2025-10-12 16:35:57
doc: 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
This commit is contained in:
273
Research Method/Quantitative Research/Normal Distribution.md
Normal file
273
Research Method/Quantitative Research/Normal Distribution.md
Normal file
@@ -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
|
Reference in New Issue
Block a user