Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 16.2 | The Standard Error of Estimate and Analysis of Regression 539
Conceptually, the standard error of estimate is very much like a standard deviation:
Both provide a measure of standard distance. Also, you will see that the calculation of the
standard error of estimate is very similar to the calculation of standard deviation.
To calculate the standard error of estimate, we first find a sum of squared deviations
(SS). Each deviation measures the distance between the actual Y value (from the data) and
the predicted Y value (from the regression line). This sum of squares is commonly called
SS residual
because it is based on the remaining distance between the actual Y scores and the
predicted values.
SS residual
= Σ(Y – Ŷ) 2 (16.8)
Recall that variance
measures the average
squared distance.
The obtained SS value is then divided by its degrees of freedom to obtain a measure of
variance. This procedure should be very familiar:
Variance = SS
df
The degrees of freedom for the standard error of estimate are df = n – 2. The reason for
having n – 2 degrees of freedom, rather than the customary n – 1, is that we now are measuring
deviations from a line rather than deviations from a mean. To find the equation for the
regression line, you must know the means for both the X and the Y scores. Specifying these
two means places two restrictions on the variability of the data, with the result that the scores
have only n – 2 degrees of freedom. (Note: the df = n – 2 for SS residual
is the same df = n – 2
that we encountered when testing the significance of the Pearson correlation on p. 508.)
The final step in the calculation of the standard error of estimate is to take the square
root of the variance to obtain a measure of standard distance. The final equation is
standard error of estimate = Î SS residual
df
5Î S (Y 2 Y ⁄ ) 2
n 2 2
(16.9)
The following example demonstrates the calculation of this standard error.
EXAMPLE 16.3
The same data that were used in Example 16.1 are used here to demonstrate the calculation
of the standard error of estimate. These data have the regression equation
Ŷ = 2X – 1
Using this regression equation, we have computed the predicted Y value, the residual, and
the squared residual for each individual, using the data from Example 16.1.
Data
Predicted
Y value
Residual
Squared
Residual
X Y Ŷ = 2X 2 1 Y 2 Ŷ (Y 2 Ŷ) 2
5 10 9 1 1
1 4 1 3 9
4 5 7 –2 4
7 11 13 –2 4
6 15 11 4 16
4 6 7 –1 1
3 5 5 0 0
2 0 3 –3 9
0 SS residual
= 44