diff --git a/Research Method/Quantitative Research/Descriptive Analysis.md b/Research Method/Quantitative Research/Descriptive Analysis.md
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----
-Course:
-  - PSYC10100 Introduction to Statistics for Psychological Sciences
----
-## Central Tendency
-
diff --git a/Research Method/Quantitative Research/Descriptive Statistics.md b/Research Method/Quantitative Research/Descriptive Statistics.md
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+---
+Course:
+  - PSYC10100 Introduction to Statistics for Psychological Sciences
+---
+## 1. Size
+
+We note the size of:
+
+- Population: $N$
+- Sample: $n$
+
+To distinguish the difference between Population and Sample. For difference between which, see more in [Population and Sample](Population%20and%20Sample.md). 
+
+## 2. Central Tendency
+
+Central tendency provides a general description of the data set, using mean, mode, and median.
+
+| Measure    | Advantages                          | Disadvantages                           | Best for             |
+| ---------- | ----------------------------------- | --------------------------------------- | -------------------- |
+| **Mean**   | Uses all data, algebraic properties | Sensitive to outliers                   | Normal distributions |
+| **Median** | Robust to outliers                  | Ignores magnitude of extreme values     | Skewed distributions |
+| **Mode**   | Works for categorical data          | May not be unique, ignores other values | Nominal data         |
+
+### 2.1. Mean
+
+Mean is the average of a specific variable in a data set. To tell apart, a population mean is noted as $\mu$, while a sample mean is noted as $\bar{x}$.
+
+$$
+\mu = \frac{\sum_{i}^{N}{x}}{N}
+$$
+$$
+\bar{x} = \frac{\sum_{i}^{n}{x}}{n}
+$$
+
+In some article, it is also considered as the expectation of a variable, notated as $E[x] = \sigma$. In most situation, people estimate $\mu$ by $E[x]$, but which leads to a degree of freedom (DoF) lost.
+
+Trying to avoid the influence of outliers, a *trimmed mean* calculates the average after removing a specified percentage of the highest and lowest values to reduce the effect of outliers.
+
+``` R
+describe(data, trim = 0.1) # Set the persentace for trimed mean
+```
+
+### 2.2. Mode
+
+The **mode** is the value that appears most frequently in a data set. Unlike mean and median, a data set can have:
+
+- **Unimodal**: One mode
+- **Bimodal**: Two modes  
+- **Multimodal**: Multiple modes
+- **No mode**: All values appear with equal frequency
+
+**Characteristics**:
+- Useful for categorical data and discrete distributions
+- Not affected by extreme values (outliers)
+- May not exist or may not be unique
+- For continuous data, mode is often estimated using kernel density estimation
+
+**Example**: For data set {1, 2, 2, 3, 4, 4, 4, 5}, the mode is 4.
+
+### 2.3. Median
+
+The **median** is the middle value in an ordered data set. It divides the data into two equal halves:
+
+- For odd number of observations: Middle value
+- For even number of observations: Average of two middle values
+
+**Calculation**:
+- Order the data from smallest to largest
+- If $n$ is odd: $\text{median} = x_{(n+1)/2}$
+- If $n$ is even: $\text{median} = \frac{x_{n/2} + x_{n/2 + 1}}{2}$
+
+**Characteristics**:
+- Robust to outliers and skewed distributions
+- More appropriate than mean for ordinal data
+- Represents the 50th percentile (second quartile)
+
+**Example**: 
+- Data: {3, 1, 7, 4, 2} → Ordered: {1, 2, 3, 4, 7} → Median = 3
+- Data: {3, 1, 7, 4} → Ordered: {1, 3, 4, 7} → Median = (3+4)/2 = 3.5
+
+## 3. Measures of Dispersion
+
+### 3.1. Variance
+
+Variance measures the average squared deviation from the mean. It distinguishes between population and sample variance due to degrees of freedom considerations.
+
+#### 3.1.1. Population Variance ($\sigma^2$)
+The variance of the entire population:
+
+$$
+\sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}
+$$
+
+where:
+
+- $N$ = population size
+- $\mu$ = population mean
+- $x_i$ = individual values
+
+**Example**:
+Population {1, 3, 5, 7, 9}, $\mu = 5$.
+  $\sigma^2 = \frac{(1-5)^2 + (3-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2}{5} = \frac{16+4+0+4+16}{5} = 8$
+#### 3.1.2. Sample Variance ($s^2$)
+The variance estimated from a sample, using $n-1$ in denominator for unbiased estimation:
+
+$$
+s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}
+$$
+
+where:
+- $n$ = sample size
+- $\bar{x}$ = sample mean
+- $x_i$ = individual values
+
+**Key Differences**:
+- **Denominator**: Population uses $N$, sample uses $n-1$ (Bessel's correction)
+- **Purpose**: Population variance describes the entire population, sample variance estimates population variance
+- **Bias**: Using $n$ instead of $n-1$ in sample variance creates a biased estimator
+
+**Why $n-1$ for sample variance?**
+
+Using $n-1$ (degrees of freedom) corrects for the fact that we're estimating the population mean from the sample, making $s^2$ an unbiased estimator of $\sigma^2$. 
+
+**Example**:
+Subset {1, 3, 5} from population, $\bar{x} = 3$.
+  $s^2 = \frac{(1-3)^2 + (3-3)^2 + (5-3)^2}{3-1} = \frac{4+0+4}{2} = 4$
+
+**R Implementation**:
+
+```R
+sample_var <- var(data) # Sample variance (built-in function)
+```
+
+### 3.2. Standard Deviation
+
+Standard deviation is the square root of variance, providing a measure of dispersion in the original units of the data. Like variance, it distinguishes between population and sample standard deviation.
+
+#### 3.2.1. Population Standard Deviation ($\sigma$)
+The standard deviation of the entire population:
+
+$$
+\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}
+$$
+
+#### 3.2.2. Sample Standard Deviation ($s$)
+The standard deviation estimated from a sample:
+
+$$
+s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}}
+$$
+
+**Interpretation**:
+
+- Approximately 68% of data falls within $\pm1$ standard deviation from the mean (for normal distributions)
+- Approximately 95% of data falls within $\pm2$ standard deviations from the mean
+- Approximately 99.7% of data falls within $\pm3$ standard deviations from the mean
+
+**Advantages over Variance**:
+
+- Same units as original data (easier to interpret)
+- More intuitive measure of spread
+- Directly comparable to the mean
+
+**Example** (continuing from variance examples):
+
+- Population: $\sigma = \sqrt{8} \approx 2.83$
+- Sample: $s = \sqrt{4} = 2$
+
+**R Implementation**:
+```R
+sample_sd <- sd(data) # Sample standard deviation (built-in function)
+```
+
+### 3.3. Median Absolute Deviation
+
+Median Absolute Deviation (MAD) is a robust measure of statistical dispersion that uses the median rather than the mean, making it resistant to outliers.
+
+**Definition**:
+MAD is the median of the absolute deviations from the data's median:
+
+$$
+\text{MAD} = \text{median}(|x_i - \text{median}(x)|)
+$$
+
+**Scale Factor**:
+For normally distributed data, MAD needs to be scaled by approximately 1.4826 to be comparable to standard deviation:
+
+$$
+\text{MAD}_{\text{normal}} = 1.4826 \times \text{MAD}
+$$
+
+**Why Use MAD?**
+
+- **Robustness**: Not influenced by extreme values
+- **Breakdown point**: 50% - up to half the data can be outliers without affecting MAD
+- **Interpretation**: Similar to standard deviation but for median-based analysis
+
+**Comparison with Standard Deviation**:
+
+| Aspect          | Standard Deviation          | MAD                 |
+| --------------- | --------------------------- | ------------------- |
+| **Sensitivity** | Sensitive to outliers       | Robust to outliers  |
+| **Center**      | Uses mean                   | Uses median         |
+| **Breakdown**   | 0% (any outlier affects it) | 50%                 |
+| **Units**       | Original data units         | Original data units |
+
+**Example**:
+Data: {1, 2, 3, 4, 100} (100 is an outlier)
+
+- Median = 3
+- Absolute deviations: |1-3|=2, |2-3|=1, |3-3|=0, |4-3|=1, |100-3|=97
+- Sorted deviations: {0, 1, 1, 2, 97}
+- MAD = median{0, 1, 1, 2, 97} = 1
+
+**R Implementation**:
+```R
+# Calculate MAD
+mad_value <- mad(data)
+
+# Scaled MAD for normal distributions
+scaled_mad <- mad(data, constant = 1.4826)
+```
+### 3.4. Interquartile Range
+
+Interquartile Range (IQR) is a robust measure of statistical dispersion that describes the spread of the middle 50% of the data. It is particularly useful for identifying outliers and understanding data distribution.
+
+**Definition**:
+IQR is the difference between the third quartile (Q3) and the first quartile (Q1):
+
+$$
+\text{IQR} = Q_3 - Q_1
+$$
+
+where:
+- $Q_1$ = First quartile (25th percentile)
+- $Q_3$ = Third quartile (75th percentile)
+
+**Quartile Calculation Methods**:
+There are several methods to calculate quartiles. Common approaches include:
+
+1. **Method 1 (Inclusive)**: $Q_1$ at position $(n+1)/4$, $Q_3$ at position $3(n+1)/4$
+2. **Method 2 (Exclusive)**: $Q_1$ at position $n/4$, $Q_3$ at position $3n/4$
+3. **Tukey's Method**: Median of lower half and upper half
+
+**Outlier Detection**:
+IQR is commonly used to identify outliers using the "1.5×IQR rule":
+- **Lower fence**: $Q_1 - 1.5 \times \text{IQR}$
+- **Upper fence**: $Q_3 + 1.5 \times \text{IQR}$
+- Values outside these fences are considered potential outliers
+
+**Box Plot Relationship**:
+IQR forms the "box" in box plots:
+- Box extends from Q1 to Q3
+- Line inside box represents median
+- Whiskers extend to most extreme non-outlier values
+
+**Advantages**:
+- **Robust**: Not affected by extreme values
+- **Intuitive**: Easy to interpret and visualize
+- **Standardized**: Widely used in exploratory data analysis
+
+**Example**:
+Data: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
+
+- Q1 (25th percentile) = 3.25
+- Q3 (75th percentile) = 7.75
+- IQR = 7.75 - 3.25 = 4.5
+- Lower fence = 3.25 - 1.5×4.5 = -3.5
+- Upper fence = 7.75 + 1.5×4.5 = 14.5
+- No outliers in this dataset
+
+**R Implementation**:
+```R
+# Calculate IQR (built-in function)
+iqr_value <- IQR(data)
+
+# Calculate quartiles
+q1 <- quantile(data, 0.25)
+q3 <- quantile(data, 0.75)
+manual_iqr <- q3 - q1
+
+# Outlier detection
+lower_fence <- q1 - 1.5 * iqr_value
+upper_fence <- q3 + 1.5 * iqr_value
+outliers <- data[data < lower_fence | data > upper_fence]
+
+# Box plot visualization
+boxplot(data, main = "Box Plot with IQR")
+```
+
+**Comparison with Other Dispersion Measures**:
+
+| Measure | Robustness | Interpretation | Best Use |
+|---------|------------|----------------|----------|
+| **Range** | Not robust | Total spread | Quick overview |
+| **Variance/SD** | Not robust | Average deviation | Normal distributions |
+| **MAD** | Very robust | Median-based spread | Outlier-prone data |
+| **IQR** | Robust | Middle 50% spread | Exploratory analysis |
+
+## 4. Distribution Shape
+
+### 4.1. Skewness
+
+Skewness measures the asymmetry of a probability distribution around its mean. It indicates whether data are concentrated more on one side of the distribution.
+
+**Types of Skewness**:
+
+- **Positive Skew (Right Skew)**: Tail extends to the right, mean > median > mode
+- **Negative Skew (Left Skew)**: Tail extends to the left, mean < median < mode
+- **Zero Skew**: Symmetrical distribution, mean = median = mode
+
+**Calculation**:
+The most common measure is Pearson's moment coefficient of skewness:
+
+$$
+\text{Skewness} = \frac{E[(X - \mu)^3]}{\sigma^3}
+$$
+
+For sample data:
+
+$$
+g_1 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^3}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\right)^{3/2}}
+$$
+
+**Interpretation**:
+
+- **|Skewness| < 0.5**: Approximately symmetric
+- **0.5 ≤ |Skewness| < 1**: Moderately skewed
+- **|Skewness| ≥ 1**: Highly skewed
+
+**R Implementation**:
+```R
+# Using moments package
+library(moments)
+skewness_value <- skewness(data)
+
+# Using psych package
+library(psych)
+skewness_value <- describe(data)$skew
+
+# Manual calculation
+manual_skew <- function(x) {
+  n <- length(x)
+  mean_x <- mean(x)
+  sd_x <- sd(x)
+  (sum((x - mean_x)^3) / n) / (sd_x^3)
+}
+```
+
+### 4.2. Kurtosis
+
+Kurtosis measures the "tailedness" of a probability distribution, indicating how much data are in the tails compared to a normal distribution.
+
+**Types of Kurtosis**:
+
+- **Mesokurtic**: Normal distribution, kurtosis = 3 (excess kurtosis = 0)
+- **Leptokurtic**: Heavy tails and sharp peak, kurtosis > 3 (excess kurtosis > 0)
+- **Platykurtic**: Light tails and flat peak, kurtosis < 3 (excess kurtosis < 0)
+
+**Calculation**:
+Pearson's moment coefficient of kurtosis:
+
+$$
+\text{Kurtosis} = \frac{E[(X - \mu)^4]}{\sigma^4}
+$$
+
+Excess kurtosis (more commonly used):
+
+$$
+\text{Excess Kurtosis} = \frac{E[(X - \mu)^4]}{\sigma^4} - 3
+$$
+
+For sample data:
+
+$$
+g_2 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^4}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2\right)^2} - 3
+$$
+
+**Interpretation**:
+- **Excess Kurtosis = 0**: Normal distribution
+- **Excess Kurtosis > 0**: Heavy tails (more outliers)
+- **Excess Kurtosis < 0**: Light tails (fewer outliers)
+
+**R Implementation**:
+```R
+# Using moments package
+library(moments)
+kurtosis_value <- kurtosis(data)  # Returns excess kurtosis
+
+# Using psych package
+library(psych)
+kurtosis_value <- describe(data)$kurtosis
+
+# Manual calculation
+manual_kurtosis <- function(x) {
+  n <- length(x)
+  mean_x <- mean(x)
+  sd_x <- sd(x)
+  (sum((x - mean_x)^4) / n) / (sd_x^4) - 3
+}
+```
+
+## 5. Estimation Accuracy
+
+### 5.1. Standard Error
+
+Standard Error (SE) measures the precision of a sample statistic as an estimate of the population parameter. It quantifies the variability of the sampling distribution.
+
+**Standard Error of the Mean (SEM)**:
+The most common standard error, measuring how much the sample mean varies from sample to sample:
+
+$$
+\text{SEM} = \frac{s}{\sqrt{n}}
+$$
+
+where:
+
+- $s$ = sample standard deviation
+- $n$ = sample size
+
+**Interpretation**:
+
+- Smaller SE indicates more precise estimate
+- SE decreases as sample size increases
+- Used to construct confidence intervals
+
+**Other Standard Errors**:
+
+- **Standard Error of Proportion**: $\text{SE}_p = \sqrt{\frac{p(1-p)}{n}}$
+- **Standard Error of Difference**: $\text{SE}_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
+
+**R Implementation**:
+```R
+# Standard Error of the Mean
+sem <- function(x) {
+  sd(x) / sqrt(length(x))
+}
+
+# Using built-in functions
+sample_mean <- mean(data)
+sample_sd <- sd(data)
+n <- length(data)
+sem_manual <- sample_sd / sqrt(n)
+
+# For proportions
+se_prop <- function(p, n) {
+  sqrt(p * (1 - p) / n)
+}
+```
+
+### 5.2. Confidence Interval
+
+Confidence Interval (CI) provides a range of values that likely contains the population parameter with a specified level of confidence.
+
+**Confidence Interval for Mean**:
+For large samples or known population variance:
+
+$$
+\text{CI} = \bar{x} \pm z_{\alpha/2} \times \frac{s}{\sqrt{n}}
+$$
+
+For small samples with unknown population variance:
+
+$$
+\text{CI} = \bar{x} \pm t_{\alpha/2, df} \times \frac{s}{\sqrt{n}}
+$$
+
+where:
+- $\bar{x}$ = sample mean
+- $s$ = sample standard deviation
+- $n$ = sample size
+- $z_{\alpha/2}$ = critical z-value
+- $t_{\alpha/2, df}$ = critical t-value with $df = n-1$
+
+**Common Confidence Levels**:
+- **90% CI**: $\alpha = 0.10$, $z_{0.05} = 1.645$
+- **95% CI**: $\alpha = 0.05$, $z_{0.025} = 1.96$
+- **99% CI**: $\alpha = 0.01$, $z_{0.005} = 2.576$
+
+**Interpretation**:
+- "We are 95% confident that the true population mean lies between [lower, upper]"
+- Does NOT mean there's a 95% probability that the specific interval contains the parameter
+
+**R Implementation**:
+```R
+# Using t.test for confidence interval
+result <- t.test(data)
+ci <- result$conf.int
+
+# Manual calculation for 95% CI
+manual_ci <- function(x, conf_level = 0.95) {
+  n <- length(x)
+  mean_x <- mean(x)
+  sd_x <- sd(x)
+  se <- sd_x / sqrt(n)
+  t_critical <- qt(1 - (1 - conf_level)/2, df = n - 1)
+  lower <- mean_x - t_critical * se
+  upper <- mean_x + t_critical * se
+  return(c(lower, upper))
+}
+
+# For proportions
+prop_ci <- function(x, conf_level = 0.95) {
+  p <- mean(x)
+  n <- length(x)
+  se <- sqrt(p * (1 - p) / n)
+  z_critical <- qnorm(1 - (1 - conf_level)/2)
+  lower <- p - z_critical * se
+  upper <- p + z_critical * se
+  return(c(lower, upper))
+}
+```
+
+## 6. Implementation in R
+
+The package `psych` provides pretty descriptive statistic:
+
+``` R
+library(psych)
+data <- c(1, 1, 6, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9)
+View(describe(data))
+```
+
+But which package does not calculate the mode of data set, you could calculate it by:
+
+``` R
+# Calculate mode
+mode_val <- as.numeric(names(sort(table(data), decreasing = TRUE)[1]))
+```
\ No newline at end of file
diff --git a/Research Method/Quantitative Research/Population and Sample.md b/Research Method/Quantitative Research/Population and Sample.md
new file mode 100644
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+++ b/Research Method/Quantitative Research/Population and Sample.md	
@@ -0,0 +1,14 @@
+---
+Course:
+  - PSYC10100 Introduction to Statistics for Psychological Sciences
+---
+## Population
+
+Population is 
+
+- Population size: $N$
+## Sample 
+
+- Sample size: $n$
+
+## Sampling