1. 2.2 Normal Distributions

2. 2.2
Density Curves and Normal Distributions

3. Describing Location in a Distribution
Density Curves
In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy.
So far our strategy for exploring data is :
1. Graph the data to get an idea of the overall pattern
2. Calculate an appropriate numerical summary to describe the center and spread of the distribution.
Sometimes the overall pattern of a large number of observations is so regular, that we can describe it by a smooth curve, called a density curve.
Describing Location in a Distribution

4. Describing Location in a Distribution
Density Curve
Definition:
A density curve is a curve that
is always on or above the horizontal axis, and
has area exactly 1 underneath it.
A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval.
Describing Location in a Distribution
The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

5. Here’s the idea behind density curves:

6. NOTE: No real set of data is described by a density curve
NOTE: No real set of data is described by a density curve. It is simply an approximation that is easy to use and accurate enough for practical use.
Describing Density Curves
The median of the density curve is the “equal-areas” point.
The point at which half of the data is on one side and half is on the other
The mean of the density curve is the “balance” point.
This is your book’s definition
Density Curves

7. Section 2.2 Normal Distributions
Learning Objectives
After this section, you should be able to…
DESCRIBE and APPLY the Rule
DESCRIBE the standard Normal Distribution
PERFORM Normal distribution calculations
ASSESS Normality

8. Normal Distributions
One particularly important class of density curves are the Normal curves, which describe Normal distributions.
All Normal curves are symmetric, single-peaked, and bell-shaped
A Specific Normal curve is described by giving its mean µ and standard deviation σ.
Two Normal curves, showing the mean µ and standard deviation σ.

9. Normal Distributions Definition:
A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ.
The mean of a Normal distribution is the center of the symmetric Normal curve.
The standard deviation is the distance from the center to the change-of-curvature points on either side.
We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).
Normal distributions are good descriptions for some distributions of real data.
Normal distributions are good approximations of the results of many kinds of chance outcomes.
Many statistical inference procedures are based on Normal distributions.

10. The Rule
Although there are many Normal curves, they all have properties in common.
Definition: The Rule (“The Empirical Rule”)
In the Normal distribution with mean µ and standard deviation σ:
Approximately 68% of the observations fall within σ of µ.
Approximately 95% of the observations fall within 2σ of µ.
Approximately 99.7% of the observations fall within 3σ of µ.

11. The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55).
Sketch the Normal density curve for this distribution.
What percent of ITBS vocabulary scores are less than 3.74?
What percent of the scores are between 5.29 and 9.94?
Example, p. 113
Normal Distributions

12. An Example
On our first test, the mean raw score was approximately a 68, and the standard deviation was approximately 14, and scores were approximately normally distributed.
What percent of students scored between 54 and 82?
What percent of students scored between 40 and 96?
What percent of students scored between 68 and 100?
What percent of students scored between 54 and 100?

13. An Example
On our first test, the mean raw score was approximately a 68, and the standard deviation was approximately 14, and scores were approximately normally distributed.
What percent of students scored between 54 and 82?
68%
What percent of students scored between 40 and 96?
95% (it was actually 100%, since our distribution was small and wasn’t perfectly normal)
What percent of students scored between 68 and 100?
50%
What percent of students scored between 54 and 100?
84%

14. The Standard Normal Distribution
All Normal distributions are the same if we measure in units of size σ from the mean µ as center.
Definition:
The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable
has the standard Normal distribution, N(0,1).

15. The Standard Normal Table
Normal Distributions
Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table.
Definition: The Standard Normal Table
Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.
Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81.
We can use Table A:
P(z < 0.81) =
.7910 Z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212

16. Finding Areas Under the Standard Normal Curve
Example, p. 117
Find the proportion of observations from the standard Normal distribution that are between and 0.81.
Can you find the same proportion using a different approach?
1 - ( ) = 1 – =

17. Normal Distribution Calculations
Normal Distributions
How to Solve Problems Involving Normal Distributions
Steps:
Draw a picture of the distribution and shade the area of interest under the curve
Standardize your variable to restate the problem in terms of z- scores (the standard Normal distribution)
Use Table A and to find the required area
Conclude: write your conclusion in the context of the problem

18. Normal Distribution Calculations
Normal Distributions
When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards?
Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13.
– = About 44% of Tiger’s drives travel between 305 and 325 yards.

19. Using Technology
There are two useful “technology corner” sections of your textbook that will help you use your calculator to do these types of calculations
Page 118 and Page 123
Let’s focus on the one on Page 123
You can use your calculator as a replacement for Table A
You will use the normalcdf command on your TI 83/84
Normalcdf(lower bound, upper bound, mean, st. dev)
NOTICE: We no longer need to convert everything to z-scores first (though we could)
Let’s confirm this using our chapter 1 test scores
Normalcdf(54,82,68,14)
Normalcdf(-1,1,0,1)

20. Let’s try it
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15
We know from the rule that 68% of people should be between 85 and 115. Let’s confirm this using our calculator
normalcdf(85,115,100,15) = or 68.27%
Let’s get more advanced. What percent of people have IQ’s between 97 and 135?

21. Let’s try it
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15
We know from the rule that 68% of people should be between 85 and 115. Let’s confirm this using our calculator
normalcdf(85,115,100,15) = or 68.27%
Let’s get more advanced. What percent of people have IQ’s between 97 and 135? 56.9%
What percent of people have IQ’s between 40 and 111?
What about between 0 and 111?

22. Let’s try it
IQ scores are normally distributed with a mean of 100 and a standard deviation of 15
Let’s get more advanced. What percent of people have IQ’s between 97 and 135? 56.9%
What percent of people have IQ’s between 40 and 111? 76.8%
What about between 0 and 111? %
Why are these two numbers the same?