Often you may want to find a linear regression equation from a table of data.

For example, suppose you are given the following table of data:

The following step-by-step example explains how to find a linear regression equation from this table of data.

Step 1: Calculate X*Y, X², and Y²

First, we’ll calculate the following metrics for each row:

• x*y
• x²
• y²

The following screenshot shows how to do so:

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

Next, we’ll calculate the sum of each column:

Step 3: Calculate b₀

The formula to calculate the intercept of the regression equation, b₀, is as follows:

• b₀ = ((Σy)(Σx²) – (Σx)(Σxy))  /  (n(Σx²) – (Σx)²)
• b₀ = ((128)(831) – (85)(1258))  /  (10(831) – (85)²)
• b₀ = -0.518

Note: In the formula, n represents the total number of observations. In this example, there were 10 total observations.

Step 4: Calculate b₁

The formula to calculate the slope of the regression equation, b₁, is as follows:

• b₁ =  (n(Σxy) – (Σx)(Σy))  /  (n(Σx²) – (Σx)²)
• b₁ = (10(1258) – (85)(128))  /  (10(831) – (85)²)
• b₁ = 1.5668

Step 5: Write Linear Regression Equation

The final linear regression equation can be written as:

• ŷ = b₀ + b₁x

Thus, our linear regression equation would be written as:

• ŷ = -0.518 + 1.5668x

We can double check that this answer is correct by plugging in the values from the table into the Simple Linear Regression Calculator:

We can see that the linear regression equation from the calculator matches the one that we calculated by hand.

Additional Resources

The following tutorials provide additional information about linear regression:

Introduction to Simple Linear Regression
Introduction to Multiple Linear Regression
How to Interpret Regression Coefficients