R is a powerful built-in functions and libraries to handle a wide range of statistical analyses. I’ll cover some basic statistical functions and distributions along with examples.

I’ll explain mean and median, as well as essential distributions like normal, binomial, and Poisson. Each concept is accompanied by simple examples, demonstrating how to implement them in R for data analysis and visualization.

With these examples, you can enhance your understanding of statistical analysis in R and apply it effectively in their research or projects.

### Statistical Functions

#### 1. Mean

The mean function calculates the arithmetic mean of a numeric vector.

**Example**

x <- c(1, 2, 3, 4, 5) mean_x <- mean(x) print(mean_x) Output: 3

#### 2. Median

The median function calculates the median of a numeric vector.

**Example**

x <- c(1, 2, 3, 4, 5) median_x <- median(x) print(median_x) #Output: 3

#### 3. Standard Deviation

The sd function computes the standard deviation of a numeric vector.

**Example**

x <- c(1, 2, 3, 4, 5) sd_x <- sd(x) print(sd_x) #Output: 1.581139

### Statistical Distributions

#### 1. Normal Distribution

- The normal distribution, or Gaussian distribution, is symmetric and bell-shaped, characterized by its mean and standard deviation.
- It is widely used in statistical analysis due to its prevalence in nature and its role in the central limit theorem.
- Common functions for working with the normal distribution in R include rnorm() for generating random numbers, dnorm() for calculating density, and pnorm() for calculating cumulative probabilities.

The rnorm function generates random numbers from a normal distribution.

**Example**

set.seed(123) #for reproducibility n <- 1000 mu <- 0 sigma <- 1 data <- rnorm(n, mean = mu, sd = sigma) hist(data, main = "Normal Distribution", prob = TRUE) curve(dnorm(x, mean = mu, sd = sigma), add = TRUE, col = "blue")

#### 2. Binomial Distribution

- The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.
- It is characterized by two parameters: the number of trials (n) and the probability of success (p).
- R provides functions like rbinom() for generating random numbers, dbinom() for calculating probabilities, and pbinom() for calculating cumulative probabilities.

The rbinom function generates random numbers from a binomial distribution.

**Example**

set.seed(123)# for reproducibility n <- 1000 size <- 10 prob <- 0.5 data <- rbinom(n, size = size, prob = prob) hist(data, main = "Binomial Distribution", prob = TRUE) x <- 0:size y <- dbinom(x, size = size, prob = prob) points(x, y, type = "h", col = "blue")

#### 3. Poisson Distribution

- The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence.
- It is characterized by a single parameter, lambda, representing the average rate of occurrence.
- In R, functions like rpois() generate random numbers, dpois() calculate probabilities, and ppois() calculate cumulative probabilities for the Poisson distribution.

The rpois function generates random numbers from a Poisson distribution.

**Example**

set.seed(123) #for reproducibility n <- 1000 lambda <- 3 data <- rpois(n, lambda = lambda) hist(data, main = "Poisson Distribution", prob = TRUE) x <- 0:15 y <- dpois(x, lambda = lambda) points(x, y, type = "h", col = "blue")

#### 4. Chi-Square Distribution

- The chi-square distribution is commonly used in hypothesis testing and confidence interval estimation for the variance of a normal distribution.
- It is characterized by a single parameter, degrees of freedom (df), which influences its shape.
- R provides functions like rchisq() for generating random numbers, dchisq() for calculating density, and pchisq() for calculating cumulative probabilities for the chi-square distribution.

The rchisq function generates random numbers from a chi-square distribution.

**Example**

set.seed(123) #for reproducibility n <- 1000 df <- 5 data <- rchisq(n, df = df) hist(data, main = "Chi-Square Distribution", prob = TRUE) x <- seq(0, 30, by = 0.1) y <- dchisq(x, df = df) lines(x, y, col = "blue")

These examples cover some of the basic statistical functions and distributions in R. R provides extensive support for statistical analysis and visualization, making it a preferred choice for data analysis tasks.