Statistics How To

Collinear Definition: What is Collinearity?

Statistics Definitions > Collinear 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 P1, P2 and P3 are collinear”, which can also be written as “point P1 is collinear with points P2 and P3“.
collinear points
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 P1 and P2, and P2 and P3); 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.
collinear points slope

Sample question: Do the points P1=(−4,0), P2=(−1,12) and P3=(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.

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