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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: CalculateÂ Î£X,Â Î£Y,Â Î£X*Y, Î£X ^{2}, and Î£Y^{2}**

**Step 3: CalculateÂ b _{0}**

The formula to calculate b_{0Â }is:Â [(Î£Y)(Î£X^{2}) â€“ (Î£X)(Î£XY)]Â /Â [n(Î£X^{2}) â€“Â (Î£X)^{2}]

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

**Step 4: CalculateÂ b _{1}**

The formula to calculate b_{1Â }is:Â [n(Î£XY) â€“ (Î£X)(Î£Y)]Â /Â [n(Î£X^{2}) â€“Â (Î£X)^{2}]

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

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

The estimated linear regression equation is:Â Å· =Â b_{0} + b_{1}*x

In our example, it isÂ **Å· = 0.32783 + (0.2001)*x**

**How to Interpret a Simple Linear Regression Equation**

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

**b _{0}Â = 32.7830**. WhenÂ weightÂ is zero pounds, theÂ predicted height is 32.783 inches.Â Sometimes the value for b

_{0}Â can be useful to know, but in this example it doesnâ€™t actually make sense to interpretÂ b

_{0}Â since a person canâ€™t weigh zero pounds.

**b _{1Â }= 0.2001**. A one pound increase inÂ weightÂ is associated with a 0.2001Â 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.