Derivation of Sxx Formula (Step by Step Proof)

In this article, we explain how to derive the expanded form of the S_xx formula from the original expression for the sum of squares S_xx. We will prove that both of the formulae for S_xx are equal. We will show that,

S_xx = ∑(X_i – X̅)² = (∑X_i²) – n*X̅²

Step 1: Write the original expression for Sxx and expand the square.

We know that (a-b)² = a² – 2ab + b². Therefore we have,

S_xx = ∑(X_i – X̅)² = ∑(X_i² – 2*X_i*X̅ + X̅²)

We then put the summation sign on each of the three terms,

S_xx = ∑X_i² – ∑2*X_i*X̅ + ∑X̅² (Equation 1)

Now, since the summation is taken from i=1 to i=n, over all of the X_i data values and since the mean X̅ is a constant, we have ∑X̅² = nX̅².

Similarly, we can pull the constant 2*X̅ out of the summation in the middlemost term to get, ∑2*X_i*X̅ = 2*X̅ ∑X_i.

Substituting these expressions for the second and third term in Equation 1 we get,

S_xx = ∑X_i² – 2*X̅ ∑X_i + nX̅² (Equation 2)

Step 2: Use the formula for the Mean X̅ to substitute in the middle term.

We know that the mean X̅ of the data values is given by the formula,

X̅ = ∑X_i/n.

This implies that,

∑X_i = n*X̅

Substituting the value of ∑X_i in Equation 2 we get,

S_xx = ∑X_i² – 2*X̅ *n*X̅ + nX̅²

S_xx = ∑X_i² – 2nX̅² + nX̅²

Since -2nX̅²+nX̅² = -nX̅² we conclude that,

S_xx = (∑X_i²) – n*X̅².

This concludes the derivation of the S_xx formula.