The t distribution, also known as Student’s t-distribution, is a type of probability distribution that is symmetric and bell-shaped, like the normal distribution, but has heavier tails. It is primarily used in hypothesis testing and constructing confidence intervals when the sample size is small, and the population standard deviation is unknown.

In this comprehensive guide, we’ll delve into the process of generating and plotting a t-distribution in R using various visualization techniques.

## 1. Understanding t-Distribution Functions in R

R provides several functions to work with t-distributions:

`rt(n, df)`

: Generates`n`

random variates from a t-distribution with`df`

degrees of freedom.`dt(x, df)`

: Gives the height of the probability density function at`x`

for a t-distribution with`df`

degrees of freedom.`pt(q, df)`

: Returns the cumulative probability at`q`

for a t-distribution with`df`

degrees of freedom.`qt(p, df)`

: Gives the`p`

-quantile for a t-distribution with`df`

degrees of freedom.

For this guide, we’ll primarily be using the `dt()`

function to plot the probability density function (PDF) of a t-distribution.

## 2. Plotting a t-Distribution

Let’s generate and plot a t-distribution with 5 degrees of freedom.

```
# Define sequence of x-values
x <- seq(-5, 5, length.out = 400)
# Define degrees of freedom
df <- 5
# Compute density values
y <- dt(x, df)
# Convert to data frame for ggplot
df <- data.frame(x, y)
# Plot the t-distribution
ggplot(df, aes(x = x, y = y)) +
geom_line(color = "darkgreen") +
labs(x = "x", y = "Density", title = "t-Distribution with 5 degrees of freedom") +
theme_minimal()
```

In the above code, we first define a sequence of x-values using the `seq()`

function. We then compute the density values for the t-distribution with 5 degrees of freedom using the `dt()`

function. Finally, we use `ggplot()`

and `geom_line()`

to create the plot.

## 3. Plotting t-Distributions with Varying Degrees of Freedom

The shape of a t-distribution is primarily influenced by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution. Let’s plot t-distributions with varying degrees of freedom to observe this behavior.

```
# Define sequence of x-values
x <- seq(-5, 5, length.out = 400)
# Define a vector of degrees of freedom
dfs <- c(1, 2, 5, 10, 30)
# Create an empty data frame to store the density values
df <- data.frame()
# Loop over degrees of freedom, compute density values, and bind to the data frame
for (df_i in dfs) {
df_temp <- data.frame(x = x, y = dt(x, df_i), df = factor(df_i))
df <- rbind(df, df_temp)
}
# Plot the t-distributions
ggplot(df, aes(x = x, y = y, color = df)) +
geom_line() +
labs(x = "x", y = "Density", color = "Degrees of freedom",
title = "t-Distributions with Varying Degrees of Freedom") +
theme_minimal()
```

In this script, we first define a vector of degrees of freedom. We then loop over the degrees of freedom, compute the density values for each, and bind them to a data frame. The `factor(df_i)`

function is used to convert the numeric degrees of freedom to a factor, which is necessary for `ggplot2`

to handle it as a categorical variable correctly. Finally, we plot the t-distributions, with the `color = df`

aesthetic in `ggplot()`

resulting in different colors for each degree of freedom.

## 4. Overlaying a t-Distribution on a Normal Distribution

Given that the t-distribution approaches the standard normal distribution as the degrees of freedom increase, it might be helpful to visualize them together. The following code overlays a t-distribution with 30 degrees of freedom on a standard normal distribution.

```
# Define sequence of x-values
x <- seq(-5, 5, length.out = 400)
# Compute density values for t-distribution
y_t <- dt(x, df = 30)
# Compute density values for normal distribution
y_n <- dnorm(x)
# Convert to data frame for ggplot
df <- data.frame(x, y_t, y_n)
# Plot the distributions
ggplot(df, aes(x = x)) +
geom_line(aes(y = y_t), color = "darkgreen", linetype = "solid") +
geom_line(aes(y = y_n), color = "red", linetype = "dashed") +
labs(x = "x", y = "Density", title = "Comparison of t-Distribution (df = 30) and Standard Normal Distribution") +
theme_minimal() +
scale_colour_manual("",
breaks = c("t-Distribution", "Normal Distribution"),
values = c("darkgreen", "red"))
```

Here, we compute the density values for both the t-distribution and the standard normal distribution. We then use `geom_line()`

twice within the `ggplot()`

function to plot both distributions on the same graph. The `linetype`

argument is used to distinguish between the two distributions.

## Conclusion

This guide provided an in-depth look at how to plot a t-distribution in R using the `ggplot2`

package. We explored how to plot a single t-distribution, multiple t-distributions with varying degrees of freedom, and a t-distribution overlaid on a standard normal distribution.

Understanding and being able to visualize the t-distribution is crucial in statistics, particularly in situations with small sample sizes or unknown population standard deviations. With R and `ggplot2`

, you can create these visualizations and customize them to your needs, making them powerful tools in your statistical analysis toolkit.