Transcript The Normal Distribution

The Normal Distribution
Using Models to Describe Data Distributions
•The Density Curve
•All density curves are based on mathematical models
(“equations”) that can be used to describe the
frequency for a given datum
•The area under all densities curves equals 1
the model
the data
A Very Useful Model
•The Normal Distribution is
a model that can be used to
describe data that is:
•Uni-modal
•Symmetric
•Approximates the “bell
curve”
•Can be described by the
equation:
1
N (  , ) 
e
 2
1 x 2
 (
)
2 
The “Rule”
•Normally distributed data has the following
critical property:
The 68-95-99.7 Rule
Example…
• You have just received the score on your
law-school admissions test (LSAT). You
got 163. The exam results are normally
distributed with N(155,2.6) for that year of
testing. In order to apply to a very
prestigious law school you must finish in
the 97th percentile or better. Can you apply
with this score?
Z-Scores and the Standard Normal Distribution
•All normal distributions share the same shape
•A simple linear transformation can convert any normal
distribution to the Standard Form
•This gives us the concept of the Z-Score
•The 68-95-99.7 rule applies here and can give us a deeper
insight into what a z-score means
z
X 

•Converting to Z-scores allows you to use Table A (inside
cover of book)
The Standard Normal Distribution
•Any normal distribution can be converted to the
SND via z-score to N(m,s)  N(0,1)
Using z-scores…
If the z-score for a data
Point is 1 then this means
That 84.13% of the samples
In the population are less than
The value of this data point
How do you
interpret
a z-score of
-1.71?
Z
Who’s the Greatest?
Ty Cobb batted 0.420
In 1911
N(0.266,0.0371)
Ted Williams batted
0.406 in 1941
N(0.267,0.0326)
z = 4.15
z = 4.26
George Brett batted
0.390 in 1980
N(0.261,0.0317)
z = 4.07
Normal Quantile Plots (digging deeper)
• If you want to use what we have just learned to
assess data then you must be sure that the data
fits a normal distribution.
• Visual inspection (stemplot or histogram is a good
start).
• The Normal Quantile Plot is even better…Plot data
against the probability value that you get for the zscore of the data. If the graph is a straightline the
data is normally distributed.
Warning! Your text “messed-up!” All of the normal quantile plots have a
deceptively labeled x-axis – instead of z-score it should be the probability
associated with the z-score.
Example:
•Here is the data from an amazing star WZ Sagittae
•Am I justified in thinking that the noise in this data is
normally distributed?
My comparison
sources – the “fuzz”
in the data is the
“noise” or error
which I attribute to
basic natural
processes of
measuring light
Look at the Data…
“Z-Score” Plot
In conclusion…
• Review the summary on pages 83-84.
Make sure you understand z-scores, what
they mean and how to use them.
• Sample problems to gauge your
understanding: 1.81, 1.93, 1.97