Draw a Quantile-Quantile Plot in R Programming

A Quantile-Quantile plot is a graphical method for comparing two probability distributions by plotting their quantiles against each other. Typically, it is used to compare the distribution of the observed data with a theoretical distribution, such as the normal distribution.

When to Use Q-Q Plot in R

Q-Q plots are often used in statistical analysis to:

• Check for Normality: They help assess whether a dataset is approximately normally distributed, which is a common assumption in many statistical tests.
• Detect Skewness or Kurtosis: If the data has heavy tails or is skewed, this will show up in the Q-Q plot.
• Compare Two Distributions: It can be used to check if two datasets come from the same distribution.

Implementation of Drawing Q-Q Plots in R

We are plotting Q-Q (Quantile-Quantile) plots to visually assess whether the sample data comes from a theoretical distribution like normal, exponential or t-distribution.

1. Installing and Loading Required Packages

We install the ggplot2 package and load it to allow advanced Q-Q plotting.

• install.packages: Installs external R packages.
• library: Loads the installed package into the R session.

install.packages("ggplot2")
library(ggplot2)

2. Drawing a Basic Q-Q Plot Using qqnorm

We are using base R's qqnorm function to create a basic Q-Q plot with a reference line.

• rnorm: Generates random values from a normal distribution.
• qqnorm: Creates a Q-Q plot against the normal distribution.
• qqline: Adds a straight reference line to the Q-Q plot.

data <- rnorm(100)
qqnorm(data)
qqline(data, col = "blue")

Output:

3. Drawing a Q-Q Plot Using ggplot2

We are creating a Q-Q plot with ggplot2 for better customization and visual clarity.

• ggplot: Initializes a plot using a data frame and aesthetic mapping.
• aes: Defines the variables used in the plot.
• stat_qq: Plots sample quantiles against theoretical quantiles.
• stat_qq_line: Adds a reference line to the Q-Q plot.
• theme_minimal: Applies a minimal theme to the plot.
• ggtitle: Adds a title to the plot.

ggplot(data = data.frame(sample = data), aes(sample = sample)) +
  stat_qq() +
  stat_qq_line(col = "blue") +
  theme_minimal() +
  ggtitle("Q-Q Plot Using ggplot2")

Output:

4. Creating a Q-Q Plot for Exponential Distribution

We are plotting a Q-Q plot to compare sample data with an exponential distribution.

• rexp: Generates random values from an exponential distribution.
• ppoints: Generates theoretical probabilities for plotting.
• qexp: Calculates theoretical quantiles for the exponential distribution.
• qqplot: Plots one set of quantiles against another.
• abline: Adds a straight reference line to the plot.

exp_data <- rexp(100, rate = 1)
qqplot(qexp(ppoints(100)), exp_data, main = "Q-Q Plot for Exponential Distribution")
abline(0, 1, col = "blue")

Output:

5. Creating a Q-Q Plot for t-Distribution

We are plotting sample data against a t-distribution to check its conformity.

• rt: Generates random values from a t-distribution.
• qt: Computes quantiles of the t-distribution.

t_data <- rt(100, df = 5)
qqplot(qt(ppoints(100), df = 5), t_data, main = "Q-Q Plot for t-Distribution")
abline(0, 1, col = "red")

Output:

The output shows a Q-Q plot comparing sample data with a t-distribution, where most points lie along the red line, indicating the data approximately follows a t-distribution.