How to Plot a Log Normal Distribution in R

In this article, we will explore how to plot a log-normal distribution in R, providing you with an understanding of its properties and the practical steps for visualizing it using R Programming Language.

Log-Normal Distribution

The log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. This means if you take the natural logarithm (log) of a variable following a log-normal distribution, the result will be normally distributed. This distribution is widely used in finance, biology, and environmental studies where data tends to be positively skewed.

• It is always positive.
• It is right-skewed.
• Its shape depends on the mean and standard deviation of the associated normal distribution.

Generating and Plotting a Log-Normal Distribution in R

To plot a log-normal distribution, we can use base R and several additional packages like ggplot2 for better visualization. We'll explore different examples, starting from basic plotting to more advanced visualizations. To plot the log-normal distribution we would require two functions namely dlnorm() and curve().

dlnorm(x, meanlog = 0, sdlog = 1)

Where,

• x: vector of quantiles
• meanlog: mean of the distribution on the log scale with a default value of 0.
• sdlog: standard deviation of the distribution on the log scale with default values of 1.

curve(expr, from = NULL, to = NULL)

Where,

• function: The name of a function, or a call or an expression written as a function of x which will evaluate to an object of the same length as x.
• from: the start range over which the function will be plotted.
• to: the end range over which the function will be plotted.

let us plot a log-normal distribution using mean 0 and standard deviation 1 over a range of 0 to 25 using curve and dlnorm function.

curve(dlnorm(x, meanlog=0, sdlog=1), from=0, to=25)

Output:

Now we will discuss step by step How to Plot a Log Normal Distribution and customized plot with different packages and methods using R Programming Language.

Step 1: Generate Log-Normal Data

R provides a built-in function rlnorm() to generate random numbers from a log-normal distribution.

# Set the parameters
set.seed(123)  # Setting seed for reproducibility
n <- 1000      # Number of observations
meanlog <- 0   # Mean of the underlying normal distribution
sdlog <- 0.5   # Standard deviation of the underlying normal distribution

# Generate log-normal data
log_normal_data <- rlnorm(n, meanlog = meanlog, sdlog = sdlog)

In this example, rlnorm(n, meanlog, sdlog) generates n log-normal random numbers with the specified mean and standard deviation.

Step 2: Plot the Log-Normal Distribution Using Base R

We can create a histogram of the generated data using the hist() function.

# Plot the histogram
hist(log_normal_data,
     breaks = 30,
     probability = TRUE,
     main = "Log-Normal Distribution",
     xlab = "Value",
     col = "skyblue",
     border = "white")

# Add a density curve
lines(density(log_normal_data), col = "red", lwd = 2)

Output:

Step 3: Plotting Log-Normal Distribution Using ggplot2

For more advanced visualization, ggplot2 offers powerful tools for plotting the log-normal distribution.

library(ggplot2)

# Create a data frame
df <- data.frame(Value = log_normal_data)

# Plot using ggplot2
ggplot(df, aes(x = Value)) +
  geom_histogram(aes(y = ..density..), bins = 30, fill = "skyblue", color = "black", alpha = 0.7) +
  geom_density(color = "red", size = 1.2) +
  labs(title = "Log-Normal Distribution with ggplot2", x = "Value", y = "Density") +
  theme_minimal()

Output:

Step 4: Overlaying the Theoretical Log-Normal Distribution Curve

We can add the theoretical log-normal probability density function (PDF) on top of the histogram to visualize how the generated data fits the log-normal distribution.

# Create a sequence of values for the x-axis
x_vals <- seq(min(log_normal_data), max(log_normal_data), length.out = 1000)

# Calculate the theoretical PDF
theoretical_pdf <- dlnorm(x_vals, meanlog = meanlog, sdlog = sdlog)

# Plot using ggplot2 with theoretical curve
ggplot(df, aes(x = Value)) +
  geom_histogram(aes(y = ..density..), bins = 30, fill = "lightblue", color = "black", alpha = 0.6) +
  geom_line(aes(x = x_vals, y = theoretical_pdf), color = "darkblue", size = 1.2, linetype = "dashed") +
  labs(title = "Log-Normal Distribution with Theoretical Curve",
       x = "Value",
       y = "Density") +
  theme_minimal()

Output:

Step 5: Comparing Log-Normal Distribution with Normal Distribution

It's often insightful to compare the log-normal distribution with a normal distribution by plotting them together.

# Generate normal data for comparison
normal_data <- rnorm(n, mean = meanlog, sd = sdlog)

# Create a combined data frame
combined_df <- data.frame(Value = c(log_normal_data, normal_data),
                          Distribution = rep(c("Log-Normal", "Normal"), each = n))

# Plot the comparison
ggplot(combined_df, aes(x = Value, fill = Distribution)) +
  geom_histogram(aes(y = ..density..), bins = 30, position = "identity", alpha = 0.4) +
  geom_density(aes(color = Distribution), size = 1.2) +
  labs(title = "Comparison of Log-Normal and Normal Distributions",
       x = "Value",
       y = "Density") +
  theme_minimal() +
  scale_color_manual(values = c("blue", "red")) +
  scale_fill_manual(values = c("blue", "red"))

Output:

Conclusion

The log-normal distribution is widely used in different fields and is an essential concept in statistics. In this article, we demonstrated how to generate and plot a log-normal distribution using both base R and ggplot2. We also showed how to overlay a theoretical density curve, visualize the distribution on a log scale, and compare it with a normal distribution.