Plotting A Normal Distribution In R

Plotting a Normal Distribution in R: A Comprehensive Guide

The normal distribution, also known as the Gaussian distribution, is a fundamental concept in statistics and probability. Its bell-shaped curve is ubiquitous across various fields, from finance and physics to biology and social sciences. Understanding and visualizing this distribution is crucial for data analysis and interpretation. This comprehensive guide will walk you through plotting a normal distribution in R, covering various aspects, from basic plotting to advanced customizations. We’ll explore different approaches, incorporating various R packages and functionalities to cater to diverse needs and skill levels.

Understanding the Normal Distribution

Before diving into the plotting aspects, let's briefly recap the key characteristics of a normal distribution:

• Symmetry: The distribution is perfectly symmetrical around its mean (μ).
• Mean, Median, and Mode: The mean, median, and mode are all equal in a normal distribution.
• Standard Deviation (σ): This parameter determines the spread or dispersion of the data. A larger standard deviation indicates a wider spread, while a smaller standard deviation indicates a narrower spread.
• Empirical Rule (68-95-99.7 Rule): Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

Basic Plotting with R's Built-in Functions

R offers several built-in functions for generating and plotting normal distributions. The most straightforward approach uses the dnorm() function, which calculates the probability density function (PDF) of the normal distribution. Let's create a basic plot:

# Define the range of x-values
x <- seq(-3, 3, length.out = 100)

# Calculate the probability density function for a standard normal distribution (mean = 0, sd = 1)
y <- dnorm(x, mean = 0, sd = 1)

# Create the plot
plot(x, y, type = "l", col = "blue", lwd = 2,
     xlab = "Z-score", ylab = "Density",
     main = "Standard Normal Distribution")

This code generates a plot of the standard normal distribution (mean = 0, standard deviation = 1). seq() creates a sequence of x-values, dnorm() calculates the corresponding y-values (density), and plot() generates the line plot. type = "l" specifies a line plot, col sets the line color, lwd adjusts line width, and xlab, ylab, and main set the axis labels and title respectively.

Plotting with Different Means and Standard Deviations

Modifying the mean and sd arguments in the dnorm() function allows us to plot normal distributions with different parameters. Let's plot several distributions to visualize the impact of changing the mean and standard deviation:

x <- seq(-6, 6, length.out = 100)

# Standard Normal Distribution
y1 <- dnorm(x, mean = 0, sd = 1)

# Distribution with mean = 2, sd = 1
y2 <- dnorm(x, mean = 2, sd = 1)

# Distribution with mean = 0, sd = 2
y3 <- dnorm(x, mean = 0, sd = 2)

# Plot all three distributions
plot(x, y1, type = "l", col = "blue", lwd = 2, ylim = c(0, 0.45),
     xlab = "X", ylab = "Density", main = "Normal Distributions with Varying Parameters")
lines(x, y2, col = "red", lwd = 2)
lines(x, y3, col = "green", lwd = 2)

legend("topright", legend = c("Mean=0, SD=1", "Mean=2, SD=1", "Mean=0, SD=2"),
       col = c("blue", "red", "green"), lty = 1, lwd = 2)

This code demonstrates how changes in mean shift the curve horizontally, and changes in standard deviation affect the curve's spread. The ylim argument ensures all curves are visible on the same plot, and the legend() function adds a clear legend.

Using ggplot2 for Enhanced Visualization

The ggplot2 package provides a powerful and flexible grammar of graphics for creating visually appealing plots. Let's recreate the above plot using ggplot2:

library(ggplot2)

# Create a data frame
df <- data.frame(x = x,
                 y1 = y1,
                 y2 = y2,
                 y3 = y3)

# Reshape the data for ggplot2
df_long <- tidyr::pivot_longer(df, cols = starts_with("y"), names_to = "distribution", values_to = "density")

# Create the ggplot2 plot
ggplot(df_long, aes(x = x, y = density, color = distribution)) +
  geom_line(size = 1.2) +
  labs(x = "X", y = "Density", title = "Normal Distributions with Varying Parameters") +
  scale_color_manual(values = c("y1" = "blue", "y2" = "red", "y3" = "green"),
                     labels = c("Mean=0, SD=1", "Mean=2, SD=1", "Mean=0, SD=2")) +
  theme_bw()

This ggplot2 code is more concise and allows for greater customization. tidyr::pivot_longer reshapes the data into a format suitable for ggplot2, making it easier to handle multiple distributions. The scale_color_manual function allows precise control over colors and legends. theme_bw() applies a clean black-and-white theme.

Adding Shaded Regions to Highlight Probabilities

Visualizing probabilities under the normal curve is often essential. We can achieve this by shading specific areas using ggplot2 and geom_area():

library(ggplot2)

# Data for standard normal distribution
x <- seq(-3, 3, length.out = 100)
y <- dnorm(x, mean = 0, sd = 1)

# Calculate probabilities
prob_within_1sd <- pnorm(1) - pnorm(-1)
prob_within_2sd <- pnorm(2) - pnorm(-2)

# Create the plot
ggplot() +
  geom_area(aes(x = x, y = y), fill = "lightblue", data = data.frame(x = x[x >= -1 & x <=1], y = y[x >= -1 & x <= 1])) +
  geom_area(aes(x = x, y = y), fill = "lightgreen", data = data.frame(x = x[x >= -2 & x <=2], y = y[x >= -2 & x <= 2])) +
  geom_line(aes(x = x, y = y), color = "blue", size = 1.2) +
  labs(x = "Z-score", y = "Density", title = "Normal Distribution with Shaded Probabilities") +
  annotate("text", x = 0, y = 0.15, label = paste("68%", "Within 1 SD"), color = "blue") +
  annotate("text", x = 0, y = 0.3, label = paste("95%", "Within 2 SD"), color = "blue") +
  theme_bw()

This code highlights the areas within one and two standard deviations of the mean, visually representing the 68-95-99.7 rule. geom_area() fills the specified regions, and annotate() adds text labels to clarify the probabilities.

Advanced Customizations and Considerations

Numerous customization options exist for enhancing your normal distribution plots. These include:

• Adding vertical lines: Mark specific points (e.g., mean, standard deviations) using geom_vline() in ggplot2.
• Changing colors and themes: Experiment with different color palettes and ggplot2 themes (theme_classic(), theme_minimal(), etc.) for visual appeal.
• Adding data points: Overlay your actual data points on the normal distribution curve to visually assess normality.
• Using different plotting functions: Explore alternative functions like curve() for a more concise approach, particularly for simple plots.
• Handling skewed distributions: If your data deviates significantly from normality, consider transformations (e.g., logarithmic, Box-Cox) before plotting. Visualizing the transformed data's distribution will assist with understanding normality.

Conclusion

Plotting a normal distribution in R is a versatile tool for statistical visualization. This guide provides a foundation for creating various plots ranging from basic representations to visually rich and informative graphics. Mastering these techniques will greatly aid your data analysis and interpretation, enabling you to better understand and communicate statistical findings. Remember to choose the plotting method that best suits your needs and the level of customization you require. Whether you opt for base R functions or the more advanced capabilities of ggplot2, the key is clarity and effective communication of the data. Continuously exploring the rich functionalities within R will only enhance your data visualization skills and your ability to effectively share insights derived from your analyses.