In this article, we explain how to derive the expanded form of the Sxx formula from the original expression for the sum of squares Sxx. We will prove that both of the formulae for Sxx are equal. We will show that,
Sxx = ∑(Xi – X̅)2 = (∑Xi2) – 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,
Sxx = ∑(Xi – X̅)2 = ∑(Xi2 – 2*Xi*X̅ + X̅2)
We then put the summation sign on each of the three terms,
Sxx = ∑Xi2 – ∑2*Xi*X̅ + ∑X̅2 (Equation 1)
Now, since the summation is taken from i=1 to i=n, over all of the Xi 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*Xi*X̅ = 2*X̅ ∑Xi.
Substituting these expressions for the second and third term in Equation 1 we get,
Sxx = ∑Xi2 – 2*X̅ ∑Xi + 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 = ∑Xi/n.
This implies that,
∑Xi = n*X̅
Substituting the value of ∑Xi in Equation 2 we get,
Sxx = ∑Xi2 – 2*X̅ *n*X̅ + nX̅2
Sxx = ∑Xi2 – 2nX̅2 + nX̅2
Since -2nX̅2+nX̅2 = -nX̅2 we conclude that,
Sxx = (∑Xi2) – n*X̅2.
This concludes the derivation of the Sxx formula.