squares regression to obtain a line of

best fit. We can use the line of best fit to estimate

values of one variable given a value for the other.

If we use values ofxin betweenthe poles, we say

we areinterpolatingbetween the poles.

If we use values ofxoutsidethe poles, we say we

areextrapolatingoutside the poles.

The accuracy of an interpolation depends on how

linear the original data was. This can be gauged by

determining the correlation coefficient and ensuring that the data is randomly scattered around the line of

best fit.

The accuracy of an extrapolation depends not only on how linear the original data was, but also on the

assumption that the linear trend will continue past the poles. The validity of this assumption depends greatly

on the situation under investigation.

As a general rule, it is reasonable to interpolate between the poles, but unreliable to extrapolate outside them.

STATISTICS

PACKAGE

STATISTICS

PACKAGE

lower

pole

upper pole

line of

best fit

y

x

extrapolation interpolation extrapolation

O

Two variable analysis (Chapter 22) 463

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