ASSESSING NORMALITY
DAVAR KHOSHNEVISAN
1. Histograms
Consider the linear model Y=Xβ +ε. The pressing question is, “is it
true that ε∼Nn(0,σ
2In)”?
To answer this, consider the “residuals,”
b
ε=Y−Xb
β.
If ε∼Nn(0,σ
2In) then one would like to think that the histogram of the
bεi’s should look like a normal pdf with mean 0 and variance σ2(why?). How
close is close? It helps to think more generally.
Consider a sample U1,...,U
n(e.g., Ui=bεi). We wish to know where the
Ui’s are coming from a normal distribution. Again, the first thing to do is
to plot the histogram. In Ryou type,
hist(u,nclass=n)
where udenotes the vector of the samples U1,...,U
nand ndenotes the
number of bins in the histogram.
For instance, consider the following exam data:
16.8 9.2 0.0 17.6 15.2 0.0 0.0 10.4 10.4 14.0 11.2 13.6 12.4
14.8 13.2 17.6 9.2 7.6 9.2 14.4 14.8 15.6 14.4 4.4 14.0 14.4 0.0
0.0 10.8 16.8 0.0 15.2 12.8 14.4 14.0 17.2 0.0 14.4 17.2 0.0 0.0
0.0 14.0 5.6 0.0 0.0 13.2 17.6 16.0 16.0 0.0 12.0 0.0 13.6 16.0
8.4 11.6 0.0 10.4 0.0 14.4 0.0 18.4 17.2 14.8 16.0 16.0 0.0 10.0
13.6 12.0 15.2
The command f1.dat,hist(nclass=15) produces Figure 1(a).1
Try this for different values of nclass to see what types of hitograms you
can obtain. You should always ask, “which one represents the truth the
best”? Is there a unique answer?
Now the data U1,...,U
nis probably not coming from a normal distribu-
tion if the histogram does not have the “right” shape. Ideally, it would be
symmetric, and the tails of the distribution taper off rapidly.
In Figure 1(a), there were many students who did not take the exam in
question. They received a ‘0’ but this grade should probably not contribute
to our knowledge of the distribution of all such grades. Figure 1(b) shows
Date: September 1, 2004.
1You can obtain this data freely from the website b elow:
http://www.math.utah.edu/˜davar/math6010/2004/notes/f1.dat.
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