In R, we have various functions to create random variables that follow specific probability distributions. Two such functions are `rnorm()`

and `runif()`

. `rnorm()`

is used to generate random numbers that follow a normal distribution, while `runif()`

is used to generate random numbers that follow a uniform distribution. While they are fundamentally similar in that they both generate random numbers, they exhibit major differences in their underlying distributions, parameters, and usage.

## Basic Definitions

Before we delve deeper into their differences, let’s first establish what `rnorm()`

and `runif()`

are in the context of R.

### rnorm()

`rnorm()`

is a function in R that generates a set of random numbers according to a normal distribution (also known as Gaussian distribution). The syntax for `rnorm()`

is as follows:

`rnorm(n, mean = 0, sd = 1)`

Here `n`

is the number of observations (random numbers) to generate, `mean`

is the mean of the normal distribution, and `sd`

is the standard deviation.

### runif()

`runif()`

is a function in R that generates a set of random numbers according to a uniform distribution. The syntax for `runif()`

is as follows:

`runif(n, min = 0, max = 1)`

Here `n`

is the number of observations (random numbers) to generate, `min`

is the minimum limit of the distribution, and `max`

is the maximum limit.

## Differences Between rnorm() and runif()

### 1. Underlying Distributions

The fundamental difference between `rnorm()`

and `runif()`

is the underlying probability distribution they use to generate the random numbers.

`rnorm()`

generates numbers from a normal distribution, which is bell-shaped and symmetric, meaning it has a peak at the mean, and the data around the mean is dense. In other words, most of the observations in a normal distribution are close to the mean, and the frequency of the observations decreases as we move away from the mean on either side.

On the other hand, `runif()`

generates numbers from a uniform distribution. In a uniform distribution, all numbers within the specified range have an equal probability of being drawn. The distribution is a flat line because no value is more likely to occur than others within the specified minimum and maximum bounds.

### 2. Parameters

The parameters required by `rnorm()`

and `runif()`

are different due to their distribution characteristics.

`rnorm()`

requires the mean (`mean`

) and standard deviation (`sd`

) of the distribution. The mean is the central value of the distribution, around which the data is symmetrically distributed. The standard deviation is a measure of the dispersion or how spread out the values are around the mean.

Conversely, `runif()`

requires the minimum (`min`

) and maximum (`max`

) values of the distribution. The generated random numbers will fall within this specified range, and each will have an equal probability of being chosen.

### 3. Usage

The `rnorm()`

and `runif()`

functions are used in different scenarios that align with their distribution characteristics.

`rnorm()`

is often used when the data is expected to have a lot of values close to a central value with symmetry on both sides, like in the case of heights of people, test scores, etc. It’s also a go-to choice for many statistical tests and procedures as they often assume that the data is normally distributed.

`runif()`

, on the other hand, is useful when there is no apparent skew or preference for particular values in the data within the specified range, such as the roll of a fair die, or drawing a card from a well-shuffled deck.

## Practical Examples

### rnorm() Example:

```
# Generate 5 random numbers from a normal distribution with mean 10 and standard deviation 2
set.seed(123)
random_numbers <- rnorm(5, mean = 10, sd = 2)
print(random_numbers)
```

### runif() Example:

```
# Generate 5 random numbers from a uniform distribution between 0 and 100
set.seed(123)
random_numbers <- runif(5, min = 0, max = 100)
print(random_numbers)
```

These examples help illustrate how `rnorm()`

and `runif()`

work and how their resulting outputs are fundamentally different.

## Conclusion

In summary, `rnorm()`

and `runif()`

are integral parts of the random number generation capabilities of R. While they both generate random numbers, they do so based on different probability distributions – normal and uniform, respectively. Their choice depends on the problem at hand, the data’s distribution characteristics, and the specific requirements of statistical analysis.