ㄹ====== 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
	95% Confidence	Lower 	1.5734	
	Interval	Upper 	1.6756	
        for Mean
	5% 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

     1.00        3 .  4
     5.00        3 .  66689
     5.00        4 .  33444
     7.00        4 .  6666799
     5.00        5 .  01224
     5.00        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]] 변량 
    * 표본변량 $ s^2 $
    * 모집단변량(전집) $ \sigma^2 $

<code>Descriptives					
			SET			Statistic	Std. Error
ATTRACT	 4	Mean				2.6445		.14651
		95% Confidence	Lower Bound	2.3379	
		Interval for 	Upper Bound	2.9511	
		Mean
		5% 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
	32	Mean				3.2615	.01541
		95% Confidence Interval for Mean	Lower Bound	3.2292	
							Upper Bound	3.2938	
		5% 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 <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 $

  * [[:Degrees of Freedom]] N-1
    * [[:Why n-1]]
===== Sampling Distribution, Standard Error =====
  * [[:Sampling]]
  * [[:Sampling Distribution]]
  * [[: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.