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Regression

In a cause and effect relationship, the independent variable is the cause, and the dependent variable is the effect. Least squares linear regression is a method for predicting the value of a dependent variable Y, based on the value of an independent variable X.

Linear regression finds the straight line, called the least squares regression line or LSRL, that best represents observations in a bivariate data set. Suppose Y is a dependent variable, and X is an independent variable. Then, the equation for the regression line would be:

ŷ = b₀ + b₁x

where b₀ is a constant, b₁ is the regression coefficient, x is the value of the independent variable, and ŷ is the predicted value of the dependent variable.

Normally, you will use a computational tool - a software package (e.g., Excel) or a graphing calculator - to find b₀ and b₁. You enter the X and Y values into your program or calculator, and the tool solves for each parameter.

In the unlikely event that you find yourself on a desert island without a computer or a graphing calculator, you can solve for b₀ and b₁ "by hand". Here are the equations.

b₁ = Σ [ (x_i - x)(y_i - y) ] / Σ [ (x_i - x)²]      

and

b₀ = y - b₁ * x

where b₀ is the constant in the regression equation, b₁ is the regression coefficient, x_i is the X value of observation i, y_i is the Y value of observation i, and x and y are the means of X and Y, respectively.

See also:

AP Statistics Tutorial: Least Squares Linear Regression