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Chapter 4 The Normal Distribution

- Density Curves and the Normal Distribution

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).

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.

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.

Shapes of Density Curves

- We have mostly discussed right skewed, left

skewed, and roughly symmetric distributions that

look like this

Other Shapes (previously discussed)

- Uniform Distributions
- Bi-modal Distributions
- Multi-modal Distributions

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!

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.

What does it look like?

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.

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.

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.

The 68-95-99.7 Rule

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.

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.

The 68-95-99.7 Rule

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.?

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?