How to Find Linear Regression Equation from a Table

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

One Reply to “How to Find Linear Regression Equation from a Table”

  1. Hello Zach,
    I came across your explanation while trying to work on regression equations for Algebra II. I love the way you have this laid out but have a question.

    Why did we need to compute y^2 (and the sum of y^2) if it isn’t used in the formulas?
    Is is it supposed to be sum of y^2 at the beginning of Step 3? (This made the most logical sense to me as a possible error.)

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