• The normal distribution, also known as the Gaussian distribution, is one of the most important probability distributions in statistics.
• It is characterized by its bell-shaped curve and is symmetric around the mean.

The probability density function (PDF) of a Normal Distribution is given by:

\[ f(x|\mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

Where \(\mu\) is the mean and \(\sigma\) is the standard deviation.

• The mean \(\mu\) determines the center of the distribution.
• The standard deviation \(\sigma\) determines the width of the bell curve.
  • A larger \(\sigma\) results in a wider and flatter curve. A smaller \(\sigma\) results in a narrower and taller curve.

68-95-99.7 Rule (Empirical Rule)

• In a normal distribution:
  • 68% of data falls within 1 standard deviation of the mean (\(\mu \pm 1\sigma\)),
  • 95% of data falls within 2 standard deviations of the mean (\(\mu \pm 2\sigma\)),
  • 99.7% of data falls within 3 standard deviations of the mean (\(\mu \pm 3\sigma\)).

We can visualize the Normal Distribution using ggplot2. Below is the plot of the Standard Normal Distribution (mean = 0 and standard deviation = 1).

The Standard Normal Distribution is a special case of the Normal Distribution. It has a mean of 0 and a standard deviation of 1.

Any Normal Distribution can be transformed into a Standard Normal Distribution using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

Where \(Z\) is the standard score (z-score), \(X\) is the value from the original Normal Distribution, \(\mu\) is the mean of the original distribution, and \(\sigma\) is the standard deviation of the original distribution

The Z-score represents how many standard deviations a data point \(X\) is from the mean \(\mu\).

The Normal Distribution is widely used. Let’s use ggplot2 to visualize the iris dataset, where the sepal lengths are normally distributed.

R Code to Generate the Plot:

# Load the iris dataset
data(iris)

# Generate data for sepal length
sepal_length <- iris$Sepal.Length
density_sepal_length <- dnorm(
  sort(sepal_length), mean=mean(sepal_length), sd=sd(sepal_length))
data_sepal_length <- data.frame(
  Sepal.Length = sort(sepal_length), 
                                Density = density_sepal_length)

library(ggplot2)
ggplot(data_sepal_length, aes(Sepal.Length, Density)) +
  geom_line(color="purple", linewidth=1) +
  labs(title="Distribution of Sepal Length in Iris Dataset",
       x="Sepal Length",
       y="Density") +
  theme_minimal() +
  geom_vline(xintercept = mean(sepal_length), linetype = "dashed", 
             color = "red") +
  geom_vline(xintercept = c(mean(sepal_length) - sd(sepal_length), 
                            mean(sepal_length) + sd(sepal_length)), 
             linetype = "dashed", color = "green") +
  geom_vline(xintercept = c(mean(sepal_length) - 2 * 
                              sd(sepal_length), mean(sepal_length) + 
                              2 * sd(sepal_length)), 
             linetype = "dashed", color = "orange") +
  annotate("text", x = mean(sepal_length), y = 0.05, 
           label = paste("Mean =", round(mean(sepal_length), 2)), 
           vjust = -1, color = "red", size = 4)

Let’s use Plotly to chart a 3D visualization of the mtcars dataset. This is the relationship between miles per gallon and horsepower. Adding a Z axis representing PDF density shows that it follows a 3D Normal Distribution.