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:

The “Rule” Normally distributed data has the following critical property: The 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 97 th 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 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)

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 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?

Who’s the Greatest? Ty Cobb batted In 1911 N(0.266,0.0371) Ted Williams batted in 1941 N(0.267,0.0326) George Brett batted 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 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