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(xi – x)(yi – y)
where:
- Σ: A symbol that means “sum”
- xi: The ith value of x
- x: The mean value of x
- yi: The ith 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