## Various Forms of the t-Test

A t-test is a procedure utilized to assess whether the means of two measurements statistically differ from one another.

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The independent samples t-test is employed when it is needed to choose samples from two different populations; the populations have a common variable but may differ concerning the values of these variables (which is to be tested). There is no overlap between the two groups concerning the membership (George & Mallery, 2016); for instance, smokers and non-smokers, Americans and Europeans, men and women, etc., might be compared.

The samples are measured (each member of a sample is only measured once), and their means are compared:

t = (M_{1} – M_{2}) / ((s_{1}^{2} / n_{1}) + (s_{2}^{2} / n_{2}))^{1/2},

where M_{1} and M_{2} are the means of the variable, s_{1} and s_{2} are the standard deviations, and the n_{1} and n_{2} are the sample size, of the first and second samples, respectively. The *df* for an independent t-test is (n_{1} + n_{2} – 2) (Warner, 2013).

In contrast, the paired samples t-test assesses the difference in the means of the same group in different conditions; each member of the group experiences “both conditions of the variables of interest” (George & Mallery, 2016, p. 149); for instance, the learners’ grade for the first exam and the second one; the body mass of people before and after stopping smoking; the levels of blood sugar in people with diabetes before and after taking a medication, etc. Thus, the members of the group are measured twice, and a pair of values are collected for each member, hence the name.

The paired samples t-test is calculated as follows:

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t = M / (s^{2} / n)^{1/2},

where M is the mean difference, s is the standard deviation, and n is the sample size. The *df* for a paired t-test is (n-1) (Warner, 2013).

Therefore, the independent samples t-test is to be used when it is needed to compare two different, non-overlapping populations, while the paired samples t-test is utilized when it is required to measure the differences in the members of the same group in different conditions.

## Two Versions of the Independent Samples t-Test

The SPSS software, when used to conduct an independent samples t-test, outputs two versions of the t-test: with equal variances assumed and equal variances not assumed (see Appendix 1).

The assumption of homogeneity of variance is an assumption according to which the two groups for which the t-test is being conducted have the same or similar variance, that is, their distribution is approximately the same (the variance is the squared standard deviation). This is important because the formula for the t-test includes the variances for both distributions it is assessing:

t = (M_{1} – M_{2}) / ((s_{1}^{2} / n_{1}) + (s_{2}^{2} / n_{2}))^{1/2},

therefore, if the variances differ much, this can adversely affect the value of the t-test for the given two samples (Warner, 2013). The violation of the assumption of homogeneity of variance may increase the likelihood of a type I error (Grissom, 2000).

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To preclude the type I errors, SPSS conducts the Levene’s test for equality of variances. This test is used to assess if there are statistically significant differences between the two variances (Gastwirth, Gel, & Miao, 2009). If a significant difference between variances has not been found, a researcher should use the more powerful version of the t-test, the one for which the equal variances are assumed.

To check it, one ought to look at the *p*-value (denoted by “Sig.”) of Levene’s test. If there is no significant difference (p>.05 or p>.01), the t-test with assumed equal variances should be used (George & Mallery, 2016, pp. 156-157); besides, the small F-value indicates the small difference in variances, so in this case the t-test with assumed equal variances should be utilized (Warner, 2013, p. 213).

However, if a significant difference has been found, the more conservative version of the t-test, where the equal variances are not assumed, should be used.

## References

George, D., & Mallery, P. (2016). *IBM SPSS Statistics 23 step by step: A simple guide and reference* (14th ed.). New York, NY: Routledge.

Warner, R. M. (2013). *Applied statistics: From bivariate through multivariate techniques* (2nd ed.). Thousand Oaks, CA: SAGE Publications.

Gastwirth, J. L., Gel, Y. R., & Miao, W. (2009). *The impact of Levene’s test of equality of variances on statistical theory and practice*. *Statistical Science, 24*(3), 343-360. Web.

George, D., & Mallery, P. (2016). *IBM SPSS Statistics 23 step by step: A simple guide and reference* (14th ed.). New York, NY: Routledge.

Grissom, R. J. (2000). Heterogeneity of variance in clinical data. *Journal of Consulting and Clinical Psychology, 68*(1), 155-165.

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## Appendix 1

The SPSS output for the independent samples t-test for the *total* variable between male and female samples; file “grades.sav.”