1. 2.1 Density Curves and the Normal Distributions
Grab warm-up and start!

2. 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.

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

4. EXAMPLE

5. 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.

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

7. What is a normal distribution?
Normal distributions are a family of distributions that have the same general shape.
They are symmetric with scores more concentrated in the middle than in the tails.
bell shaped
two parameters: the mean () and the standard deviation ().
Inflection points
The points at which the graph changes concavity (curvature)
Inflection points are  units on either side of the mean .

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

9. The 68-95-99.7 Rule The Empirical Rule
In the normal distribution with mean  and standard deviation :
68% of the observations fall within 1 of .
95% of the observations fall within 2 of .
99.7% of the observations fall within 3 of .

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

11. Standardizing and Z-scores
Standardizing normal distributions make them all the same. They are still normal.
It produces a new variable that has the standard normal distribution of N(0,1).
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.

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

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

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