2 1 Density Curves and the Normal Distributions

2. 1 Density Curves and the Normal Distributions Grab warm-up and start!

Strategies in Chapter 1 New Strategy: Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

Density Curves Characteristics: ¡ Always on or above horizontal axis ¡ Has an area of 1 underneath the curve What are they: o o Density curves are mathematical models for distributions. Outliers are not described by the curve. All curves are approximations. They are idealized.

EXAMPLE

Mean and Median of Density Curves The median of a density curve is the equal areas point. The mean is at the balance point (the point where the curve would balance if it were made of solid material). The mean and median of symmetric curves are at the center.

Normal Distributions l Normal distributions can be described with a density curve that is symmetric, single peaked, and bell-shaped. l Changing the mean without changing standard deviation moves the mean along x-axis without changing the spread. l Standard deviation controls the spread of the curve. (how flat or peaked it is!)

What is a normal distribution? l Normal distributions are a family of distributions that have the same general shape. l They are symmetric with scores more concentrated in the middle than in the tails. l bell shaped l two http: //www. ruf. rice. edu/~lane/hyperstat/A 12328. html the mean ( ) and the standard deviation ( ). l Inflection points ¡ The points at which the graph changes concavity (curvature) ¡ Inflection points are units on either side of the mean .

Notation Idealized Distribution Notation: VS. Mean: µ (greek letter mu) Standard deviation: σ (greek letter sigma) Normal Distributions: N(µ, σ) Sample Distribution Notation: Mean: x-bar Standard Deviation: s (lower case s)

The 68 -95 -99. 7 Rule The Empirical Rule l In the normal distribution with mean and standard deviation : l 68% of the observations fall within 1 of . l 95% of the observations fall within 2 of . l 99. 7% of the observations fall within 3 of .

The standard normal distribution l All normal distributions are the same if we measure in units of size about the mean as center. Changing the units is called standardizing. l z-score – tells how many standard deviations the original observation falls from the mean, and in which direction.

Standardizing and Z-scores l Standardizing normal distributions make them all the same. They are still normal. l It produces a new variable that has the standard normal distribution of N(0, 1). l When a score is expressed in standard deviation units, it is referred to as a Z-score ¡EX: A score that is one standard deviation above the mean has a Z-score of 1. ¡A score that is one standard deviation below the mean has a Z-score of -1. ¡A score that is at the mean would have a Z-score of 0.

Why do we standardize? l This makes it possible to compare to distributions easily. l Lets look at # 19 to better understand! l Remember:

Normal Distribution Calculations l Since all normal distributions are normal when we standardize, we can find the areas under any normal curve from a single table. l Table A (inside the front cover of text) gives areas under the curve for standard normal distribution. l Lets do #21 on p. 103 to better understand how to use the table!

Assignment l Read through the end of the chapter. l Do #20, 22 -26, 29