Basic Statistics

122 VARIANCES: ESTIMATION AND TESTS

We compute $/n1 = ,0535 and s:/n2 = .8333 and enter them into the formula for
the d.f.’s:

(.0535 + .8333)2

(.0535)2/(19 + 1) + (.8333)2/(12 + 1)

d.f. =

• 2 = 14.68 - 2 = 12.68

.7864

.00014 + .05342

or, rounding up, 13 d.f. In general, the d.f.’s for the approximate test are less than for
the usual t test. When the sample sizes are large, this makes little difference since
the d.f.’s will be large enough so that a reduction does not appreciably affect the size
of the tabled t value. For instance, looking at Table A.3 for the 95th percentile, if we
reduce the d.f. by 5 for a small sample, from say 10 to 5, the tabled t value increases
from 1.812 to 2.015. But if we reduce the d.f.’s for a larger sample by 5, from 55 to
50, the tabled t only increases from 1.673 to 1.676.
Statistical programs commonly include an option for using the approximate t test
(see SAS, SPSS, and Stata). The test statistic tends to be the same among the various
programs; however, slightly differing formulas are used for estimating the d.f.’s.
Some of these formulas are more conservative than others in that they result in fewer
degrees of freedom.
Other options for handling unequal variances include checking that the two dis-
tributions from the samples appear to be from normal populations and trying trans-
formations on the data before performing the t test. Sometimes the difference in
variances is associated with a lack of normality, so that the transformations discussed
in Section 6.5 should be considered.

9.4 OTHER TESTS

Sometimes the data are not normally distributed and finding a suitable transformation

is impossible, so that using a test other than the usual t test should be considered. For

example, a distribution all of whose observations are positive values and whose mode

is zero cannot be transformed to a normal distribution using power transformations.

Sometimes the data are not measured using an equal interval or ratio scale (see

Section 5.4.3) so that the use of means and standard deviations is questionable. Often,

psychological or medical data such as health status are measured using an ordinal

scale.

Distribution-free methods can be used with ordinal data that do not meet the

assumptions for the tests or confidence limits described in Chapters 7, 8, and in this

chapter. Such tests used are commonly called nonpurumetric tests. Nonparametric

tests are available that are suitable for single-sample, independent groups or paired

groups. For single samples or paired samples, either the sign test or the Wilcoxon

signed ranks test is often used. For two independent samples, the Wilcoxon-Mann-

Whitney test is commonly used. These tests are described in Chapter 13.