Statistics Definitions > Collinear Definition

## Definition

A set of points is collinear if you can draw one line through them all. The word literally means “together on a line.” Two points are *always* collinear: no matter where you draw the two points, you can always draw a straight line between them. A general way to write this is: “Points P_{1}, P_{2} and P_{3} are collinear”, which can also be written as “point P_{1} is collinear with points P_{2} and P_{3}“.

It goes without saying that points are **non-collinear** if they *do not* fall on the same line.

## How to Show Points are Collinear

It seems reasonable that if you can draw a line through a set of points, then those points are collinear. The trouble is, those points may not be *exactly* on the same line. One way to work around this is with the knowledge that **the points must satisfy the same linear equation**. For example, if you are given the linear equation y = 4x + 16, you know that the points (-4,0) and (-1,12) meet the definition because (plugging the x and y values into the equation) we get:

0 = 4*(-4) + 16

12 = 4*(-1) + 16.

## An Alternate Method

A second way is to find the slope between the points (i.e. the slopes of the line segments between points P_{1} and P_{2}, and P_{2} and P_{3}); **if the slopes are the same then the points are collinear**. For example, the set of points in the image below fit the definition if the slope of line segment A equals the slope of line segment B.

**Sample question: **Do the points P_{1}=(−4,0), P_{2}=(−1,12) and P_{3}=(4,32) show collinearity?

Step 1: Find the slope for the line segment between the first two points using rise-over-run =(y2−y1)/(x2−x1)=(12−0)/(−1−(−4))= 12/3 =4

Step 2: Find the slope for the line segment between the next two points =(y3−y2)/(x3−x2)=(32−12)/(4-(-1))= 20/5 = 4.

Step 3: Compare the slopes you calculated in Steps 1 and 2. The two slopes equal 4, so the points do show collinearity.

Check out our YouTube channel for hundreds of videos on elementary probability and statistics, from basic stats to advanced techniques using Excel, SPPS and Minitab.

If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!