Statistical Analysis for Education and Psychology Researchers

means are normally distributed. If the sample is sufficiently large then because of the

Central Limit Theorem, even if the distribution of the variables are not normal, their

sample means will be. If the samples are small and both variables are normally

distributed their means will also have an underlying normal distribution. (Verify the

normality assumption by doing a normal probability plot for the two variables.

Interpret the plot as described in Chapter 5, section 5.5, checking for normality.)

• The population variances should be equal, this is called the homogeneity of variance
  assumption. (Verify assumption of homogeneity by an approximate rule, variances are
  homogeneous if the ratio of the larger standard deviation (SD) to the smaller standard
  deviation is less than or equal to two. A folded F-test may be used, the ratio of two
  sample variances distributed with n 1 +n 2 −2 df but this is affected by non normality of
  data. This assumption is not necessary if the approximate t′-ratio (separate variance
  estimate) is used.)


• Samples are independent and selected at random. (This assumption is related to the
  research design.)

In a practical setting these assumptions are not straightforward to apply and as this is one
of the most common statistical tests some interpretation and guidance is required.

Guidelines for Practical Use of the t-test

The utility of the independent t-test is related to how far these assumptions can be relaxed
without invalidating inferences.
Normality assumption. What is important is that sample means are normally
distributed in the population. With large sample sizes (n>30 in each sample) this is not a
problem. With smaller samples the equality of sample sizes is important. The
independent t-test is robust against non normality even with small sample sizes (n<10),
provided the sample sizes are equal.
Moderately skewed distributions. If both samples are moderately skewed, have
similar shape and are approximately equal in size with the smaller sample being about 15
or more, then the t-test may be used with caution.
Severely skewed distributions. The t-test should only be considered with larger
samples, n 1 +n 2 >45 which are approximately equal in size and have similar variances. If
these assumptions are not met consider transforming the data, using an alternative
sampling distribution, use a nonparametric test or use a different analytic approach. A
general discussion about what to do when parametric assumptions are not met is
presented at the end of this chapter.
Unequal variances. If the approximate t′ (unequal variance estimate) is used then the
homogeneity assumption is not critical.
Independence. The sample observations must be independent, this is a critical
assumption.
Outlier observations. The independent t-test should not be used when there are
extreme outlier observations. These observations will greatly influence the means and
invalidate any inferences.

Inferences involving continuous data 295