• A continuous probability distribution that is symmetric about its mean
• Values close to the mean occur more frequently than values farther away from the mean

ggplot() +
  xlim(-5, 5) +
  stat_function(
    fun = dnorm, 
    args = list(mean = 0, sd = 1),
    linewidth = 1,
    color = "#004225"
  ) +
  labs(
    title = "Standard Normal Distribution",
    x = "Z Score",
    y = "Density"
  ) +
  theme_minimal()

• The Mean, Median, and Mode are all equal at the center of the distribution

• Also known as a bell curve, showing peak at the mean

• The tails of the curve never touch the x-axis, but get infinitely smaller

• Example: Height in Adults

• x = Value of Random Variable
• \(\mu\) = mean of the distribution
• \(\sigma\) = standard deviation of the distribution

\[\frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}\]

• The Empirical Rule, also known as the 68-95-99.7 rule helps understand the distribution of the graph
• Approximately 68% of the data falls within \(\sigma\) of the mean
• Approximately 95% of the data falls within 2\(\sigma\) of the mean
• Approximately 99.7% of the data falls within 3\(\sigma\) of the mean

• Skewness (Right or left)

• Mean/Mode/Median are not equal

• Heavy Data Clusters

• Example: Skewed Graph

• What did we learn?
• Normal Distributions are bell curves that shows a symmetric distribution
• Empirical Rule - 68-95-99.7
• When a distribution is not normal