Temple of Divine Mother
x = [2,4,6,8] n = len(x) sum_x = sum(x) sum_x2 = sum( xi**2 for xi in x ) Sxx = sum_x2 - (sum_x**2)/n print(Sxx) # 20.0
| Student | Score | | --- | --- | | 1 | 80 | | 2 | 70 | | 3 | 90 | | 4 | 85 | | 5 | 75 |
The computational formula Sxx = Σxᵢ² – (Σxᵢ)² / n is a single formula that can be applied even when the mean is unknown. The definitional form Sxx = Σ(xᵢ – x̄)² explicitly requires the mean. Both are correct; use the one that is more convenient for your current calculation.
—is a critical, specialized tool used to quantify the total variation of values around the mean.
x̄ = (1 + 2 + 2 + 3 + 5 + 8) / 6 = 21 / 6 = 3.5
. It is a foundational component for calculating variances and standard errors of regression coefficients.
| Student | Score | Deviation from mean | | --- | --- | --- | | 1 | 80 | 0 | | 2 | 70 | -10 | | 3 | 90 | 10 | | 4 | 85 | 5 | | 5 | 75 | -5 |
Here is a breakdown of what it is, how it works, and why it matters. 1. The Definitional Formula At its core, cap S sub x x end-sub
Variance (σ²) = E[(xi - μ)²]
represents the for a single variable,
s2=Sxxn−1s squared equals the fraction with numerator cap S sub x x end-sub and denominator n minus 1 end-fraction Using our previous example where
The definitional formula directly reflects the concept of "sum of squared deviations."
Sxx=220−180=40cap S sub x x end-sub equals 220 minus 180 equals 40 Both methods yield
The total SST is precisely ( S_xx ) for the entire response variable. And the variance estimate within groups is based on SSW/df, which is analogous to Sxx within each group summed.
s2=204−1=203≈6.67s squared equals the fraction with numerator 20 and denominator 4 minus 1 end-fraction equals 20 over 3 end-fraction is approximately equal to 6.67 To find the Standard Deviation (