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--- c:ms:2017:schedule:week03 [2017/03/20 07:16] – [Sampling Distribution, Standard Error] hkimscil
+++ c:ms:2017:schedule:week03 [2022/05/15 11:25] (current) – [Central Tendency] hkimscil
@@ Line 1: / Line 1: @@
-====== Week 3 내용 ======
+ㄹ====== Week 3 내용 ======
 ===== SPSS =====
 <del>Chapter 3</del>, Chapter 4
@@ Line 47: / Line 47: @@
 {{:hist.jpg}}
-</WRAP>
-<WRAP half column>
 data file: {{:Ex3-1.sav}} 읽지 않은 지문에 대한 답을 한 학생들의 점수 (Katz, 1990).
@@ Line 71: / Line 69: @@
   * Data file: [[http://www.uvm.edu/~dhowell/fundamentals7/DataFiles/Tab5-1.dat|Web site]] or {{:Tab5-1.sav}} p.86-7
   * [[:range]]
-  * [[:outlier]]
+  * [[:outliers]]: It is beyond our scope. Please just refer to it. Won't be appearing in tests.
   * 평균편차
   * [[:Variance]] 변량
@@ Line 78: / Line 76: @@
 <code>Descriptives
-	SET			Statistic	Std. Error
+			SET			Statistic	Std. Error
-ATTRACT	 4	Mean		2.6445	.14651
+ATTRACT	 4	Mean				2.6445		.14651
-% Confidence Interval for Mean	Lower Bound	2.3379
+% Confidence	Lower Bound	2.3379
-			Upper Bound	2.9511
+		Interval for 	Upper Bound	2.9511
-% Trimmed Mean		2.6483
+		Mean
-		Median		2.5950
+% Trimmed Mean			2.6483
-		Variance		.429
+		Median				2.5950
-		Std. Deviation		.65520
+		Variance			.429
-		Minimum		1.20
+		Std. Deviation			.65520
-		Maximum		4.02
+		Minimum				1.20
-		Range		2.82
+		Maximum				4.02
+		Range				2.82
 		Interquartile Range		.82
-		Skewness		-.001	.512
+		Skewness			-.001	.512
-		Kurtosis		.438	.992
+		Kurtosis			.438	.992
-	Mean		3.2615	.01541
+	Mean				3.2615	.01541
 % Confidence Interval for Mean	Lower Bound	3.2292
-			Upper Bound	3.2938
+							Upper Bound	3.2938
 % Trimmed Mean		3.2622
-		Median		3.2650
+		Median			3.2650
 		Variance		.005
 		Std. Deviation		.06892
-		Minimum		3.13
+		Minimum			3.13
-		Maximum		3.38
+		Maximum			3.38
-		Range		.25
+		Range			.25
 		Interquartile Range		.11
 		Skewness		-.075	.512
@@ Line 108: / Line 107: @@
   * [[:Standard Deviation]] 표준편차
-  * $\displaystyle S_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N-1} $
+  * Variance calculation formula <wrap #variance_calculation_formula />
+    * $ \displaystyle S_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N-1} $
-  * $\displaystyle \sigma_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N} = \displaystyle \frac {\Sigma X^2}{N} - \frac {(\Sigma X)^2}{N^2} = \displaystyle \frac {\Sigma X^2}{N} - \bigg(\frac {\Sigma X}{N}\bigg)^2 = \displaystyle \frac {\Sigma X^2}{N} - \mu^2   $
+    * $ \displaystyle \sigma_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N} = \displaystyle \frac {\Sigma X^2}{N} - \frac {(\Sigma X)^2}{N^2} = \displaystyle \frac {\Sigma X^2}{N} - \bigg(\frac {\Sigma X}{N}\bigg)^2 = \displaystyle \frac {\Sigma X^2}{N} - \mu^2 $
   * [[:Degrees of Freedom]] N-1
@@ Line 119: / Line 118: @@
   * [[:Central Limit Theorem]]
   * [[:Standard Error]]
+===== CLT에 관한 정리 =====
+우선, Expected value (기대값)와 Variance (분산)의 연산은 아래와 같이 계산될 수 있다.
+X,Y 가 서로 독립적이라고 할 때:
+\begin{eqnarray}
+E[aX] = a E[X] \\
+E[X+Y] = E[X] + E[Y] \\
+Var[aX] = a^{\tiny{2}} Var[X] \\
+Var[X+Y] = Var[X] + Var[Y]
+\end{eqnarray}
+이때, 한 샘플의 평균값을 $X$ 라고 하면, 평균들의 합인 $S_k$ 는
+$$ S_{k} = X_1 + X_2 + . . . + X_k $$
+와 같다.
+이렇게 얻은 샘플들(k 개의)의 평균인 $ A_k $ 는,
+$$ A_k = \displaystyle \frac{(X_1 + X_2 + . . . + X_k)}{k} = \frac{S_{k}}{k} $$
+라고 할 수 있다.
+이때,
+$$
+\begin{align*}
+E[S_k] & = E[X_1 + X_2 + . . . +X_k] \\
+   & = E[X_1] + E[X_2] + . . . + E[X_k] \\
+   & = \mu + \mu + . . . + \mu = k * \mu \\
+\end{align*}
+$$
+$$
+\begin{align*}
+Var[S_k] & = Var[X_1 + X_2 + . . . +X_k]  \\
+     & = Var[X_1] + Var[X_2] + \dots + Var[X_k] \\
+     & = k * \sigma^2
+\end{align*}
+$$
+이다.
+그렇다면, $ A_k $ 에 관한 기대값과 분산값은:
+$$
+\begin{align*}
+E[A_k] & = E[\frac{S_k}{k}] \\
+ & = \frac{1}{k}*E[S_k] \\
+ & = \frac{1}{k}*k*\mu = \mu
+\end{align*}
+$$
+이고,
+$$
+\begin{align*}
+Var[A_k] & = Var[\frac{S_k}{k}] \\
+ & = \frac{1}{k^2} Var[S_k] \\
+ & = \frac{1}{k^2}*k*\sigma^2 \\
+ & = \frac{\sigma^2}{k} \nonumber
+\end{align*}
+$$
+라고 할 수 있다.