The independent sample t test is a parametric test specifically used to compare the means of two independent groups and determine whether there is statistical evidence of significant differences in their associated population means. The variables involved when performing an independent sample t test are identified as dependent variable and independent/grouping variable.

The independent sample t test is alternatively referred to as:

- Independent t-test
- Independent Measures t-test
- Independent Two-sample t-test
- Student t-test
- Two-Sample t-test
- Uncorrelated Scores t-test
- Unpaired t-test
- Unrelated t-test

## When is Independent Sample T Test Used?

The Independent Samples t test is mainly used to test whether:

- There is a significant mean difference between two groups
- There is a significant mean difference between two independent interventions
- There is a significant mean difference between two scores (eg, whether there is a difference in performance between math and science exams).

Please note that the Independent Samples t Test is limited to comparing means for only two groups. It cannot be used to make comparisons among more than two groups. If you need to compare means across more than two groups, running an ANOVA is likely the appropriate method.

You can also check out our tutorials on how to perform a one-way ANOVA in SPSS and how to report one-way anova spss output in APA format.

## Assumptions of Independent Sample T Test

Your data must satisfy the following criteria for conducting the Independent Samples t Test:

- Dependent variable must be continuous, at the interval or ratio level.
- Independent variable must be categorical with exactly two categories.
- Cases should have values on both the dependent and independent variables.
- Samples/groups must be independent, meaning there should be no relationship between the subjects in each sample: a.
*Subjects in the first group cannot overlap with those in the second group*. b.*No subject in either group can influence subjects in the other group*. c.*Neither group can influence the other group. Violating this assumption will lead to inaccurate p-values.* - A random sample of data should be taken from the population.
- The dependent variable should have a normal distribution (approximately) within each group. Non-normal population distributions, especially those with thick-tailed or heavily skewed data, can substantially reduce the test’s power. However, in moderate or large samples, a violation of normality may still yield accurate p-values.
- Homogeneity of variances is necessary, meaning variances should be approximately equal across groups. a. When this assumption is violated and sample sizes for each group differ, the p-value is not reliable. b. The Independent Samples t Test output also includes an approximate t statistic (Welch t Test statistic) that does not assume equal population variances. This statistic can be used when equal variances among populations cannot be assumed, and it is also known as the Unequal Variance t Test or Separate Variances t Test.
- No outliers should be present in the data.

In cases where one or more assumptions for the Independent Samples t Test are not met, researchers may consider using the nonparametric Mann-Whitney U Test instead.

## Hypotheses in Independent Sample T Test

The null hypothesis (*H*_{0}) and alternative hypothesis (*H*_{1}) of the Independent Sample *t* Test can be expressed in two different but equivalent ways:

*H*_{0}: µ_{1} = µ_{2} (“the two population means are equal”)*H*_{1}: µ_{1} ≠ µ_{2} (“the two population means are not equal”)

OR

*H*_{0}: µ_{1} – µ_{2} = 0 (“the difference between the two population means is equal to 0”)*H*_{1}*: *µ_{1} – µ_{2} ≠ 0 (“the difference between the two population means is not 0”)

where µ_{1} and µ_{2} are the population means for group 1 and group 2, respectively.

## Understanding the Levene’s Test for Equality of Variances in Independent t test

In the context of the** Independent Sample t Test**, it is essential to assume homogeneity of variance, meaning both groups should have the same variance. To conveniently assess this assumption, SPSS includes Levene’s Test, which evaluates the equality of variances, whenever you run an independent sample t test.

The hypotheses for Levene’s test are:

- H0: σ
_{1}^{2}– σ_{2}^{2}= 0 (“the population variances of group 1 and 2 are equal”) - H1: σ
_{1}^{2}– σ_{2}^{2}≠ 0 (“the population variances of group 1 and 2 are not equal”)

Rejecting the null hypothesis of Levene’s Test implies that the variances of the two groups are not equal, indicating a violation of the homogeneity of variances assumption.

In the Independent Sample Test table, you will find two rows of output: “Equal variances assumed” and “Equal variances not assumed.” Depending on the outcome of Levene’s test (i.e., large p-value for equal variances, or small p-value for unequal variances), you will choose the appropriate row of output to interpret the results of the actual Independent Sample t Test (under the heading “t-test for Equality of Means”).

When equal variances are assumed (large p-value), the calculation of the independent sample t test statistic uses pooled variances. On the other hand, when equal variances cannot be assumed (small p-value), the calculation utilizes un-pooled variances along with a correction to the degrees of freedom. This distinction in the calculation accounts for the potential differences in variances between the two groups and ensures the accuracy of the test results.

## Test Statistic in Independent Sample T Test

The Independent Samples t Test employs a test statistic denoted as “t,” which has two forms, depending on whether equal variances are assumed or not. SPSS generates both forms to cater to different scenarios, and both are described below. It is important to note that the null and alternative hypotheses remain identical for both forms of the test statistic.

### EQUAL VARIANCES ASSUMED (Homogeneity of Variance Assumed)

When we assume that the two independent samples are drawn from populations with equal variances (homogeneity of variance assumed), the test statistic “t” is calculated as follows:

where:

**x̄ _{1}** is the mean of the first sample

**x̄ _{2}** is the mean of the second sample

**n _{1}** is the sample size (number of observations) of the

first sample

**n _{2}** is the sample size (number of observations) of the

second sample

**s _{p}** is the pooled standard deviation calculated as

Where

**s _{1}^{2}** is the sample variance of the first sample

**s _{2}^{2}** is the sample variance of the second sample

The calculated “t” value is then compared to the critical t value from the t-distribution table with:

Degrees of freedom (df) = n1 + n2 – 2 and the chosen confidence level.

That is, critical value is obtained from t tables using the following formula:

t_{α/2}, n_{1}+n_{2}-2, For a two-tailed t test

t_{α}, n_{1}+n_{2}-2, For a one-tailed t test

If the calculated “t” value is greater than the critical t value, we reject the null hypothesis.

Note: This form of the independent samples t test statistic assumes equal variances. The assumption allows us to “pool” the sample variances (sp). However, if this assumption is violated, the pooled variance estimate may not be accurate, which can affect the precision of our test statistic (and consequently, the p-value). In this case, we conduct a different t test that assumes unequal variance. This type of t test is widely known as the **Welch’s t test**.

### EQUAL VARIANCES NOT ASSUMED (Homogeneity of variance assumption violated)

When we assume that the two independent samples are drawn from populations with unequal variances (homogeneity of variance assumption violated), the test statistic “t” is computed as follows:

where:

**x̄ _{1}** is the mean of the first sample

**x̄ _{2}** is the mean of the second sample

**n _{1}** is the sample size (number of observations) of the first sample

**n _{2}** is the sample size (number of observations) of the second sample

s1 is the standard deviation of the first sample

s2 is the standard deviation of the second sample

The calculated “t” value is then compared to the critical t value from the t-distribution table with degrees of freedom (df) calculated as:

and the chosen confidence level. If the calculated “t” value is greater than the critical t value, we reject the null hypothesis.

Note: This form of the independent samples t test statistic does not assume equal variances. As a result, both the denominator of the test statistic and the degrees of freedom for the critical t value are different from the equal variances form of the test statistic.