estimated_standard_deviation
Differences
This shows you the differences between two versions of the page.
Both sides previous revisionPrevious revision | Next revisionBoth sides next revision | ||
estimated_standard_deviation [2019/10/19 05:05] – hkimscil | estimated_standard_deviation [2020/04/09 01:37] – hkimscil | ||
---|---|---|---|
Line 1: | Line 1: | ||
====== Why n-1 ====== | ====== Why n-1 ====== | ||
+ | |||
+ | 우선, | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | |||
+ | Var[X] & = & {E{(X-\mu)^2}} \\ | ||
+ | & = & E[(X^2 - 2 X \mu + \mu^2)] \\ | ||
+ | & = & E[X^2] - 2 \mu E[X] + E[\mu^2] \\ | ||
+ | & = & E[X^2] - 2 \mu E[X] + E[\mu^2], \;\; \text{because E[X]=} \mu \text{, \; E[} \mu^2 \text{] = } \mu^2, \\ | ||
+ | & = & E[X^2] - 2 \mu^2 + \mu^2 \\ | ||
+ | & = & E[X^2] - \mu^2 | ||
+ | \end{eqnarray*} | ||
+ | 이므로 | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | E[X^2] & = & \sigma^{2} - [\mu]^2 \;\;\; \dots\dots\dots\dots\dots [1] | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | 마찬가지로 | ||
+ | |||
+ | \begin{eqnarray*} | ||
+ | Var[\overline{X}] & = & E[\overline{X}^2] - [E(\overline{X})]^2 \\ | ||
+ | & = & E[\overline{X}^2] - [\mu]^2 | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | 따라서 | ||
+ | \begin{eqnarray*} | ||
+ | E[\overline{X}^{2}] & = & \frac {\sigma^{2}} {n} - [\mu]^2 \;\;\; \dots\dots\dots\dots\dots [2] | ||
+ | \end{eqnarray*} | ||
+ | |||
+ | |||
+ | |||
Why we use n-1 instead of n in getting standard deviation \\ | Why we use n-1 instead of n in getting standard deviation \\ | ||
http:// | http:// |
estimated_standard_deviation.txt · Last modified: 2023/09/13 11:00 by hkimscil