[ s_x^2 = \fracS_xxn-1 = \frac\sum (x_i - \barx)^2n-1 ]
∑x2=22+42+42+72+82sum of x squared equals 2 squared plus 4 squared plus 4 squared plus 7 squared plus 8 squared
The primary definition of Sxx is as the , which is simply the sum of each data point squared: (S_xx = \sum_i=1^n x_i^2). However, in the context of variance and regression, "Sxx" almost always refers to the more powerful corrected sum of squares , often represented as (S_xx) or sometimes as the sum of squared deviations (SS).
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.
If we simply summed ( (x_i - \barx) ), the result would always be zero (positive and negative deviations cancel). Squaring removes the sign, ensuring we measure of spread, not direction. Sxx Variance Formula
Sum of Squares (SSx) , often written as , is a key value used to measure the total variation of a single variable (
) . This tells us how much the members of one sex deviate from their specific group mean.
x̄=2+4+6+8+105=305=6x bar equals the fraction with numerator 2 plus 4 plus 6 plus 8 plus 10 and denominator 5 end-fraction equals 30 over 5 end-fraction equals 6 Sum the squared values:
s=Sxxn−1s equals the square root of the fraction with numerator cap S sub x x end-sub and denominator n minus 1 end-fraction end-root Using our example: [ s_x^2 = \fracS_xxn-1 = \frac\sum (x_i -
Sxx is used in the denominator of the Pearson Correlation Coefficient (
The definitional formula aligns exactly with the literal meaning of "sum of squared deviations."
[ \textVariance = \fracS_xxn-1 ]
where E denotes the expected value, and μ represents the population mean. Rounding can also cause errors if you round
[ SE(b_1) = \sqrt\fracs_e^2S_xx ]
This is the standard formula used by software algorithms, calculators, and statisticians working by hand, as it significantly reduces calculation steps and preserves decimal accuracy. Sxxcap S sub x x end-sub vs. Variance: What is the Difference? A common point of confusion is how Sxxcap S sub x x end-sub
Notice that Sxx provides the “scale” for ( x ), and Syy provides the scale for ( y ). The correlation normalizes the covariance by the geometric mean of the two corrected sums of squares.
From this, we see:
∑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