INTRODUCTION TO LINEAR REGRESSION

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Professionals often want to know how two or more numeric variables are related.

For example, is there a relationship between the grade on the second math exam a student takes and the grade on the final exam? If there is a relationship, what is the relationship and how strong is it?

In another example, your income may be determined by your education, your profession, your years of experience, and your ability.

The amount you pay a repair person for labor is often determined by an initial amount plus an hourly fee.

The type of data described in the examples is bivariate data — “bi” for two variables.

In reality, statisticians use multivariate data, meaning many variables.

In this article, you will be studying the simplest form of regression, “linear regression” with one independent variable (x).

This involves data that fits a line in two dimensions.

You will also study the correlation which measures how strong the relationship is.

Linear Equations

Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:

y = a + bx

where a and b are constant numbers.

The variable x is the independent variable, and y is the dependent variable.

Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

Scatter Plots

A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either:

• High values of one variable occurring with high values of the other variable or low values of one variable occurring
with low values of the other variable.

• High values of one variable occurring with low values of the other variable.

You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function.

For a linear relationship, there is an exception.
Consider a scatter plot where all the points fall on a horizontal line providing a “perfect fit.”

The horizontal line would in fact show no relationship.

 

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