## 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 , 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 and standard deviation (this is the “basic” normal distribution – we often use as a variable to set it apart): .

Normal distributions with different mean and standard deviation

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

The normal distribution with mean and standard deviation : .

### 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 from the left bound to the right bound.

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

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