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 data

the model

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:

21( )2

1( , )

2

x

N e

The “Rule”•Normally distributed data has the following critical property:

The 68-95-99.7 RuleThe 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

•Converting to Z-scores allows you to use Table A (inside cover of book)

Xz

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…

Z

If the z-score for a dataPoint is 1 then this means That 84.13% of the samplesIn the population are less thanThe value of this data point

How do you interpreta z-score of -1.71?

Who’s the Greatest?Ty Cobb batted 0.420

In 1911N(0.266,0.0371)

Ted Williams batted0.406 in 1941

N(0.267,0.0326)

George Brett batted 0.390 in 1980

N(0.261,0.0317)

z = 4.15

z = 4.26

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 z-score 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