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M Research Method/Quantitative Research/Descriptive Statistics.md Social Psychology/Aggression.md Social Psychology/Altruism.md, A Research Method/Quantitative Research/Normal Distribution.md Research Method/Quantitative Research/Systematic Comparison of Student's t, Welch's t, and Mann-Whitney U Tests.md
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Research Method/Quantitative Research/Descriptive Statistics.md
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@@ -28,6 +28,7 @@ Mean is the average of a specific variable in a data set. To tell apart, a popul
$$
\mu = \frac{\sum_{i}^{N}{x}}{N}
$$
$$
\bar{x} = \frac{\sum_{i}^{n}{x}}{n}
$$

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Research Method/Quantitative Research/Normal Distribution.md Normal file
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---
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

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Research Method/Quantitative Research/Systematic Comparison of Student's t, Welch's t, and Mann-Whitney U Tests.md Normal file
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---
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

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Social Psychology/Aggression.md
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---
Course: PSYG2504 Social psychology
tags:
- Psychology/Social
---
## 1. Definition of Aggression

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Social Psychology/Altruism.md
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---
Course: PSYG2504 Social psychology
tags:
- Psychology/Social
---
## 1. Definitions