The reciprocals of these times are
[0.100, 0.091, 0.077, 0.071, 0.067, 0.022, 0.002]
Notice that the differences among the longer times are much reduced from what they
were in the original units. Moreover, the outliers will have considerably less effect on the
size of the standard deviation than they had before the transformation. Similar kinds of ef-
fects are found when we apply reciprocal transformations to reaction times, where long re-
action times probably indicate less about information-processing speeds than they do about
the fact that the subject was momentarily not paying attention or missed the response key
that she was supposed to hit.The Arcsine Transformation
In Chapter 5 we saw that for the binomial distribution, m5Npand. In this case,
then, because both the mean and the variance are dependent on p, the variance will be a di-
rect function of the mean. Suppose that for some experiment our dependent variable was
the proportion of items recalled correctly. Then each item can be thought of as a Bernoulli
trial with probability pof being correct (and probability 1 2 pof being incorrect), and the
whole set of items can be thought of as a series of Bernoulli trials. In other words, the re-
sults would have a binomial distribution where the variance is dependent on the mean. If
this is so, groups with different means would necessarily have different variances, and we
would have a problem. For this situation, the arcsine transformation is often helpful. The
usual form of this transformation is. In this case pis the proportion cor-
rect and Ywill be twice the angle whose sine equals the square root of p.^6 The arcsine
transformation can be obtained with most calculators (labeled sin^21 ) and is presented in
any handbook of statistical tables.
Both the square-root and arcsine transformations are suitable when the variance is pro-
portional to the mean. There is, however, a difference between them. The square-root trans-
formation compresses the upper tail of the distribution, whereas the arcsine transformation
stretches out both tails relative to the middle. Normally the arcsine is more helpful when
you are dealing with proportions.Trimmed Samples
Rather than transforming each of your raw scores to achieve homogeneity of variance or
normality, an alternative approach with heavy-tailed distributions(relatively flat distribu-
tions that have an unusual number of observations in the tails) is to use trimmed samples.
In Chapter 2 a trimmed samplewas defined as a sample from which a fixed percentage of
the extreme values in each tail has been removed. Thus, with 40 cases, a 5% trimmed sam-
ple will be the sample with two of the observations in each tail eliminated. When compar-
ing several groups, as in the analysis of variance, you would trim each sample by the same
amount. Although trimmed samples have been around in statistics for a very long time,
they have recently received a lot of attention because of their usefulness in dealing with
distributions with occasional outliers. You will probably see more of them in the future.
Closely related to trimmed samples are Winsorized samples,in which the trimmed values
are replaced by the most extreme value remaining in each tail. Thus, a 10% Winsorization of3 7 12 15 17 17 18 19 19 19
20 22 24 26 30 32 32 33 36 50Y=2 arcsin 2 ps^2 =NpqSection 11.9 Transformations 341(^6) The arcsine transformation is often referred to as an “angular” transformation because of this property. When p
is close to 0 or 1, we often take , where the plus is used when pis close to 0, and the minus
when pis close to 1.
2 arcsin 1 p 61 > 2 n
heavy-tailed
distributions
Winsorized
samples