How to Calculate Sxy in Statistics (With Example)

In statistics, Sxy represents the sum of the product of the differences between x values and the mean of x and the differences between y values and the mean of y.

This value is often calculated when fitting a simple linear regression model by hand.

We use the following formula to calculate Sxy:

Sxy = Σ(x_i – x)(y_i – y)

where:

• Σ: A symbol that means “sum”
• x_i: The i^th value of x
• x: The mean value of x
• y_i: The i^th value of y
• y: The mean value of y

The following example shows how to use this formula in practice.

Example: Calculating Sxy by Hand

Suppose we would like to fit a simple linear regression model to the following dataset:

Suppose we would like to calculate Sxy for this dataset.

First, we must calculate the mean value of x:

• x = (1 + 2 + 2 + 3 + 5 + 8) / 6 = 3.5

Then, we must calculate the mean value of y:

• y = (8 + 12 + 14 + 19 + 22 + 21) / 6 = 16

Using these values, the following screenshot shows how to calculate the value for Sxy:

The value for Sxy turns out to be 59.

Note that we could also use the Sxy Calculator to automatically calculate the value of Sxy for this model as well:

The calculator returns a value of 59, which matches the value that we calculated by hand.

Note that we use the following formulas to perform simple linear regression by hand:

y = a + bx

where:

• a = y – bx
• b = Sxy / Sxx

The calculation for Sxy is just one calculation that we must perform in order to fit a simple linear regression model.

Related: How to Calculate Sxx in Statistics

Additional Resources

The following tutorials explain how to perform other common tasks in statistics:

How to Perform Simple Linear Regression by Hand
How to Perform Multiple Linear Regression by Hand