CK-12 Probability and Statistics - Advanced

9.2. Least-Squares Regression http://www.ck12.org

variables is associated with a 1% change in the outcome or the criterion variable. For example, if we had a regression

coefficient of 10.76, we would say that a “10.76% change inXis associated with a 1% change inY.” To calculate

this regression coefficient we can use the formulas:

b=

n∑XY−∑X∑Y

n∑X^2 −(∑X)^2

or

b= (r)

sy

sx

where:

r=correlation between variablesXandY

sy=standard deviation of theYscores

sx=standard deviation of theXscores

In addition to calculating the regression coefficient, we also need to calculate the regression constant. The regression

constant is also they-intercept and is the place where the line crosses they-axis. For example, if we had an equation

with a regression constant of 4.58, we would conclude that the regression line crosses they-axis at 4.58. We use the

following formula to calculate the regression constant:

a=∑

Y−b∑X

n

=Y ̄−bX ̄

Example:

Find the least squared regression line (also known as the regression line or the line of best fit) for the example

measuring the verbal SAT score and GPA that was used in the previous section.

TABLE 9.6: SAT and GPA data including intermediate computations for computing a linear
regression.

Student SAT Score(X) GPA(Y) XY X^2 Y^2

1 595 3. 4 2023 354025 11. 56

2 520 3. 2 1664 270400 10. 24

3 715 3. 9 2789 511225 15. 21

4 405 2. 3 932 164025 5. 29

5 680 3. 9 2652 462400 15. 21

6 490 2. 5 1225 240100 6. 25

7 565 3. 5 1978 319225 12. 25

Sum 3970 22. 7 13262 2321400 76. 01

Using these data, we first calculate the regression coefficient and the regression constant: