diff --git a/Research Method/Quantitative Research/Descriptive Statistics.md b/Research Method/Quantitative Research/Descriptive Statistics.md
index ab6075c..9c717c7 100644
--- a/Research Method/Quantitative Research/Descriptive Statistics.md	
+++ b/Research Method/Quantitative Research/Descriptive Statistics.md	
@@ -28,6 +28,7 @@ Mean is the average of a specific variable in a data set. To tell apart, a popul
 $$
 \mu = \frac{\sum_{i}^{N}{x}}{N}
 $$
+
 $$
 \bar{x} = \frac{\sum_{i}^{n}{x}}{n}
 $$
diff --git a/Research Method/Quantitative Research/Normal Distribution.md b/Research Method/Quantitative Research/Normal Distribution.md
new file mode 100644
index 0000000..9bda6c3
--- /dev/null
+++ b/Research Method/Quantitative Research/Normal Distribution.md	
@@ -0,0 +1,273 @@
+---
+Course:
+  - PSYC10100 Introduction to Statistics for Psychological Sciences
+tags:
+  - statistics
+  - probability
+  - distributions
+  - normal-distribution
+---
+## 1. Introduction to Probability Distributions
+
+### 1.1. What is a Probability Distribution?
+
+A probability distribution describes how probabilities are distributed over the values of a random variable. It specifies the likelihood of different outcomes in an experiment or observation.
+
+### 1.2. Types of Probability Distributions
+
+- **Discrete Distributions**: For countable outcomes (e.g., binomial, Poisson)
+- **Continuous Distributions**: For measurable outcomes (e.g., normal, exponential)
+
+## 2. The Normal Distribution
+
+### 2.1. Definition and Properties
+The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve.
+
+**Key Properties**:
+- Symmetrical about the mean
+- Mean = Median = Mode
+- Defined by two parameters: mean ($\mu$) and standard deviation ($\sigma$)
+- Total area under the curve equals 1
+- Follows the Empirical Rule (68-95-99.7 rule)
+
+### 2.2. Probability Density Function
+The probability density function (PDF) of the normal distribution is:
+
+$$
+f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}
+$$
+
+Where:
+
+- $\mu$ = mean
+- $\sigma$ = standard deviation
+- $\pi$ ≈ 3.14159
+- $e$ ≈ 2.71828
+
+### 2.3. Empirical Rule (68-95-99.7 Rule)
+
+For normally distributed data:
+
+- Approximately 68% of data falls within $\pm1$ standard deviation from the mean
+- 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
+
+## 3. Distribution Shape Characteristics
+
+### 3.1. 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**: See [[Descriptive Statistics]]
+
+### 3.2. 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**: See [[Descriptive Statistics]]
+
+## 4. Standard Normal Distribution (Z-Distribution)
+
+### 4.1. Definition
+
+The standard normal distribution is a special case of the normal distribution with:
+
+- Mean ($\mu$) = 0
+- Standard deviation ($\sigma$) = 1
+
+### 4.2. Z-Scores
+
+A z-score (standard score) measures how many standard deviations an observation is from the mean:
+
+$$
+z = \frac{x - \mu}{\sigma}
+$$
+
+**Interpretation**:
+
+- $z = 0$: Value equals the mean
+- $z > 0$: Value above the mean
+- $z < 0$: Value below the mean
+
+### 4.3. Z-Table and Probability Calculations
+
+Z-tables provide the cumulative probability from $-\infty$ to a given z-value. Common z-values and their probabilities:
+
+| Z-Score | Cumulative Probability |
+| ------- | ---------------------- |
+| -3.0    | 0.0013                 |
+| -2.0    | 0.0228                 |
+| -1.0    | 0.1587                 |
+| 0.0     | 0.5000                 |
+| 1.0     | 0.8413                 |
+| 2.0     | 0.9772                 |
+| 3.0     | 0.9987                 |
+
+## 5. Student's t-Distribution
+
+### 5.1. Definition and Purpose
+The t-distribution is used when:
+- Sample sizes are small ($n < 30$)
+- Population standard deviation is unknown
+- We need to estimate population parameters from sample data
+
+### 5.2. Properties
+
+- Similar bell shape to normal distribution
+- Heavier tails than normal distribution (more probability in extremes)
+- Approaches normal distribution as degrees of freedom increase
+- Defined by degrees of freedom ($df = n - 1$)
+
+### 5.3. Degrees of Freedom
+
+Degrees of freedom represent the number of independent pieces of information available to estimate a parameter:
+
+$$
+df = n - 1
+$$
+
+Where $n$ is the sample size.
+
+### 5.4. T-Scores
+
+T-scores are calculated similarly to z-scores but use sample standard deviation:
+
+$$
+t = \frac{\bar{x} - \mu}{s/\sqrt{n}}
+$$
+
+Where:
+
+- $\bar{x}$ = sample mean
+- $\mu$ = population mean (hypothesized)
+- $s$ = sample standard deviation
+- $n$ = sample size
+
+## 6. Comparing Z and T Distributions
+
+| Characteristic | Z-Distribution | T-Distribution |
+|----------------|----------------|----------------|
+| **When to Use** | $\sigma$ known, large $n$ | $\sigma$ unknown, small $n$ |
+| **Parameters** | $\mu$, $\sigma$ | $\mu$, $s$, $df$ |
+| **Shape** | Fixed bell curve | Varies with $df$ |
+| **Tails** | Lighter | Heavier |
+| **Applications** | Hypothesis testing, confidence intervals | Same, but for small samples |
+
+## 7. Other Important Distributions
+
+### 7.1. Bimodal Distribution
+
+- Has two distinct peaks or modes
+- Often indicates two different populations or processes
+- Common in mixed data sets
+
+### 7.2. Uniform Distribution
+
+- All outcomes equally likely
+- Rectangular shape
+- Constant probability density function
+
+### 7.3. Other Common Distributions
+
+- **Binomial**: For binary outcomes
+- **Poisson**: For count data
+- **Exponential**: For time between events
+
+## 8. Applications in Psychological Research
+
+### 8.1. Hypothesis Testing
+
+- Using z-tests for large samples with known population parameters
+- Using t-tests for small samples or unknown population parameters
+
+### 8.2. Confidence Intervals
+
+- Constructing intervals for population means
+- Determining margin of error
+
+### 8.3. Effect Size Calculations
+
+- Standardizing measures for comparison across studies
+- Cohen's d and other effect size metrics
+
+## 9. Practical Examples
+
+### 9.1. Example 1: Z-Score Calculation
+
+Given: $\mu = 100$, $\sigma = 15$, $x = 130$
+
+$$
+z = \frac{130 - 100}{15} = 2.0
+$$
+
+Interpretation: This score is 2 standard deviations above the mean.
+
+### 9.2. Example 2: T-Score Calculation
+
+Given: $\mu = 50$, $\bar{x} = 55$, $s = 8$, $n = 25$
+
+$$
+t = \frac{55 - 50}{8/\sqrt{25}} = \frac{5}{1.6} = 3.125
+$$
+
+$df = 25 - 1 = 24$
+
+## 10. R Implementation
+
+### 10.1. Normal Distribution Functions
+
+```R
+# Probability density
+dnorm(x, mean = 0, sd = 1)
+
+# Cumulative probability
+pnorm(q, mean = 0, sd = 1)
+
+# Quantile function
+qnorm(p, mean = 0, sd = 1)
+
+# Random generation
+rnorm(n, mean = 0, sd = 1)
+```
+
+### 10.2. T-Distribution Functions
+
+```R
+# Probability density
+dt(x, df)
+
+# Cumulative probability
+pt(q, df)
+
+# Quantile function
+qt(p, df)
+
+# Random generation
+rt(n, df)
+```
+
+### 10.3. Sample Standard Deviation
+
+```R
+sample_sd <- sd(data) # Sample standard deviation
+```
+
+## 11. Summary
+
+- The normal distribution is fundamental in statistics with predictable properties
+- Z-distribution is used when population parameters are known
+- T-distribution is used for small samples with unknown population parameters
+- Understanding distribution shapes (skewness, kurtosis) helps interpret data patterns
+- These distributions form the basis for many statistical tests in psychological research
diff --git a/Research Method/Quantitative Research/Systematic Comparison of Student's t, Welch's t, and Mann-Whitney U Tests.md b/Research Method/Quantitative Research/Systematic Comparison of Student's t, Welch's t, and Mann-Whitney U Tests.md
new file mode 100644
index 0000000..396a15a
--- /dev/null
+++ b/Research Method/Quantitative Research/Systematic Comparison of Student's t, Welch's t, and Mann-Whitney U Tests.md	
@@ -0,0 +1,381 @@
+---
+Course:
+tags:
+  - statistics
+  - hypothesis-testing
+  - t-test
+  - welch
+  - mann-whitney
+  - nonparametric
+  - parametric
+  - comparison
+---
+## 1. Overview and Purpose
+
+This systematic note provides a comprehensive comparison of three commonly used statistical tests for comparing two independent groups: Student's t-test, Welch's t-test, and the Mann-Whitney U test. Each test serves different purposes and has specific assumptions and applications.
+
+## 2. Quick Reference Table
+
+| Test | Type | Key Assumptions | When to Use | Effect Size |
+|------|------|----------------|-------------|-------------|
+| **Student's t-test** | Parametric | Normality, equal variances, independence | Normal data with equal variances | Cohen's d |
+| **Welch's t-test** | Parametric | Normality, independence | Normal data with unequal variances | Cohen's d |
+| **Mann-Whitney U** | Nonparametric | Independence, ordinal/continuous data | Non-normal data, ordinal data | Rank-biserial correlation |
+
+## 3. Detailed Test Characteristics
+
+### 3.1. Student's t-test (Independent Samples)
+
+**Definition**: A parametric test comparing means of two independent groups assuming equal population variances.
+
+**Test Statistic**:
+$$
+t = \frac{\bar{X}_1 - \bar{X}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
+$$
+
+Where:
+
+- $\bar{X}_1$, $\bar{X}_2$ = sample means
+- $n_1$, $n_2$ = sample sizes
+- $s_p$ = pooled standard deviation
+
+**Pooled Standard Deviation**:
+$$
+s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}
+$$
+
+**Degrees of Freedom**:
+$$
+df = n_1 + n_2 - 2
+$$
+
+**Key Assumptions**:
+
+1. **Normality**: Data in each group are normally distributed
+2. **Homogeneity of variances**: Population variances are equal
+3. **Independence**: Observations are independent
+4. **Interval/ratio scale**: Data are continuous
+
+**R Implementation**:
+```R
+# Student's t-test (equal variances assumed)
+result <- t.test(group1, group2, var.equal = TRUE)
+
+# With formula interface
+result <- t.test(score ~ group, data = dataset, var.equal = TRUE)
+```
+
+### 3.2. Welch's t-test
+
+**Definition**: A parametric test comparing means without assuming equal variances between groups.
+
+**Test Statistic**:
+$$
+t = \frac{\bar{X}_1 - \bar{X}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
+$$
+
+**Degrees of Freedom** (Welch-Satterthwaite equation):
+$$
+df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1-1} + \frac{(s_2^2/n_2)^2}{n_2-1}}
+$$
+
+**Key Assumptions**:
+
+1. **Normality**: Data in each group are normally distributed
+2. **Independence**: Observations are independent
+3. **Interval/ratio scale**: Data are continuous
+4. **Unequal variances allowed**: No homogeneity of variances assumption
+
+**R Implementation**:
+```R
+# Welch's t-test (default in R)
+result <- t.test(group1, group2, var.equal = FALSE)
+
+# Explicit specification
+result <- t.test(group1, group2)
+
+# With formula interface
+result <- t.test(score ~ group, data = dataset)
+```
+
+### 3.3. Mann-Whitney U Test (Wilcoxon Rank-Sum Test)
+
+**Definition**: A nonparametric test determining if one group tends to have larger values than another.
+
+**Test Procedure**:
+
+1. Combine all observations from both groups
+2. Rank them from smallest to largest
+3. Calculate U statistics:
+   - $U_1 = R_1 - \frac{n_1(n_1+1)}{2}$
+   - $U_2 = R_2 - \frac{n_2(n_2+1)}{2}$
+4. Test statistic: $U = \min(U_1, U_2)$
+
+**Key Assumptions**:
+
+1. **Independence**: Observations are independent
+2. **Ordinal/continuous data**: Data can be ranked
+3. **Similar shape distributions**: For location shift interpretation
+4. **No normality assumption**: Distribution-free
+
+**R Implementation**:
+
+```R
+# Mann-Whitney U test
+result <- wilcox.test(group1, group2)
+
+# With formula interface
+result <- wilcox.test(score ~ group, data = dataset)
+
+# Extract results
+U_statistic <- result$statistic
+p_value <- result$p.value
+```
+
+## 4. Decision Framework
+
+### 4.1. Test Selection Algorithm
+
+```mermaid
+graph TD
+    A[Start: Compare Two Independent Groups] --> B{Data Normal?};
+    B -->|Yes| C{Equal Variances?};
+    B -->|No| D[Mann-Whitney U Test];
+    C -->|Yes| E[Student's t-test];
+    C -->|No| F[Welch's t-test];
+    
+    style D fill:#e1f5fe
+    style E fill:#f3e5f5
+    style F fill:#e8f5e8
+```
+
+### 4.2. Detailed Selection Criteria
+
+| Scenario | Recommended Test | Rationale |
+|----------|-----------------|-----------|
+| **Normal data, equal variances** | Student's t-test | Maximizes power when assumptions met |
+| **Normal data, unequal variances** | Welch's t-test | Robust to variance heterogeneity |
+| **Non-normal data** | Mann-Whitney U test | Distribution-free, handles outliers |
+| **Ordinal data** | Mann-Whitney U test | Designed for ranked data |
+| **Small samples** | Mann-Whitney U test | Less sensitive to distribution |
+| **Unequal sample sizes** | Welch's t-test | Handles unequal n better |
+| **Default choice** | Welch's t-test | More robust, recommended by many statisticians |
+
+## 5. Assumption Checking Procedures
+
+### 5.1. Normality Testing
+
+**Shapiro-Wilk Test**:
+
+```R
+# Test normality for each group
+shapiro.test(group1)
+shapiro.test(group2)
+```
+
+**Visual Inspection**:
+
+- Q-Q plots
+- Histograms
+- Density plots
+
+### 5.2. Homogeneity of Variances
+
+**Levene's Test**:
+
+```R
+library(car)
+leveneTest(score ~ group, data = dataset)
+```
+
+**F-test**:
+
+```R
+var.test(group1, group2)
+```
+
+**Bartlett's Test**:
+
+```R
+bartlett.test(score ~ group, data = dataset)
+```
+
+### 5.3. Independence
+
+- Research design consideration
+- No statistical test available
+- Ensure random sampling and assignment
+
+## 6. Effect Size Measures
+
+### 6.1. For Parametric Tests (Student's and Welch's t-tests)
+
+**Cohen's d**:
+$$
+d = \frac{\bar{X}_1 - \bar{X}_2}{s_{pooled}}
+$$
+
+Where:
+$$
+s_{pooled} = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}
+$$
+
+**Interpretation**:
+
+- Small: $d = 0.2$
+- Medium: $d = 0.5$
+- Large: $d = 0.8$
+
+### 6.2. For Mann-Whitney U Test
+
+**Rank-biserial correlation**:
+$$
+r = 1 - \frac{2U}{n_1n_2}
+$$
+
+**Common language effect size**:
+
+- Probability that random observation from group 1 > group 2
+- $CL = \frac{U}{n_1n_2}$
+
+## 7. Practical Examples
+
+### 7.1. Example 1: Student's t-test
+
+**Scenario**: Comparing exam scores between two classes with similar variance.
+
+```R
+# Data
+class_A <- c(78, 82, 85, 76, 79, 81, 83, 77, 80, 84)
+class_B <- c(75, 78, 72, 79, 76, 74, 77, 73, 75, 78)
+
+# Assumption checking
+shapiro.test(class_A)  # p = 0.423 (normal)
+shapiro.test(class_B)  # p = 0.356 (normal)
+var.test(class_A, class_B)  # p = 0.218 (equal variances)
+
+# Student's t-test
+t.test(class_A, class_B, var.equal = TRUE)
+```
+
+### 7.2. Example 2: Welch's t-test
+
+**Scenario**: Comparing reaction times between two age groups with different variances.
+
+```R
+# Data
+young <- c(210, 195, 225, 240, 205, 215, 230, 220, 200, 210)
+elderly <- c(280, 295, 270, 310, 320, 290, 300, 285, 315, 305)
+
+# Assumption checking
+shapiro.test(young)    # p = 0.512 (normal)
+shapiro.test(elderly)  # p = 0.487 (normal)
+var.test(young, elderly)  # p = 0.023 (unequal variances)
+
+# Welch's t-test
+t.test(young, elderly)  # var.equal = FALSE by default
+```
+
+### 7.3. Example 3: Mann-Whitney U Test
+
+**Scenario**: Comparing customer satisfaction ratings (ordinal scale 1-5).
+
+```R
+# Data
+store_A <- c(4, 3, 5, 2, 4, 3, 5, 4, 3, 4)
+store_B <- c(3, 2, 3, 1, 2, 3, 2, 1, 3, 2)
+
+# Mann-Whitney U test
+wilcox.test(store_A, store_B)
+```
+
+## 8. Power and Sample Size Considerations
+
+### 8.1. Relative Power
+
+- **Student's t-test**: Most powerful when assumptions are perfectly met
+- **Welch's t-test**: Slightly less power than Student's when variances equal, but better Type I error control
+- **Mann-Whitney U**: About 95% as powerful as t-tests for normal data, often more powerful for non-normal data
+
+### 8.2. Sample Size Guidelines
+
+| Test | Minimum Sample Size | Recommended per Group |
+|------|---------------------|----------------------|
+| Student's t-test | 15-20 | 30+ |
+| Welch's t-test | 15-20 | 30+ |
+| Mann-Whitney U | 5-10 | 20+ |
+
+## 9. Common Pitfalls and Best Practices
+
+### 9.1. Common Mistakes
+
+1. **Using Student's t-test without checking variances**
+2. **Applying parametric tests to non-normal data**
+3. **Ignoring effect sizes**
+4. **Not reporting assumption checks**
+5. **Using multiple tests without correction**
+
+### 9.2. Best Practices
+
+1. **Always check assumptions first**
+2. **Use Welch's t-test as default for parametric comparisons**
+3. **Report both p-values and effect sizes**
+4. **Use visualizations to support statistical findings**
+5. **Consider the research question when choosing tests**
+
+## 10. Advanced Considerations
+
+### 10.1. Transformations
+
+When data violate normality assumptions:
+
+- **Log transformation**: For right-skewed data
+- **Square root transformation**: For count data
+- **Arcsin transformation**: For proportions
+
+### 10.2. Robust Alternatives
+
+- **Trimmed means**: Remove extreme values
+- **Bootstrap methods**: Resampling approaches
+- **Permutation tests**: Exact nonparametric tests
+
+### 10.3. Software Implementation
+
+**Python**:
+```python
+from scipy import stats
+# Student's t-test
+stats.ttest_ind(group1, group2, equal_var=True)
+# Welch's t-test
+stats.ttest_ind(group1, group2, equal_var=False)
+# Mann-Whitney U test
+stats.mannwhitneyu(group1, group2)
+```
+
+## 11. Summary and Recommendations
+
+### 11.1. Key Takeaways
+
+1. **Student's t-test**: Use only when normality and equal variances are confirmed
+2. **Welch's t-test**: Recommended default for parametric comparisons
+3. **Mann-Whitney U**: Go-to choice for non-normal or ordinal data
+4. **Always validate assumptions** before test selection
+5. **Report comprehensive results** including effect sizes and assumption checks
+
+### 11.2. Final Decision Matrix
+
+| Data Characteristic | Preferred Test |
+|---------------------|----------------|
+| Normal + equal variances | Student's t-test |
+| Normal + unequal variances | Welch's t-test |
+| Non-normal data | Mann-Whitney U test |
+| Ordinal data | Mann-Whitney U test |
+| Small samples | Mann-Whitney U test |
+| Default choice | Welch's t-test |
+
+### 11.3. Related Tests
+
+- **Paired t-test**: For dependent samples
+- **One-way ANOVA**: For comparing >2 groups
+- **Kruskal-Wallis test**: Nonparametric alternative to ANOVA
+- **Bootstrapping**: For complex data situations
diff --git a/Social Psychology/Aggression.md b/Social Psychology/Aggression.md
index ffcc839..970dcad 100644
--- a/Social Psychology/Aggression.md	
+++ b/Social Psychology/Aggression.md	
@@ -1,5 +1,7 @@
 ---
 Course: PSYG2504  Social psychology
+tags:
+  - Psychology/Social
 ---
 
 ## 1. Definition of Aggression
diff --git a/Social Psychology/Altruism.md b/Social Psychology/Altruism.md
index 2280f39..ba606a8 100644
--- a/Social Psychology/Altruism.md	
+++ b/Social Psychology/Altruism.md	
@@ -1,5 +1,7 @@
 ---
 Course: PSYG2504  Social psychology
+tags:
+  - Psychology/Social
 ---
 
 ## 1. Definitions