Sxx Variance Formula

In simple linear regression ( y = a + bx ):

[ b = \fracS_xyS_xx ] [ S_xy = \sum (x_i - \barx)(y_i - \bary) ]

Also, the standard error of the slope uses Sxx: Sxx Variance Formula

[ SE(b) = \sqrt\fracs_e^2S_xx ] where ( s_e^2 ) is the residual variance.

Thus Sxx measures the spread of x – larger Sxx → smaller standard error → more precise slope estimate. In simple linear regression ( y = a

[ S_xx = \sum x_i^2 - n\barx^2 ]

All three yield the same result. The computational form (Formula 2) is preferred when using a calculator or spreadsheet because it avoids computing each deviation separately. [ S_xx = \sum x_i^2 - n\barx^2 ]

Let’s start with the most common definition. Given a set of ( n ) observations for a variable ( x ): ( x_1, x_2, x_3, \dots, x_n ), the quantity Sxx is defined as:

[ S_xx = \sum_i=1^n (x_i - \barx)^2 ]

Where:

This is often called the corrected sum of squares (or sum of squares about the mean). It measures the total squared deviation of each data point from the average.