Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 11 - 11.1 - Sequences and Series - 11.1 Exercises - Page 779: 103


$Σ(x_i-x ̅)^2=Σx_i^2-\frac{1}{n}(Σx_i)^2$

Work Step by Step

$Σ(x_i-x ̅)^2=Σ(x_i^2-2x_ix ̅+x ̅^2)=Σx_i^2-2x ̅Σx_i+x ̅^2Σ(1)$ But, $Σ(1)=1+1+1+...+1=n$ $\frac{1}{n}Σx_i=x ̅$. So: $\frac{1}{n^2}(Σx_i)^2=x ̅^2$ $Σ(x_i-x ̅)^2=Σx_i^2-2x ̅nx ̅+nx ̅^2=Σx_i^2-nx ̅^2=Σx_i^2-n\frac{1}{n^2}(Σx_i)^2=Σx_i^2-\frac{1}{n}(Σx_i)^2$
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