Chapter 3

Section 2Normal Distributions

Normal Distributions as Density Curves

• Normal Curves: Symmetric, Single-Peaked and Bell-Shaped. They describe Normal Distributions.

Normal Distributions: Not so “normal” and play a large role in statistics. We will capitalize Normal to remind us how important they are.

3 Reasons Normal Distributions are Important

1. Good descriptions for some distributions of real data. Ex.

1. Scores on large tests (SAT and IQ tests)2. Repeated measurements of the same

quantity (diameter of a tennis ball)3. Characteristics of biological population

(lengths of animal pregnancy)

2. Good approximations to the results of chance outcomes. Example: Tossing a coin many times

3. Many statistical inference procedures are based on Normal distributions

3 Reasons Normal Distributions are Important

Normal curvesChanging the mean of a Normal curve does NOT change the shape of the curve.Changing the Standard Deviation DOES change the shape of a Normal curve.

Normal curves

• We can locate the S.D. by eye on the curve– Imagine you are skiing down a mountain that has

the shape of a Normal curve– At first, you descend at a steep angle as you go out

from the peak– Before you find yourself going straight down, the

slope begins to get flatter rather than steeper– The points at which this change of curvature takes

place are located one SD on either side of the mean

Normal Density Curves

• Normal Curves: – Symmetric– Bell-Shaped– Completely described by giving its mean and SD– Mean determines the center– SD determines the shape

Caution!

• Skewed Distributions are not normal• All Normal distributions are symmetric• Not all symmetric distributions are normalNormal distribution Not Normal distributions

68-95-99.7 Rule

• In any Normal distribution, approximately:– 68% of observations fall within one SD of the

mean– 95% of observations fall within two SD of the

mean– 99.7% Of observations fall within three SD of the

mean

68-95-99.7 Rule

Example 3.6: Performing on SAT

• Jennie scored 600 on the Critical Reading section of the SAT Reasoning test. How good a score is this? That depends on where a score of 600 lies in the distributionof all scores. The SAT exams are scaled so that scores on the Critical Reading section should roughly follow the Normal distribution with mean 500 and standard deviation 100.

• Write down what you know.

Example 3.6: Performing on SAT

• Jennie’s 600 is one standard deviation above the mean. The 68-95-99.7 rule now tells us where she stands.

• What percent of students who took the test did Jennie do better than?

Population Distribution

• For the formula of a z-score for, we use Greek letters to represent mean and standard deviation for a population distribution.

• Mean = µ (“mu”)• Standard Deviation = σ (“sigma”)

3.2 Homework

• Page 121-122• 3.22, 24, 26• This is the start of Homework 3.2