Stat 4473 – Data Analysis
Robustness of t-methods
The t-methods for : and :d are exactly correct when the population from which we sample
is normally distributed. Practically speaking, there’s no way to be certain this is true. (In
fact, it’s almost certainly not true, since the normal distribution is an idealized population
model.) The usefulness of the t-methods is dependent on how strongly they are affected
by lack of normality. Luckily, the t-methods for : and :d are quite robust against non-
normality of the population except when outliers or strong skewness are present.
Defn: A confidence interval or hypothesis test is called robust if the confidence level or p-
value does not change much when the conditions for use of the procedure are violated.
We might say that before using the t-methods, we should check the “nearly normal
assumption” that the data appear close to normal — symmetric, single peak, no outliers.
We will use box plots, histograms and/or stem plots, and formal tests of normality to
check the nearly normal assumption. Modified boxplots are often useful in spotting
outliers.
Ways to not be normal
!More than one peak: The nearly normal assumption is violated if a histogram or stem
plot of the data shows two or more peaks. When you see this, look for the possibility
that your data come from two groups. If so, try to separate the data into its separate
groups, then analyze each group separately.
!Skewness: If the box plot, histogram, or stem plot indicates the data are strongly
skewed, you can’t use the t-methods unless your sample is large.
!Outliers: The nearly normal assumption is also violated if the data have outliers.
Outliers should be investigated. Sometimes, it’s obvious that a data value is wrong and
the justification for removing or correcting it is clear. When there’s no clear
justification for removing outliers, you might want to run the analysis both with and
without the outliers and note any differences in your conclusions. Any time data values
are set aside, you must report on them individually. An analysis of the non-outlying
points, along with a separate discussion of the outliers, is often very informative and
can reveal important aspects of the data. See more about handling outliers on the next
page.
P.S. As tempting as it is to get rid of annoying values, you can’t just throw away
outliers and not discuss them. It isn’t appropriate to lop off the highest or lowest
values just to improve your results.
