Question

Given $\sum x=99,n=9$ and $\sum (x-10{)}^2=79$. Find $\sum x^2$ and $\sum (x-\overline{x}{)}^2$.

Answer 1

x̄ = $\frac{\Sigma x}{n}$

= $\frac{99}{9}$

= 11

Σ (x - 10)² = 79

Σ (x² + 10² - 2 x (10)) = 79

Σ (x² + 100 - 20 x) = 79

Σ x² + 100 Σ - 20 Σ x = 79

Σ x² + 100 (9) - 20 (99) = 79

Σ x² + 900 - 1980 = 79

Σ x² - 1080 = 79

Σ x² = 79 + 1080

Σ x² = 1159

Σ (x - x̄)² = Σ (x - 11)²

= Σ (x² + 11² - 2 x (11))

= Σ (x² + 121 - 22 x)

= Σx² + 121Σ - 22 Σx

= 1159 + 121(9) - 22 (99)

= 1159 + 1089 - 2178

= 2248 - 2178

= 70