AP Statistics

Today's Topic: The (Continuous) Normal Distribution

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
Big image

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

Isn't there an easier way?

NormalPDF(x, m, s)!!!

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