Sxx, Sxy, and Syy Equations

The Sxx, Sxy, and Syy equations are given below. These formulas can be used to calculate the values of Sxx, Sxy, and Syy which are very useful quantities.

Sxx, Sxy, and Syy Equations (Formula):

The Sxx, Sxy, and Syy Equations are,

S_xx = ∑(X_i – X̅)².

S_xy = Σ (X_i – X̄)(Y_i – Ȳ).

S_yy = Σ(Y_i – Ȳ)².

• Here, the sum Σ is taken over all the x_i and y_i data values from i=1 to i=n.
• x_i denotes the given data values of the variable X.
• X̅ denotes the mean/average of the X data values.
• y_i denotes the given data values of the variable Y.
• Ȳ denotes the mean/average of the x data values.

S_xx Formula:

As stated above the Sxx value can be calculated using the formula,

S_xx = ∑(X_i – X̅)².

Alternatively, we can also use the following alternative formula,

S_xx = (∑X_i²) – n*X̅².

The proof of the fact that both these formulas are equal can be found here.

S_xy Formula:

The Sxy value can be calculated using the formula,

Sxy = Σ (X_i – X̄)(Y_i – Ȳ).

Alternatively, we can also use the following alternative formula,

S_xy = (∑ X_iY_i) – n*X̅Ȳ.

It is a fact that both these formulas give the same result.

S_yy Formula:

The Syy formula is the same as the Sxx formula with the variable Y in place of the variable X.

S_yy = ∑(Y_i – Ȳ)².

Alternatively, we have

S_yy = (∑Y_i²) – n*Ȳ².

We summarize the formulas below.

Related Articles:

1. Sxx, Sxy, and Syy Calculator.
2. How to Find Sxx (with Examples & Formula).
3. How to Calculate Sxy (Formula & Examples).

Summary

Article Name

Sxx, Sxy, and Syy Equations

Description

The Sxx, Sxy, and Syy Equations are: Sxx = ∑(Xi - X̅)^2 Sxy = Σ (Xi – X̄)(Yi - Ȳ) Syy = Σ(Yi - Ȳ)^2.