# Chapter 4: The Normal Distribution - PowerPoint PPT Presentation

PPT – Chapter 4: The Normal Distribution PowerPoint presentation | free to download - id: 82de73-NTFkM

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

## Chapter 4: The Normal Distribution

Description:

### Title: Chapter 2: The Normal Distribution Author: Stoney Pryor Last modified by: Belton ISD Created Date: 8/25/2005 1:14:28 AM Document presentation format – PowerPoint PPT presentation

Number of Views:12
Avg rating:3.0/5.0
Slides: 19
Provided by: Ston64
Category:
Transcript and Presenter's Notes

Title: Chapter 4: The Normal Distribution

1
Chapter 4 The Normal Distribution
• Density Curves and the Normal Distribution

2
Density Curves
• A density curve is similar to a histogram, but
there are several important distinctions.
• 1. Obviously, a smooth curve is used to represent
data rather than bars. However, a density curve
describes the proportions of the observations
that fall in each range rather than the actual
number of observations.
• 2. The scale should be adjusted so that the total
area under the curve is exactly 1. This
represents the proportion 1 (or 100).

3
Density Curves
• 3. While a histogram represents actual data
(i.e., a sample set), a density curve represents
an idealized sample or population distribution.
(describes the proportion of the observations)
• 4. Always on or above the horizontal axis
• 5. We will still utilize mu m for mean and sigma
s for standard deviation.

4
Density Curves Mean Median
• Three points that have been previously made are
especially relevant to density curves.
• 1. The median is the "equal areas" point.
Likewise, the quartiles can be found by dividing
the area under the curve into 4 equal parts.
• 2. The mean of the data is the "balancing" point.
• 3. The mean and median are the same for a
symmetric density curve.

5
Shapes of Density Curves
• We have mostly discussed right skewed, left
skewed, and roughly symmetric distributions that
look like this

6
Other Shapes (previously discussed)
• Uniform Distributions
• Bi-modal Distributions
• Multi-modal Distributions

7
Other Distributions
• Many other distributions exist, and some do not
clearly fall under a certain label. Frequently
these are the most interesting, and we will
discuss them later.
• 1 RULE ALWAYS MAKE A PICTURE
• It is the only way to see what is really going on!

8
Normal Curves
• Curves that are symmetric, single-peaked, and
bell-shaped are often called normal curves and
describe normal distributions.
• All normal distributions have the same overall
shape. They may be "taller" or more spread out,
but the idea is the same.

9
What does it look like?
10
Normal Curves µ and s
• The "control factors" are the mean µ and the
standard deviation s.
• Changing only µ will move the curve along the
horizontal axis.
• The standard deviation s controls the spread of
the distribution. Remember that a large s implies
that the data is spread out.

11
Finding µ and s
• You can locate the mean µ by finding the middle
of the distribution. Because it is symmetric, the
mean is at the peak.
• The standard deviation s can be found by
locating the points where the graph changes
curvature (inflection points). These points are
located a distance s from the mean.

12
The 68-95-99.7 (Empirical)Rule
• In a NORMAL DISTRIBUTIONS with mean µ and
standard deviation s
• 68 of the observations are within s of the mean
µ.
• 95 of the observations are within 2 s of the
mean µ.
• 99.7 of the observations are within 3 s of the
mean µ.

13
The 68-95-99.7 Rule
14
Why Use the Normal Distribution???
• 1. They occur frequently in large data sets (all
SAT scores), repeated measurements of the same
quantity, and in biological populations (lengths
of roaches).
• 2. They are often good approximations to chance
outcomes (like coin flipping).
• 3. We can apply things we learn in studying
normal distributions to other distributions.

15
Heights of Young Women
• The distribution of heights of young women aged
18 to 24 is approximately normally distributed
with mean ? 64.5 inches and standard deviation
? 2.5 inches.

16
The 68-95-99.7 Rule
17
Use the previous chart...
• Where do the middle 95 of heights fall?
• What percent of the heights are above 69.5
inches?
• A height of 62 inches is what percentile?
• What percent of the heights are between 62 and 67
inches?
• What percent of heights are less than 57 in.?

18
Example
• Suppose, on average, it takes you 20 minutes to
drive to school, with a standard deviation of 2
minutes. Suppose a normal model is appropriate
for the distribution of drivers times.
• How often will you arrive at school in less than
20 minutes?
• How often will it take you more than 24 minutes?