Correlation Coefficient
How well does your regression equation truly represent
your set of data?
One of the ways to determine the answer to this question is to
exam the correlation coefficient and the coefficient of determination.
| The correlation coefficient, r, and the coefficient of determination, r 2 , will appear on the screen that shows the regression equation information |
In addition to appearing with the regression information, the values r and r 2 can be found under VARS, #5 Statistics → EQ #7 r and #8 r 2 .
Correlation Coefficient, r :
the direction of a linear relationship between two variables. The linear correlation
coefficient is sometimes referred to as the Pearson product moment correlation coefficient in
honor of its developer Karl Pearson.
where n is the number of pairs of data.
(Aren't you glad you have a graphing calculator that computes this formula?)
linear correlations and negative linear correlations, respectively.
to +1. An r value of exactly +1 indicates a perfect positive fit. Positive values
indicate a relationship between x and y variables such that as values for x increases,
values for y also increase.
to -1. An r value of exactly -1 indicates a perfect negative fit. Negative values
indicate a relationship between x and y such that as values for x increase, values
for y decrease.
close to 0. A value near zero means that there is a random, nonlinear relationship
between the two variables
employed.
straight line. If r = +1, the slope of this line is positive. If r = -1, the slope of this
line is negative.
less than 0.5 is generally described as weak. These values can vary based upon the
"type" of data being examined. A study utilizing scientific data may require a stronger
correlation than a study using social science data.
Coefficient of Determination, r 2 or R2 :
the variance (fluctuation) of one variable that is predictable from the other variable.
It is a measure that allows us to determine how certain one can be in making
predictions from a certain model/graph.
variation.
of the linear association between x and y.
to the line of best
fit. For example, if r = 0.922, then r 2 = 0.850, which means that
85% of the total variation in y can be explained by the linear relationship between x
and y (as described by the regression equation). The other 15% of the total variation
in y remains unexplained.
represents the data. If the regression line passes exactly through every point on the
scatter plot, it would be able to explain all of the variation. The further the line is
away from
the points, the less it is able to explain.