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--- c:ms:2018:schedule:week03 [2018/03/21 07:57] – created hkimscil
+++ c:ms:2018:schedule:week03 [2018/03/21 08:09] (current) – [Central Tendency] hkimscil
@@ Line 1: / Line 1: @@
-ㄹ====== Week 3 내용 ======
+====== Week 3 내용 ======
-===== SPSS =====
-<del>Chapter 3</del>, Chapter 4
-  * SPSS
-  * [[http://www.uvm.edu/~dhowell/fundamentals7/DataFiles/MentalRotation.dat|Table 3.1 data file]]. for SPSS, excel format, see the below.
-    * Explanation: Read the textbook for yourself (Chapter 3)
-  * frequency distribution
-  * histogram
-  * stem and leaf display.
-    * watch [[https://www.youtube.com/watch?v=6JM80zb2fes|How to create a Stem and Leaf Plot in Microsoft Excel]]
-    * watch [[https://www.youtube.com/watch?v=atWwZmIEZ9Q|Spss]]
 ===== Central Tendency =====
   * Central Tendency (집중경향)
-    * data: {{:data_rtsec.sav|SPSS data file, rtsec}} or {{:data_rtsec.xlsx|Excel file}}
-<code>
-Statistics
-RTsec
-N	Valid	600
-	Missing	0
-Mean		1.6245
-Median		1.5300
-Mode		1.33
-</code>
-<code>Descriptives
-				StatisticStd. Error
-RTsec	Mean			1.6245	.02603
-% Confidence	Lower 	1.5734
-	Interval	Upper 	1.6756
-        for Mean
-% Trimmed Mean		1.5672
-	Median			1.5300
-	Variance		.407
-	Std. Deviation		.63772
-	Minimum			.72
-	Maximum			4.44
-	Range			3.72
-	Interquartile Range	.77
-	Skewness		1.465	.100
-	Kurtosis		2.849	.199
-</code>
-{{:hist.jpg}}
-data file: {{:Ex3-1.sav}} 읽지 않은 지문에 대한 답을 한 학생들의 점수 (Katz, 1990).
-<code>NOPASSAG Stem-and-Leaf Plot
- Frequency    Stem &  Leaf
-.00        3 .  4
-.00        3 .  66689
-.00        4 .  33444
-.00        4 .  6666799
-.00        5 .  01224
-.00        5 .  55577
- Stem width:   10.00
- Each leaf:       1 case(s)
-</code>
-{{:Fig.4.1.jpg}}
-Chapter 5
   * Dispersion (variability) -- 분산(변산성)
-  * Data file: [[http://www.uvm.edu/~dhowell/fundamentals7/DataFiles/Tab5-1.dat|Web site]] or {{:Tab5-1.sav}} p.86-7
   * [[:range]]
   * [[:outliers]]: It is beyond our scope. Please just refer to it. Won't be appearing in tests.
-  * 평균편차
+  * [[:Variance]] 분산 혹은 변량
-  * [[:Variance]] 변량
     * 표본변량 $ s^2 $
     * 모집단변량(전집) $ \sigma^2 $
-<code>Descriptives
-			SET			Statistic	Std. Error
-ATTRACT	 4	Mean				2.6445		.14651
-% Confidence	Lower Bound	2.3379
-		Interval for 	Upper Bound	2.9511
-		Mean
-% Trimmed Mean			2.6483
-		Median				2.5950
-		Variance			.429
-		Std. Deviation			.65520
-		Minimum				1.20
-		Maximum				4.02
-		Range				2.82
-		Interquartile Range		.82
-		Skewness			-.001	.512
-		Kurtosis			.438	.992
-	Mean				3.2615	.01541
-% Confidence Interval for Mean	Lower Bound	3.2292
-							Upper Bound	3.2938
-% Trimmed Mean		3.2622
-		Median			3.2650
-		Variance		.005
-		Std. Deviation		.06892
-		Minimum			3.13
-		Maximum			3.38
-		Range			.25
-		Interquartile Range		.11
-		Skewness		-.075	.512
-		Kurtosis		-.863	.992
-</code>
   * [[:Standard Deviation]] 표준편차
   * Variance calculation formula
     * {{anchor: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 $
   * [[:Degrees of Freedom]] N-1
     * [[:Why n-1]]
@@ Line 119: / Line 20: @@
   * [[: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.