When a data set is normally distributed it is a normal distribution that displays the shape of a bell curve.

Some real life applications include:

• Height and weight of people

• Blood pressure measurements

• Stock Values

A random variable follows a normal distribution it is probability density function(pdf) is given by: \[f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

-\(\mu\) is the mean (center of distribution)

-\(\sigma\) is the variance(spread of the distribution)

• When \(\mu\) is high the curve shifts to the right

• When \(\mu\) is low the curve shifts to the left

• Note the bell curve shape stays the same it only shifts

• When \(\sigma\) is high the spread becomes wider

• When \(\sigma\) is low the spread becomes narrower

• Note the bell curve shape stays centered and the spread expands or narrows

For any normal distribution:

Approximately 68% \[P(\mu - 1\sigma \leq X \leq \mu + 1\sigma)\]

Approximately 95% \[P(\mu - 2\sigma \leq X \leq \mu + 2\sigma)\]

Approximately 99.7% \[P(\mu - 3\sigma \leq X \leq \mu + 3\sigma)\]

This 3D visual is based on the axes:

X-axis: x is a random variable

Y-axis: σ is the standard deviation from narrow to wide

Z-axis: f(x) is the probability density function

The surface shows us that the bell curve become wider and flatter as the standard deviation increases.

x <- seq(-4, 4, length = 100)
y <- (1/sqrt(2*pi)) * exp(-x^2/2)

plot(x, y, type = "l", 
     main = "Simple Normal Distribution",
     xlab = "x",
     ylab = "f(x)")