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Topic: normal probability plots
Replies: 0

 KINGB@WCSUB.CTSTATEU.EDU Posts: 144 Registered: 12/6/04
normal probability plots
Posted: Mar 4, 1997 8:07 PM

Rex Boggs said:

> I must confess that, while I have some understanding about how to
> interpret a normal probability plot, I have absolutely no idea how to
> construct one for a particular data set. As a teacher who may has used
> them to justify using the t-test, this makes me very uncomfortable,
> especially when being asked how this plot was constructed and having to
> profess ignorance.
>
> Is it possible to explain how to do this in an email? Or is there a
> website that I can visit?

Here's a simple data set that my notes say came from the Minitab Reference
Manual, release 10.5. (I don't have the manual here.)
X: .1, .9, 1.1, 1.8, 2.3
Refer to these data points, in order, as x_i, with i = 1,2,3,4,5.

Now sketch a picture of the normal curve. If the X values are normally
distributed, you might expect them to occur at, say, the 10th, 30th, 50th,
70th, and 90th percentiles (i.e., at the z-values with cum probs of .1, .3,
.5, .7, and .9). Use a normal table or the TI-83 to look up the z-values
for these percentiles. I get about -1.28, -.52, 0, .52, and 1.28.
Construct, by hand, an ordinary "x-y plot" of the five points (X,z).
That's a normal probability plot.

Now compare the hand-drawn result with the following Minitab plot; they
should be pretty similar.

MTB > set c1
DATA> .1 .9 1.1 1.8 2.3
DATA> end
MTB > nscore c1 c2
MTB > name c1 'X' c2 'Nscore'
MTB > print c1-c2

< data display temporarily omitted; see below >

MTB > GStd.
MTB > Plot 'Nscore' 'X';
SUBC> Symbol 'x'.

Character Plot

Nscore -
- x
-
0.80+
-
- x
-
-
0.00+ x
-
-
- x
-
-0.80+
-
- x
-
--------+---------+---------+---------+---------+--------X
0.40 0.80 1.20 1.60 2.00

MTB > GPro.
MTB > nooutfile

One last matter... For a data set of size n = 5, as given above, think
about a formula that produces the percentiles in this example:

i 1 2 3 4 5
j .1 .3 .5 .7 .9

A little thought shows that j = (i-.5)/n. Various groups have chosen
slightly different formulas for j, yielding slightly different normal
scores for the plot. According to my notes, Minitab uses
(i - 3/8)/(n + 1/4) and Data Desk uses (i - 1/3)/(n + 1/3). Each of
these choices yields slightly different cum probs; for example,
here are the Nscores generated by Minitab (which I moved from the plot
above):

Data Display

Row X Nscore

1 0.1 -1.17877
2 0.9 -0.49532
3 1.1 0.00000
4 1.8 0.49532
5 2.3 1.17877

To summarize,

my example above: -1.28 -0.52 0.00 0.52 1.52
Minitab: -1.18 -0.50 0.00 0.50 1.18
Data Desk: -1.15 -0.49 0.00 0.49 1.15

But I think you will see that each of these three choices gives essentially
the same plot.

Hope I've got that right, and that it helps--

==============================================
Bruce King
Department of Mathematics and Computer Science
Western Connecticut State University
181 White Street
Danbury, CT 06810
(kingb@wcsu.ctstateu.edu)