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, X ^{2}, and Y^{2}**

**Step 2: Calculate ΣX, ΣY, ΣX*Y, ΣX ^{2}, and ΣY^{2}**

**Step 3: Calculate b _{0}**

The formula to calculate b_{0 }is: [(ΣY)(ΣX^{2}) – (ΣX)(ΣXY)] / [n(ΣX^{2}) – (ΣX)^{2}]

In this example, b_{0 }= [(477)(222755) – (1237)(85125)] / [7(222755) – (1237)^{2}] = **32.783**

**Step 4: Calculate b _{1}**

The formula to calculate b_{1 }is: [n(ΣXY) – (ΣX)(ΣY)] / [n(ΣX^{2}) – (ΣX)^{2}]

In this example, b_{1 }= [7(85125) – (1237)(477)] / [7(222755) – (1237)^{2}] = **0.2001**

**Step 5: Place b _{0 }and b_{1} in the estimated linear regression equation.**

The estimated linear regression equation is: ŷ = b_{0} + b_{1}*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

**b _{0} = 32.7830**. When weight is zero pounds, the predicted height is 32.783 inches. Sometimes the value for b

_{0}can be useful to know, but in this example it doesn’t actually make sense to interpret b

_{0}since a person can’t weigh zero pounds.

**b _{1 }= 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.