Sxx Variance Formula

Since (\sum x_i = n\barx), substitute:

Sxx=∑i=1n(xi−x̄)2cap S sub x x end-sub equals sum from i equals 1 to n of open paren x sub i minus x bar close paren squared

The definitional formula aligns exactly with the literal meaning of "sum of squared deviations." Sxx Variance Formula

To avoid rounding errors and reduce computation, Sxx can be expressed in an algebraically equivalent form using the sum of squares and the sum of the data:

cap S sub x x end-sub equals sum of x squared minus the fraction with numerator open paren sum of x close paren squared and denominator n end-fraction 2. Step-by-Step Calculation If you have a small data set, like , here is how you find cap S sub x x end-sub using the definitional method: Find the Mean ( Subtract Mean from each point: Square those results: Sum them up ( cap S sub x x end-sub cap S sub x x end-sub vs. Sample Variance ( It is important to note that cap S sub x x end-sub is not the final variance . It is the numerator used to find it. To get the Sample Variance ( , you divide cap S sub x x end-sub To get the Population Variance ( sigma squared , you divide cap S sub x x end-sub In our example above ( Sample Variance: 4. Why "Squared"? It is the numerator used to find it

To avoid rounding errors or needing to calculate ( \barx ) first, use:

Sxx=∑xi2−(∑xi)2ncap S x x equals sum of x sub i squared minus the fraction with numerator open paren sum of x sub i close paren squared and denominator n end-fraction To avoid rounding errors or needing to calculate

The ( \beta_1 ) is estimated as: [ \hat\beta 1 = \fracS xyS_xx ] where ( S_xy = \sum (x_i - \barx)(y_i - \bary) ).

∑xi2=22+42+62+82+102sum of x sub i squared equals 2 squared plus 4 squared plus 6 squared plus 8 squared plus 10 squared

Once you have the variance, you take the square root to find the standard deviation. is used to calculate the slope of a regression line