Accuracy of Prediction How accurate are predictions based on a correlation?

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Presentation transcript:

Accuracy of Prediction How accurate are predictions based on a correlation?

Accuracy depends on r XY  If we know nothing about an individual (e.g., we try to predict the IQ of a randomly selected person), we should guess the mean.  If we always guess the mean, then the variance tells us the average “cost” of our guesses.  However, if we use X to predict Y, we can reduce this cost by r-squared.

On Sale: How Accurate?  By squaring the correlation, we know what percentage of variance will be reduced by using X to predict Y.  If r = 1 or r = -1, the squared value is 1. These are both cases of perfect prediction, like 100% off.  If r = ½ or r = -½, the squared correlation is ¼ or.25. This means that a correlation of.5 only reduces the cost by 25%.

Variance of Residuals: the “standard error of regression”  The average squared deviation between the guess and the actual value of Y is called the variance of residuals (errors)  You compute it by multiplying the original variance of Y by (1 – r 2 ), where r is the correlation between X and Y.  The standard error of regression is the square root of this variance.

Sample Problem  Suppose we use sister’s IQ to predict brother’s IQ. The means of X and Y are both 100, and the standard deviations are both 15.  The variance of predicting Joe’s IQ if we don’t know Jane’s IQ is 225.  The correlation is.5, so the variance of the residuals is (1-.25)(225) =

Standard Deviation of Errors  Take the square root of the variance of residuals to compute the standard error of regression, i.e., the standard deviation of differences between predicted and obtained.  For our problem, the square root is 12.99, approximately 13.  Knowing Sister’s IQ reduces the standard deviation of residuals from 15 to 13.

Summary  If Jane has an IQ of 130, we predict her brother to have an IQ of 115.  However, not all brothers of sisters with such IQ will be exactly 115.  On average, they will have a mean IQ of 115, with a standard deviation of 13.  The probability that Joe has a higher IQ than his sister is only about 12%.