**Categorical vs quantitative variables**

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

### Example

Survey 100 students, record:

- height
- gender
- hair color
- age

Which variables are categorical? Which are quantitative?

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 $\sigma$

$$f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$

### Example 1

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.

$$\int_a^b \frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}} dx$$

**What is the 68-95-99.7 rule?**

- About 68% of values fall within one standard deviation of the mean.
- About 95% of the values fall within two standard deviations from the mean.
- Almost all of the values—about 99.7%—fall within three standard deviations from the mean.

### Example 2

### Example 3

### Example 4

### Resources on Probability and Statistics

- The Bear in Moonlight – Math With Bad Drawings’ 7-part series on probability (disguised in story form)
- OpenLab course hub for MAT 1372 (Probability and Statistics)
- Introduction to Probability from OpenStax textbook on Probability
- Adjustable spinner (change # of categories and probability of each, then simulate spins)

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