MATLAB Normality Distribution Hypothesis Testing: Testing Whether Data Follows a Normal Distribution

# MATLAB Normality Assumption Test: Verifying Data Normality Distribution

## 1. Overview of Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by a bell-shaped curve for its probability density function. It is widely present in nature and statistics, describing the distribution of many random variables.

Characteristics of the normal distribution include:

- Symmetry: The probability density function is symmetrical about the mean.
- Unimodality: The probability density function has a single peak located at the mean.
- Asymptoticity: When the sample size is large enough, the normal distribution can approximate many other distributions.

## 2. Normality Assumption Testing Methods

### 2.1 Principle of Normality Testing

#### 2.1.1 Features of the Normal Distribution

The normal distribution, also called the Gaussian distribution, is a continuous probability distribution with a probability density function described by:

f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)² / (2σ²))

Where:

* μ: The mean of the normal distribution
* σ: The standard deviation of the normal distribution

The normal distribution has the following features:

* Symmetry: The normal distribution curve is symmetrical about the mean.
* Bell-shaped curve: The normal distribution curve is bell-shaped, with sides that gradually decline.
* 68-95-99.7 Rule: Approximately 68% of the data is included within one standard deviation from the mean, about 95% within two standard deviations, and about 99.7% within three standard deviations.

#### 2.1.2 Significance of Normality Testing

Normality testing is a statistical method used to determine whether data conforms to a normal distribution. It is significant for the following reasons:

***Validity of Statistical Inference:** Many statistical inference methods, such as t-tests and ANOVA, assume that the data follows a normal distribution. If the data does not conform to the normal distribution, these methods may lead to erroneous conclusions.
***Model Selection and Evaluation:** The performance of machine learning models can be affected by the data distribution. Normality testing can help select the model that best fits the data and assess the performance of the model.

### 2.2 Common Normality Testing Methods

There are various methods available to assess whether data conforms to a normal distribution. Some commonly used methods include:

#### 2.2.1 Shapiro-Wilk Test

The Shapiro-Wilk test is a non-parametric test used to determine if data conforms to a normal distribution. It is based on the following statistic:

W = (b₁x₁ + b₂x₂ + ... + bnxn) / √(Σ(xᵢ - x̄)²)

Where:

* x₁, x₂, ..., xn: Sample data
* x̄: Sample mean
* b₁, b₂, ..., bn: Constants calculated from the sample data

The value of W ranges between 0 and 1, with values closer to 1 indicating a better fit to the normal distribution.

#### 2.2.2 Jarque-Bera Test

The Jarque-Bera test is a normality test based on sample skewness and kurtosis. It is based on the following statistic:

JB = n * [(S² / 6) + (K³ / 24)]

Where:

* n: Sample size
* S: Sample skewness
* K: Sample kurtosis

The JB statistic follows a chi-squared distribution with 2 degrees of freedom. If the value of JB is greater than the critical value, the normal distribution assumption is rejected.

#### 2.2.3 Lilliefors Test

The Lilliefors test is a normality test based on the maxim