Accuracy of PredictionAccuracy of Prediction

How accurate are predictions How accurate are predictions based on a correlation?based on a correlation?

Accuracy depends on Accuracy depends on rrXYXY

If we know nothing about an individual (e.g., If we know nothing about an individual (e.g., we try to predict the IQ of a randomly we try to predict the IQ of a randomly selected person), we should guess the selected person), we should guess the mean.mean.

If we always guess the mean, then the If we always guess the mean, then the variance tells us the average “cost” of our variance tells us the average “cost” of our guesses.guesses.

However, if we use X to predict Y, we can However, if we use X to predict Y, we can reduce this cost by reduce this cost by rr-squared.-squared.

On Sale: How Accurate?On Sale: How Accurate?

By squaring the correlation, we know what By squaring the correlation, we know what percentage of variance will be reduced by percentage of variance will be reduced by using X to predict Y.using X to predict Y.

If r = 1 or r = -1, the squared value is 1. If r = 1 or r = -1, the squared value is 1. These are both cases of perfect prediction, These are both cases of perfect prediction, like 100% off.like 100% off.

If r = ½ or r = -½, the squared correlation is If r = ½ or r = -½, the squared correlation is ¼ or .25. This means that a correlation of .5 ¼ or .25. This means that a correlation of .5 only reduces the cost by 25%.only reduces the cost by 25%.

Variance of Residuals: the “standard Variance of Residuals: the “standard error of regression”error of regression”

The average squared deviation between the The average squared deviation between the guess and the actual value of Y is called the guess and the actual value of Y is called the variance of residuals (errors)variance of residuals (errors)

You compute it by multiplying the original You compute it by multiplying the original variance of Y by (1 – rvariance of Y by (1 – r22), where r is the ), where r is the correlation between X and Y.correlation between X and Y.

The standard error of regression is the The standard error of regression is the square root of this variance.square root of this variance.

Sample ProblemSample Problem

Suppose we use sister’s IQ to predict Suppose we use sister’s IQ to predict brother’s IQ. The means of X and Y are brother’s IQ. The means of X and Y are both 100, and the standard deviations are both 100, and the standard deviations are both 15. both 15.

The variance of predicting Joe’s IQ if we The variance of predicting Joe’s IQ if we don’t know Jane’s IQ is 225.don’t know Jane’s IQ is 225.

The correlation is .5, so the variance of the The correlation is .5, so the variance of the residuals is (1-.25)(225) = 168.75.residuals is (1-.25)(225) = 168.75.

Standard Deviation of ErrorsStandard Deviation of Errors

Take the square root of the variance of Take the square root of the variance of residuals to compute the standard error of residuals to compute the standard error of regression, i.e., the standard deviation of regression, i.e., the standard deviation of differences between predicted and obtained.differences between predicted and obtained.

For our problem, the square root is 12.99, For our problem, the square root is 12.99, approximately 13.approximately 13.

Knowing Sister’s IQ reduces the standard Knowing Sister’s IQ reduces the standard deviation of residuals from 15 to 13.deviation of residuals from 15 to 13.

SummarySummary

If Jane has an IQ of 130, we predict her If Jane has an IQ of 130, we predict her brother to have an IQ of 115.brother to have an IQ of 115.

However, not all brothers of sisters with However, not all brothers of sisters with such IQ will be exactly 115.such IQ will be exactly 115.

On average, they will have a mean IQ of On average, they will have a mean IQ of 115, with a standard deviation of 13.115, with a standard deviation of 13.

The probability that Joe has a higher IQ than The probability that Joe has a higher IQ than his sister is only about 12%.his sister is only about 12%.