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Correlation Coefficient
One of the ways to determine the answer to this question is to exam the correlation coefficient and the coefficient of
determination.
The quantity where n is the number of pairs of data.(Aren't you glad you have a graphing calculator that computes this formula?) The value of r is such that -1 < r < +1. The + and – signs
are used for positivelinear correlations and negative linear correlations, respectively. Positive correlation: If x and y have a strong positive linear correlation, r is closeto +1. An r value of exactly +1 indicates a
perfect positive fit. Positive valuesindicate a relationship between x and y variables such that as
values for x increases,values for y also
increase. Negative correlation:
If x and y have a strong negative linear correlation, r is closeto -1. An r value of exactly -1 indicates a perfect negative fit. Negative valuesindicate a relationship between x and y such that as values for x increase,
valuesfor y decrease. No correlation:
If there is no linear correlation or a weak linear correlation, r isclose to 0. A value near zero means that there is a random, nonlinear relationship between the two variables Note that r is a dimensionless quantity;
that is, it does not depend on the units employed. A perfect correlation of ± 1 occurs only when the data points all lie exactly
on astraight line. If r = +1, the slope of
this line is positive. If r = -1, the slope of thisline is negative. A correlation greater than 0.8 is generally described as strong, whereas a
correlationless 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.
The y (as described by the regression equation). The
other 15% of the total variationin y remains unexplained.The coefficient of determination is a measure of how well the
regression linerepresents 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. |

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