Here’s a proper, self-contained guide to the Sxx variance formula – what it is, where it comes from, how to compute it, and how it connects to variance and regression.
1. What is Sxx? Sxx is a sum of squares notation commonly used in statistics , especially in:
Simple linear regression (calculating the slope) Correlation analysis Variance calculations
It represents the corrected sum of squares for a variable ( x ). Definition: [ S_{xx} = \sum_{i=1}^{n} (x_i - \bar{x})^2 ] Where: Sxx Variance Formula
( x_i ) = individual data points ( \bar{x} ) = sample mean of ( x ) ( n ) = sample size
2. Relationship to Variance The sample variance ( s_x^2 ) is defined as: [ s_x^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 ] Therefore: [ S_{xx} = (n-1) \cdot s_x^2 ] So Sxx is just the numerator of the variance (before dividing by ( n-1 )).
✅ Key point: Variance = Sxx / (n-1)
3. Alternative Computational Formulas To avoid rounding errors or needing to calculate ( \bar{x} ) first, use: (a) Direct definition: [ S_{xx} = \sum (x_i - \bar{x})^2 ] (b) Computational formula (easier for manual calculation): [ S_{xx} = \sum x_i^2 - \frac{(\sum x_i)^2}{n} ]
4. Worked Example Let’s calculate Sxx for ( x = {2, 4, 6, 8} ). Step 1: Find sums
( \sum x_i = 2+4+6+8 = 20 ) ( \sum x_i^2 = 4 + 16 + 36 + 64 = 120 ) ( n = 4 ) Here’s a proper, self-contained guide to the Sxx
Step 2: Apply computational formula [ S_{xx} = 120 - \frac{20^2}{4} = 120 - \frac{400}{4} = 120 - 100 = 20 ] Step 3: Check with definition
( \bar{x} = 20/4 = 5 ) ( (2-5)^2 = 9,\ (4-5)^2=1,\ (6-5)^2=1,\ (8-5)^2=9 ) Sum = ( 9+1+1+9 = 20 ) ✅