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Appendix I
Normal Probability Plot
The standard normal random variable is a normal variable with
mean () = 0 and
standard deviation () = 1. Its value is represented by the
symbol, z.
=0 and =1
All normal random variables can be converted to the standard
normal variable, z.
At any z value of the standard normal probability distribution
the reference
table gives the cumulative probability (area under the curve) up
to that value.
Example, the probability ofz = -3.01 from the table is
0.000967671.
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Steps for Constructing Normal Probability Plot:
y Arrange the data in ascending order from i=1 to i=n,y Find the
value of P(z) for each each data value i=1.n, find zi such
that:
y Obtain the Percentage values for each P(Z>z i).y Plot these
values against the data to obtain the normal probability plot.
Example: We know use Excel to plot our Normal probability plot
of residuals in
our project.
e=y- Residual I P(Z>zi) %P(Z>zi)
2.275 -1.625 1 0.042397 4.239672
-0.825 -1.175 2 0.102811 10.28109
-0.425 -0.975 3 0.163917 16.39169
-0.725 -0.975 4 0.225023 22.50229
-0.675 -0.825 5 0.286129 28.61289
-0.975 -0.725 6 0.347235 34.7235
-0.975 -0.675 7 0.408341 40.8341
2.125 -0.425 8 0.469447 46.9447
-1.625 -0.275 9 0.530553 53.0553
-0.125 -0.125 10 0.591659 59.1659
0.475 0.475 11 0.652765 65.2765
0.975 0.725 12 0.713871 71.38711
1.225 0.975 13 0.774977 77.49771
0.725 1.225 14 0.836083 83.60831
-0.275 2.125 15 0.897189 89.71891
-1.175 2.275 16 0.957603 95.76033
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We can see in the table that we first arrange the residuals in
ascending order and
then follow each step to plot the normal probability graph.
What does normal probability graph show?
The sample values are plottedagainst what we wouldexpect to see
if it was
strictlyconsistent with the normaldistribution.
If the data is consistent with a sample from a normal
distribution the points
should lie close to a straight line. As a reference, a straight
line can be fit to thepoints. The further the points vary from this
line, the greater the indication of
departure from normality. If the sample has mean 0, standard
deviation 1 then a
line through 0 with slope 1 could be used.
What does it mean that residuals are normally distributed?
The error term in our first order model, we have assumed it to
be normally
distributed meaning the majority of error lie near the mean i.e.
0 and that very
few error terms are far away. So, we can say that the model we
assumed is fit and
has error term or residuals normally distributed.
The value of probability P(Z) obtained gives the measure of its
location with
reference to the mean i.e zero .Probability near 50% means value
is near mean
and high probability or very low probability shows value lies
toward end of the
probability density curve .
0
20
40
60
80
100
120
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Probability
Residuals
