: Sxx is not just an intermediate calculation. It is the numerical embodiment of spread . Whether you are estimating variance, fitting a line, or testing a hypothesis, Sxx provides the scale against which all relationships are measured.
❌ Using ( n ) instead of ( n-1 ) when calculating sample variance from Sxx. ❌ Forgetting that Sxx only involves ( x ), not ( y ). ❌ Mixing up Sxx with Sxy (cross-product). ❌ Using the computational formula without checking for large rounding errors when subtracting two large numbers.
Where (x_i) are individual observations, (\barx) is the sample mean, and (n) is the sample size. This essay explores the meaning, derivation, alternative forms, and applications of Sxx in the context of variance.
[ \beginaligned & (4-5.2)^2 = (-1.2)^2 = 1.44 \ & (8-5.2)^2 = (2.8)^2 = 7.84 \ & (6-5.2)^2 = (0.8)^2 = 0.64 \ & (5-5.2)^2 = (-0.2)^2 = 0.04 \ & (3-5.2)^2 = (-2.2)^2 = 4.84 \ \endaligned ] Sum: ( 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8 ) [ S_xx = 14.8 ]
The formula ( S_xx = \sum x_i^2 - (\sum x_i)^2 / n ) is correct, but be careful with parentheses. Rounding can also cause errors if you round intermediate sums too early.
The correlation ( r ) is: [ r = \fracS_xy\sqrtS_xx S_yy ] Here, ( S_yy = \sum (y_i - \bary)^2 ) is the same concept applied to variable y. Thus, Sxx and Syy normalize the covariance ( S_xy ).