How to Calculate a Regression Line

In statistics, you can calculate a regression line for two variables if their scatterplot shows a linear pattern and the correlation between the variables is very strong (for example, r = 0.98). A regression line is simply a single line that best fits the data (in terms of having the smallest overall distance from the line to the points). Statisticians call this technique for finding the best-fitting line a simple linear regression analysis using the least squares method. (Phew! That's a mouthful!) The formula for the best-fitting line (or regression line) is y = mx + b, where m is the slope of the line and b is the y-intercept. This equation itself is the same one used to find a line in algebra; but remember, in statistics the points don’t lie perfectly on a line — the line is a model around which the data lie if a strong linear pattern exists.

• The slope of a line is the change in Y over the change in X. For example, a slope of

  

  means as the x-value increases (moves right) by 3 units, the y-value moves up by 10 units on average.

• The y-intercept is the value on the y-axis where the line crosses. For example, in the equation y=2x – 6, the line crosses the y-axis at the value b= –6. The coordinates of this point are (0, –6); when a line crosses the y-axis, the x-value is always 0.

You may be thinking that you have to try lots and lots of different lines to see which one fits best. Fortunately, you have a more straightforward option (although eyeballing a line on the scatterplot does help you think about what you’d expect the answer to be). The best-fitting line has a distinct slope and y-intercept that can be calculated using formulas (and these formulas 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

   

2. The mean of the y values

   

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 of a regression line

The formula for the slope, m, of the best-fitting line is

where r is the correlation between X and Y, and s_x and s_y are the standard deviations of the x-values and the y-values, respectively. You simply divide s_y by s_x and multiply the result by r.

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, if an increase in community center programs is related to a decrease in the number of crimes in a linear fashion; then 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 formula for slope takes the correlation (a unitless measurement) and attaches units to it. Think of s_y divided by s_x as the variation (resembling change) in Y over 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 of a regression line

The formula for the y-intercept, b, of the best-fitting line is b = y̅ -mx̅, where x̅ and y̅ are the means of the x-values and the y-values, respectively, and m is the slope.

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 above steps. Then to find the y-intercept, you multiply m by x̅ and subtract your result from y̅.

Always calculate the slope before the y-intercept. The formula for the y-intercept contains the slope!