How to Calculate Sxy (Formula & Examples)

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The Sxy value can be calculated using the formula,

Sxy = Σ (Xi – X̄)(Yi – Ȳ).

The quantity Sxy in statistics is used to calculate the regression coefficient and correlation coefficient. In this article, we explain how to calculate Sxy and give a shortcut method for the calculation.

Sxy Formula and Calculation:

Suppose that you are given two values of two variables X and Y. The subscript xy in the quantity Sxy refers to the fact that the sum is taken over the possible values of x and y.

The quantity Sxy can be calculated using the below formula.

Sxy = Σ (Xi – X̄)(Yi – Ȳ).

Here refers to the mean of the X values and Ȳ refers to the mean of the Y values. We now give an example and explain how the quantity can be calculated.

Example:

Consider the following set of 5 data values for the variables X and Y,

Step 1: We first calculate the mean for the data values as follows,

The mean is calculated using the formula,

X̄ = ΣXi/n = 15/5 = 3.

Ȳ = ΣYi/n = 40/5 = 8.

Step 2: We now calculate the quantity Sxy using the formula. Multiply the third and fourth columns and then sum up the values of the last column below,

Sxy = Σ (Xi – X̄)(Yi – Ȳ) = 10.

Shortcut Method for Calculating Sxy:

We can also calculate Sxy using a shortcut. We can use the alternative formula that is given below.

Sxy = (∑ XiYi) – n*X̅Ȳ.

So in the above example, we calculate only one extra column,

Sxy = (∑ XiYi) – n*X̅Ȳ = 130 – 5*3*8 = 130 – 120 = 10.

What does Sxy Mean in Statistics and Maths?

By Sxy, we mean the sum of the product of deviations of X and y values from their mean/average values.

We first calculate the differences between the x and y values and then multiply them. Then we sum up all of these product values in order to obtain the desired quantity.

Is Sxy a correlation?

Although Sxy is itself not a correlation, it can indeed be used to calculate the correlation coefficient. The correlation coefficient can be calculated using the formula,

Correlation = Sxy/√(Sxx*Syy).

Summary
Article Name
How to Calculate Sxy (Formula & Examples)
Description
The Sxy value can be calculated using the formula, Sxy = Σ (Xi – X̄)(Yi - Ȳ).