The normal distribution

Categorical vs quantitative variables

Quantitative variables are any variables where the data represent amounts.
Categorical variables are any variables where the data represent groups.

Typical question: What is the probability that a randomly selected student is Filipino?

Typical question: What is the probability that a randomly selected student is between 5 and 6 feet?

The normal distribution

Cool and handy fact: An enormous number of different kinds of continuous real-world variables have the same shape – the bell curve (or normal distribution).

This curve is based on the function $y=e^{-x^2}$, with constants added in appropriate places to make the values work out correctly (total area = 1, inflection points at +-1, and so on).

The normal distribution with mean $\mu=0$ and standard deviation $\sigma=1$ (this is the “standard” normal distribution – we often use $z$ as a variable to set it apart):

$$f(z)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}z^2}$$

Normal distributions with different mean and standard deviation

What if our random variable is measuring height? Then the mean will not be $\mu=0$ ft — it’s more likely the mean will be something like $\mu=5.8$ft. And the standard deviation may not turn out to be $\sigma=1$.

The normal distribution with mean $\mu$ and standard deviation $\sigma$
$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$

How do we actually calculate probabilities using the normal distribution?

Option 1. For certain simple values, use the 68-95-99.7 rule.

Option 2. Use the TI-84+ calculator.

• Press 2nd Distr
• Press 2:normalcdf(lower bound, upper bound, mean, standard deviation)
• Example: normalcdf(-1000,-64.9,79,7)

Option 3. Use the formula. In particular, calculate the area under the normal distribution curve $f(x)$ from the left bound to the right bound.
$$\int_a^b \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}} dx$$