Statistics for People Who (Think They) Hate Statistics

by Neil J. Salkind

But being practical, we can also look at the difference between the predicted value (Y ′) and the actual value (Y) when we first compute the formula of the regression line. For example, if the formula for the regression line is Y ′ = 0.704X + 0.719, the predicted Y (or Y ′) for an X value of 2.8 is 0.704(2.8) + 0.719, or 2.69. We know that the actual Y value that corresponds to an X value is 3.5 (from the data set shown in Table 16.1). The difference between 3.5 and 2.69 is 0.81, and that’s the size of the error in prediction. Another measure of error that you could use is the coefficient of determination (see Chapter 5), which is the percentage of error that is reduced in the relationship between variables. For example, if the correlation between two variables is .4 and the coefficient of determination is 16% or .42, the reduction in error is 16% since initially we suspect the relationship between the two variables starts at 0 or 100% error (no predictive value at all). If we take all of these differences, we can compute the average amount that each data point differs from the predicted data point, or the standard error of estimate. This is a kind of standard deviation that reflects average error along the line of regression. The value tells us how much imprecision there is in our estimate. As you might expect, the higher the correlation between the two values (and the better the prediction), the lower this standard error of estimate will be. In fact, if the correlation b...