Categorical vs quantitative variables
Quantitative variables are any variables where the data represent amounts.
Categorical variables are any variables where the data represent groups.
Survey 100 students, record:
- hair color
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 : .
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.
2:normalcdf(lower bound, upper bound, mean, standard deviation)
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?
- 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.
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)