How to Perform Linear Regression by Hand

Simple linear regression is a statistical method you can use to quantify the relationship between a predictor variable and a response variable.

This tutorial explains how to perform simple linear regression by hand.

Example: Simple Linear Regression by Hand

Suppose we have the following dataset that shows the weight and height of seven individuals:

Use the following steps to fit a linear regression model to this dataset, using weight as the predictor variable and height as the response variable.

Step 1: Calculate X*Y, X², and Y²

Step 2: Calculate ΣX, ΣY, ΣX*Y, ΣX², and ΣY²

Step 3: Calculate b₀

The formula to calculate b₀ is: [(ΣY)(ΣX²) – (ΣX)(ΣXY)]  /  [n(ΣX²) – (ΣX)²]

In this example, b₀ = [(477)(222755) – (1237)(85125)]  /  [7(222755) – (1237)²] = 32.783

Step 4: Calculate b₁

The formula to calculate b₁ is: [n(ΣXY) – (ΣX)(ΣY)]  /  [n(ΣX²) – (ΣX)²]

In this example, b₁ = [7(85125) – (1237)(477)]  /  [7(222755) – (1237)²] = 0.2001

Step 5: Place b₀ and b₁ in the estimated linear regression equation.

The estimated linear regression equation is: ŷ = b₀ + b₁*x

In our example, it is ŷ = 0.32783 + (0.2001)*x

How to Interpret a Simple Linear Regression Equation

Here is how to interpret this estimated linear regression equation: ŷ = 32.783 + 0.2001x

b₀ = 32.7830. When weight is zero pounds, the predicted height is 32.783 inches. Sometimes the value for b₀ can be useful to know, but in this example it doesn’t actually make sense to interpret b₀ since a person can’t weigh zero pounds.

b₁ = 0.2001. A one pound increase in weight is associated with a 0.2001 inch increase in height.

Simple Linear Regression Calculator

We can double check our results by inputting our data into the simple linear regression calculator:

This equation matches the one that we calculated by hand.