If the assumptions for the t test hold,
the sample values should come from a
normal distribution.
Departures from normality can suggest the presence of
outliers
in the data, or of a nonnormal distribution
in one or more of the samples.
The normality test will give an indication of whether the
population
from which the sample was drawn
appears to be normally distributed, but will not indicate the cause(s)
of the nonnormality. The smaller the sample size, the less
likely the normality test will be able to detect
nonnormality.
Because the residuals from a one-sample t test are simply the
original observed sample values minus the sample mean, the
normality test P value for the residuals is identical
to that for the sample values.
The histogram
for the sample has a reference
normal distribution
curve for a normal distribution with the same mean and variance
as the sample. This provides a reference for detecting gross
nonnormality when the sample size is large.
Because the residuals from a one-sample t test are simply the
original observed sample values minus the sample mean, the
histogram for the residuals would be identical
to that for the sample values, except for the range of
the X axis.
Suspected
outliers
appear in a
boxplot
as individual points o or x outside
the box. If these appear on both sides of the box, they also suggest the
possibility of a
heavy-tailed
distribution. If they appear on only one side,
they also suggest the possibility of a
skewed
distribution. Skewness is also
suggested if the mean (+) does not lie on or near the central line of the
boxplot, or if the central line of the boxplot does not evenly divide the box.
Examples of these plots
will help illustrate the various situations.
Because the residuals from a one-sample t test are simply the
original observed sample values minus the sample mean, the
boxplot for the residuals would be identical
to that for the sample values, except for the range of
the Y axis.
For values sampled from a
normal distribution,
the
normal probability plot,
(normal Q-Q plot)
has the points all lying on or near the straight line drawn
through the middle half of the points. Scattered points
lying away from the line are suspected
outliers.
Examples of these plots
will help illustrate the various situations.
Because the residuals from a one-sample t test are simply the
original observed sample values minus the sample mean, the
normal probability plot for the residuals would be identical
to that for the sample values.
Because there is only one unique fitted value,
the sample mean, the
graph of residuals against fitted values will
consist of a vertical "stack" of residuals.
Except for the subtraction of a constant value (the mean)
from all the sample values, this graph is identical
in appearance to a distribution graph of the sample values.
Outliers
may appear as anomalous points in the graph (although an outlier
may not turn up in the residuals plot by virtue of
affecting the mean so that its fitted value lies near it).
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