Regression equation: Overview

A regression equation is used in statistics to find out what relationship, if any, exists between data sets. For example, if you measure the height of a child each year you might find that it grows about 3 inches a year. This trend (which grows by three inches per year) can be modeled using a regression equation. In fact, most things in the real world (from gas prices to hurricanes) can be modeled with some equation; this allows us to predict future events.

A regression line is the “most suitable” line for your data. Basically, you draw a line that best represents the data points. It is like an average of where all the points align. In linear regression, the regression line is a perfectly straight line:

The regression line is represented by an equation. In this case, the equation is -2.2923x + 4624.4. This means that if you were to graph the equation -2.2923x + 4624.4, the line would be a rough approximation for your data.

It is not very common for all data points to actually fall on the regression line. In the image above, the points are slightly scattered around the line. In this next image, the points fall on the line. The curved shape of this line is the result of polynomial regression, which fits the points in a polynomial equation.

Regression equation: What it is and how to use it

Statistical definitions > What is a regression equation?

Regression Equation: Overview

A regression equation is used in statistics to find out what relationship, if any, exists between data sets. For example, if you measure the height of a child each year you might find that it grows about 3 inches a year. This trend (which grows by three inches per year) can be modeled using a regression equation. In fact, most things in the real world (from gas prices to hurricanes) can be modeled with some equation; this allows us to predict future events.

A regression line is the “most suitable” line for your data. Basically, you draw a line that best represents the data points. It is like an average of where all the points align. In linear regression, the regression line is a perfectly straight line:

regression line

A linear regression line.

The regression line is represented by an equation. In this case, the equation is -2.2923x + 4624.4. This means that if you were to graph the equation -2.2923x + 4624.4, the line would be a rough approximation for your data.

It is not very common for all data points to actually fall on the regression line. In the image above, the points are slightly scattered around the line. In this next image, the points fall on the line. The curved shape of this line is the result of polynomial regression, which fits the points in a polynomial equation.

The polynomial regression results in a curved line.

The polynomial regression results in a curved line.

Regression and prediction lines

Regression is useful as it allows you to make predictions about the data. The first graph above is from 1995 to 2015. If you wanted to predict what will happen in 2020, you could put it into the equation:

-2.2923(2020)+4626.4 = -4.046.

Having negative rain doesn’t make much sense, but you can say that precipitation will fall to 0 inches before 2020. According to this particular regression line, it is actually expected to happen sooner or later in 2018:

-2.2923(2018)+4626.4 = 0.5386

-2.2923(2019)+4626.4 = -1.7537

What’s a regression equation for?

Regression equations can help you understand whether your data may be suitable for an equation. This is extremely useful if you want to make predictions from your data – both future predictions and indications of past behaviour. For example, you may want to know how much your savings will be worth in the future. Or, you may want to predict how long it will take to recover from an illness.

There are different types of regression equations. Some of the most common include Exponential Linear Regression and Simple Linear Regression (to adapt data to an exponential equation or linear equation). In elementary statistics, the regression equation you are most likely to encounter is the linear form.

Calculation of linear regression

There are several ways to find a regression line, even by hand and with technology, like Excel (see below). Finding a regression line is very boring by hand. The following video illustrates the steps:

You can also find a regression line on the TI calculators:

TI 83 Regression.

How to perform TI-89 Regression.

The linear regression equation is shown below.

The downside of regression analysis

In order for the data to fit into an equation, you must first understand which general scheme fits the data. The general steps to perform regression include making a dispersion diagram and then making a hypothesis about which type of equation might be the most suitable. Then you can select the best regression equation for the job.

However, as the following picture shows, it is not always entirely easy to select the appropriate regression equation, especially when dealing with real data. Sometimes you get “noisy” data that doesn’t seem to fit any equation. If most of the data seems to follow a pattern, you might omit outliers. In fact, if you ignore the outliers, the data seems to be modeled by an exponential equation.