Page 305 - Statistics for Dummies

Chapter 18: Looking for Links: Correlation and Regression
y-intercept that can be calculated using formulas (and, I may add, these for-
mulas aren’t too hard to calculate).

To save a great deal of time calculating the best fitting line, first find the “big
five,” five summary statistics that you’ll need in your calculations:
1. The mean of the x values (denoted )

2. The mean of the y values (denoted )
3. The standard deviation of the x values (denoted s )

x

4. The standard deviation of the y values (denoted s )
y

5. The correlation between X and Y (denoted r)
Finding the slope
The formula for the slope, m, of the best-fitting line is
where r is the correlation between X and Y, and s and s are the standard 289
x y
deviations of the x-values and the y-values, respectively. You simply divide s
y
by s and multiply the result by r.
x
Note that the slope of the best-fitting line can be a negative number because
the correlation can be a negative number. A negative slope indicates that the
line is going downhill. For example, an increase in police officers is related
to a decrease in the number of crimes in a linear fashion; the correlation and
hence the slope of the best-fitting line is negative in this case.
The correlation and the slope of the best-fitting line are not the same. The for-
mula for slope takes the correlation (a unitless measurement) and attaches
units to it. Think of s ÷ s as the variation (resembling change) in Y over
y x
the variation in X, in units of X and Y. For example, variation in temperature
(degrees Fahrenheit) over the variation in number of cricket chirps (in 15
seconds).
Finding the y-intercept
The formula for the y-intercept, b, of the best-fitting line is , where
and are the means of the x-values and the y-values, respectively, and m is
the slope (the formula for which is given in the preceding section).
So to calculate the y-intercept, b, of the best-fitting line, you start by finding
the slope, m, of the best-fitting line using the steps listed in the preceding sec-
tion. You then multiply m by and subtract your result from .

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