# AP Statistics

## OBJECTIVES

• Understand that there are infinitely many continuous probability distributions such as the Uniform, Normal, and Chi-squared distributions
• Understand that the normal curve is a parameterized function of x with parameters mu and sigma (mean and standard deviation)
• Use the Empirical Rule to calculate probabilities under the normal curve
• Calculate z-scores and use them to compare quantities

## Non-uniform Probability Density Functions

• Area under the curve always equals one (for 100%)
• Can be any shape as long as the above holds true
• Not common

## The Uniform Distribution

• Also called the rectangular distribution
• It has a constant probability over its interval
• Mean = average of max and min
• To find probabilities, just find the area of the rectangle defined by the limits given

## The Gaussian Distribution (ie. The Standard Normal Curve)

• Perfectly Symmetric
• Centered at a mean of zero
• Standard Deviation of one
• Can be shifted and scaled by the parameters in order to model various scenarios that are approximately normal

## The Empirical Rule

Example 1:

Input y = (normal curve function) using a mean of 0 and standard deviation of 1.

• Set your window to x: [-3, 3] y: [0, 0.5]
• Integrate between +1 SD, +2 SD, and +3 SD to find the probability of landing in each region

Example 2:

Change the SD to 2.

• Integrate between each set of standard deviations to find the probability of landing in each region

## Z-scores

A Z-score measures how many standard deviations an observation is from the mean. It is the standardized measure of the score.

Easiest formula in this class: Z = (x - m)/s