R dnorm(): Probability Density Function

The dnorm() function in R calculates the value of the probability density function (pdf) of the normal distribution of a given value or vector of values.

It determines the probability density of a continuous random variable following a normal distribution, characterized by a mean (μ) and standard deviation (σ).

How do you define the normal distribution? Well, think like a mountain that is tallest in the middle and slopes down on both sides.

Now, if someone asks, ‘How tall is the mountain at this specific point?’ Well, the answer is the dnorm() function. It indicates the height of the hill at a specific location.

The output of this function is non-negative but not a probability (densities can exceed 1, unlike probabilities).

Syntax

dnorm(x, mean = 0, sd = 1, log = FALSE)

Parameters

Standard normal distribution

atomic <- -1

result <- dnorm(atomic, mean = 0, sd = 1)

print(result)

# Output: [1] 0.2419707

Vectorized input

What if an input is a vector? Well, it is a vectorized function, which means it will operate element-wise on vectors, making it efficient for multiple inputs.

vec <- c(-1, 0, 1)

dnorm(vec)

# Output: [1] 0.2419707 0.3989423 0.2419707

Custom mean and standard deviation

Let’s not use the default values of mean and standard deviation; instead, we will pass custom values and evaluate the density for a normal distribution.

dnorm(21, mean = 21, sd = 11)
# Output: [1] 0.03626748

# Vector input
vec <- c(8, 11, 14)

dnorm(vec, mean = 10, sd = 2)
# Output: [1] 0.12098536 0.17603266 0.02699548

Smaller SD values create a sharper peak. Wider SD values create a blunt hill.

dnorm(0)

dt <- dnorm(0)

dt

# Output: [1] 0.3989423

For computation stability, we can obtain the log of the density.

dnorm(0, log = TRUE)
# Output: [1] -0.9189385 (log of 0.3989423)

dnorm(c(-1, 0, 1), log = TRUE)
# Output: [1] -1.4189385 -0.9189385 -1.4189385

The log = TRUE argument is extremely helpful for numerical stability when density values are minimal.

Plotting

To plot a probability distribution function, we can use the plot() method.

seq(-4, 4, by = 0.05)

result <- dnorm(seq(-4, 4, by = 0.05))

plot(result)

Using ggplot2

library(ggplot2)

# Generate x values
x_values <- seq(-5, 5, by = 0.025)

# Compute y values using the dnorm function
y_values <- dnorm(x_values, mean = 0, sd = 1)

# Point of interest
poi_x <- 0
poi_y <- dnorm(poi_x, mean = 0, sd = 1)

# Plotting
df <- data.frame(x = x_values, y = y_values)
ggplot(df, aes(x, y)) +
  geom_line(color = "blue") +
  geom_point(aes(x = poi_x, y = poi_y), color = "red", size = 4) +
  labs(
    title = "Density of Standard Normal Distribution at x=0",
    x = "x", y = "Density"
  ) +
  theme_minimal() +
  theme(legend.position = "none")

Calculating probability density for a range of values

You can calculate the probability density for a range of values using this function in combination with the sapply() function.

x <- c(-3, -2, -1, 0, 1, 2, 3)
mean <- 0
sd <- 1
log <- FALSE

sapply(x, dnorm, mean=mean, sd=sd, log=log)

Output

[1] 0.004431848 0.053990967 0.241970725 0.398942280 0.241970725 0.053990967
[7] 0.004431848

dnorm(0, mean = 0, sd = 0)

# Output: [1] Inf

dnorm(0, mean = 0, sd = -1)

# Output: [1] NaN