【Advanced】Basic Statistical Analysis in MATLAB: Mean, Variance, and Probability Distributions

# [Advanced] Basic Statistical Analysis in MATLAB: Mean, Variance, and Probability Distributions

## 2.1 Mean and Variance

### 2.1.1 Definition and Calculation of Mean

The mean, also known as the average, is a vital indicator for measuring the central tendency of a set of data in statistics. It represents the average level of the data set. The formula for calculating the mean is:

Mean = (x1 + x2 + ... + xn) / n

Where:

- x1, x2, ..., xn are the values of each element in the data set
- n is the number of elements in the data set

### 2.1.2 Definition and Calculation of Variance

Variance is an indicator used to measure the degree of dispersion in a set of data. It represents the average squared deviation of each element in the data set from the mean. The formula for calculating variance is:

Variance = Σ(xi - μ)^2 / (n - 1)

Where:

- xi is the value of each element in the data set
- μ is the mean of the data
- n is the number of elements in the data set

## 2. Statistical Basics Theory

### 2.1 Mean and Variance

#### 2.1.1 Definition and Calculation of Mean

Mean, also known as arithmetic mean, is the total sum of all values in a data set divided by the number of values. It represents the central location of all values in the data set. The formula for calculating the mean is:

Mean = (x1 + x2 + ... + xn) / n

Where:

* x1, x2, ..., xn are the individual values in the data set
* n is the number of values in the data set

For example, for the data set {1, 3, 5, 7, 9}, the mean is:

Mean = (1 + 3 + 5 + 7 + 9) / 5 = 5

#### 2.1.2 Definition and Calculation of Variance

Variance measures the degree of dispersion of values in a data set from the mean. It represents the average squared deviation of each value in the data set from the mean. The formula for calculating variance is:

Variance = Σ[(xi - Mean)^2] / (n - 1)

Where:

* xi is each value in the data set
* Mean is the average of all values in the data set
* n is the number of values in the data set

For example, for the data set {1, 3, 5, 7, 9}, the variance is:

Variance = [(1 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (9 - 5)^2] / (5 - 1) = 8

### 2.2 Probability Distributions

#### 2.2.1 Concept of Probability Distributions

Probability distributions describe the likelihood of possible values of a random variable. It is a function that maps each possible value of the random variable to a probability between 0 and 1. Probability distributions help us predict future values of a random variable.

#### 2.2.2 Types of Common Probability Distributions

Common types of probability distributions include:

***Normal Distribution:** Also known as Gaussian distribution, is a sym