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.

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