The pnorm() function in R returns the Cumulative Density Function (CDF) value of the normal distribution given a specific random variable q, the population mean μ, and the population standard deviation σ.
Syntax
pnorm(q, mean, sd, lower.tail = TRUE, log.p = FALSE)Parameters
- q: It is a vector of quantiles.
- mean: It is a vector of means.
- sd: It is a vector of standard deviations.
- lower.tail: It is logical; if TRUE (default), probabilities are otherwise.
- log, log.p: It is a logical argument.
Return value
It returns the Cumulative Density Function (CDF) value of the normal distribution. If mean or sd are not specified, they assume the default values of 0 and 1, respectively.
Example 1: How to Use the pnorm() function
Find the percentage of males taller than 78 inches in a population with mean = 74 and sd = 2.
pnorm(78, mean = 74, sd = 2, lower.tail = FALSE)Output
[1] 0.02275013Graphical representation
To visualize the pnorm() function, we can plot the probability density function (PDF) of the normal distribution and shade the area to the right of x=78, representing the right-tail probability.
We will use the RStudio to plot the chart and ggplot2 library. Install if you have not already!
# Load necessary libraries
library(ggplot2)
# Define parameters
mean_val <- 74
sd_val <- 2
# Create a sequence of x values
x <- seq(68, 80, by = 0.1)
# Calculate the PDF for the x values
y <- pnorm(x, mean = mean_val, sd = sd_val)
# Plot the PDF
p <- ggplot(data.frame(x = x, y = y), aes(x = x, y = y)) +
geom_line() +
geom_area(
data = data.frame(x = x[x >= 78], y = y[x >= 78]),
aes(x = x, y = y), fill = "skyblue"
) +
geom_vline(aes(xintercept = 78), linetype = "dashed", color = "red") +
labs(
title = "Right-tail of Normal Distribution",
x = "x",
y = "Density"
) +
theme_minimal()
# Display the plot
print(p)
Example 2: Using pnorm() with different parameters
pnorm(33, mean=40, sd = 8)Output
[1] 0.190787Graphical representation
# Load necessary libraries
library(ggplot2)
# Define parameters
mean_val <- 40
sd_val <- 8
# Create a sequence of x values
x <- seq(20, 60, by = 0.1)
# Calculate the CDF for the x values
y <- pnorm(x, mean = mean_val, sd = sd_val)
# Plot the CDF
p <- ggplot(data.frame(x = x, y = y), aes(x = x, y = y)) +
geom_line() +
geom_vline(aes(xintercept = 33), linetype = "dashed", color = "red") +
geom_point(aes(x = 33, y = pnorm(33, mean = mean_val, sd = sd_val)), color = "blue") +
labs(title = "CDF of Normal Distribution",
x = "x",
y = "pnorm(x, mean = 40, sd = 8)") +
theme_minimal()
# Display the plot
print(p)
Example 3: Passing the lower.tail = TRUE
# Load necessary libraries
library(ggplot2)
# Define parameters
mean_val <- 0
sd_val <- 1
# Create a sequence of x values
x <- seq(-4, 4, by = 0.1)
# Calculate the CDF for the x values
y <- pnorm(x, mean = mean_val, sd = sd_val, lower.tail = TRUE)
# Plot the CDF
p <- ggplot(data.frame(x = x, y = y), aes(x = x, y = y)) +
geom_line() +
geom_area(data = data.frame(x = x[x <= 1], y = y[x <= 1]),
aes(x = x, y = y), fill = "skyblue") +
geom_vline(aes(xintercept = 1), linetype = "dashed", color = "red") +
labs(title = "Left-tail of Standard Normal CDF",
x = "x",
y = "CDF") +
theme_minimal()
# Display the plot
print(p)
Example 4: Calculating pnorm() with a mean of 0
# Import the stats package
library(stats)
# Calculate the cumulative probability of a normal distribution with a mean of 0,
# a standard deviation of 1, and a quantile of 0.5
prob <- pnorm(q = 0.5, mean = 0, sd = 1)
# Print the cumulative probability
print(prob)Output
[1] 0.6914625Example 5: Visualization
# Load necessary libraries
library(ggplot2)
# Define the range and compute the CDF values
x_values <- seq(-5, 5, by=0.01)
y_values <- pnorm(x_values)
print(y_values)
# Create the plot
plot <- ggplot(data.frame(x=x_values, y=y_values), aes(x=x, y=y)) +
geom_line(color="blue") +
ggtitle("CDF of Standard Normal Distribution") +
xlab("x") +
ylab("Probability") +
theme_minimal()
# Display the plot
plot
Output
That’s it.
Krunal Lathiya is a seasoned Computer Science expert with over eight years in the tech industry. He boasts deep knowledge in Data Science and Machine Learning. Versed in Python, JavaScript, PHP, R, and Golang. Skilled in frameworks like Angular and React and platforms such as Node.js. His expertise spans both front-end and back-end development. His proficiency in the Python language stands as a testament to his versatility and commitment to the craft.