Basic Statistics

LINEAR REGRESSION: SINGLE SAMPLE 169

Table 12.2 Calculations for Regression Line

Number Weight SBP" (X - x)' (Y - 7)' (X - x)(Y - Y)

1 2 3 4 5 6 7 8 9

10

165 134

243 155

180 137

152 124

163 128

210 131

203 143

195 136

178 127

218 146

660.49

2735.29

114.49

1497.69

767.29

372.49

151.29

18.49

161.29

745.29

4.41

357.21

.81

146.41

65.61

26.01

47.61

0.01

82.81

98.01

53.49

988.47

-9.63

468.27

224.37

-98.43

84.87

-.43

115.57

270.27

c 1907 1361 7224.10 828.90 2097.30

aSystolic blood pressure.

and

Then, Y equals

u = 7 - bX = 136.1 - .2903(190.7) = 80.74.

Y = 80.74 + .2903X

or

SBP = 80.74 + .2903 weight

Note that if we divide c(X - z)' by n - 1, we obtain the variance of X, or

s$. Similarly dividing c(Y - Y)' by n - 1, or in this example by 9, we get the

variance of Y, or s$. Thus, the sum in those two columns is simply the variance

of X multiplied by n - 1 and the variance of Y multiplied by n - 1. These sums

are always positive since they are the sum of squared numbers. The sum of the last

column in Table 12.2 introduces a formula that is new, c(X - x)(Y - Y). If we

divide this sum by n - 1 = 9, we obtain what is called the covariance, or sZy. The

covariance will be positive if increasing values of X result in increasing values of Y

(a positive relationship holds). In Table 12.2 we have a positive sum of 2097.3. Note

that only three of the values in the last column are negative, and two of those (-9.63

and -.43) are small numbers. In other words, when X is >x, then Y also tends to

be >Y, resulting in a positive product in this example. Whenever large values of X

tend to occur with small values of Y, c(X - X) (Y - Y) is negative.

In the formula for the slope b, we divide c (X - X) (Y - 7) by a positive number;

thus the resulting sign of b depends solely on the sign of c(X - 51) (Y - Y). If it

has a positive sign, b has a positive sign and the relationship is called positive. If it

has a negative sign, b has a negative sign and the relationship is negative. Note that

statistical programs will provide scatter plots with the linear regression line included

on the plot. In these programs scatter the numerical results for the coefficients and

information concerning the coefficients is included, usually in a table under the plot.