Is to find out whether the two independent estimates of population variance differ significantly?

What is an Independent Samples T Test?

Use an independent samples t test when you want to compare the means of precisely two groups—no more and no less! Typically, you perform this test to determine whether two population means are different. This procedure is an inferential statistical hypothesis test, meaning it uses samples to draw conclusions about populations. The independent samples t test is also known as the two sample t test.

For an example of an independent t test, do students who learn using Method A have a different mean score than those who learn using Method B?

In this post, you’ll learn about the hypotheses, assumptions, and how to interpret the results for independent samples t tests.

Related post: Difference between Descriptive and Inferential Statistics

Independent Samples T Tests Hypotheses

Independent samples t tests have the following hypotheses:

• Null hypothesis: The means for the two populations are equal.
• Alternative hypothesis: The means for the two populations are not equal.

If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the two means is statistically significant. Your sample provides strong enough evidence to conclude that the two population means are not equal.

Notice how the hypotheses for the two sample t test relate to independent populations. They do not contain the same subjects.

Related post: How to Interpret P Values

Independent Samples T Test Assumptions

For reliable independent samples t test results, your data should satisfy the following assumptions:

You have a random sample

Drawing a random sample from the population you are studying helps ensure that your data represent the population. Representative samples are vital when you want to make inferences about the population. If your data do not represent the population, your analysis results will not be valid for that population.

You must draw a random sample from your population of interest. Each item or person in the population must have an equal probability of being selected.

Related post: Populations, Parameters, and Samples in Inferential Statistics

Your data must be continuous

T tests require continuous data. Continuous variables can take on any numeric value, and the scale can be meaningfully divided into smaller increments, including fractional and decimal values. There are an infinite number of possible values between any two values. Typically, you measure continuous variables on a scale. For example, when you measure temperature, weight, and height, you have continuous data.

Other hypothesis tests can handle different types of data. For more information, read Comparing Hypothesis Tests for Continuous, Binary, and Count Data.

Your sample data should follow a normal distribution or each group has more than 15 observations

All t-tests assume that your data follow the normal distribution. However, you can waive this assumption if your sample size is large enough thanks to the central limit theorem.

For the independent samples t test, when each group is larger than 15, your data can be skewed and the test results will still be valid. However, if your sample size is less than 15 per group, graph your data and determine whether the two distributions are skewed or has outliers. Either condition can cause the test results to be invalid. In this case, you might need to use a nonparametric test.

Fortunately, if you have more than 15 observations in each group for a two sample t test, you don’t have to worry about the normality assumption too much.

Related post: Central Limit Theorem and Skewed Distributions

The groups are independent

Independent samples contain different sets of items in each sample. Independent samples t tests compare two distinct samples. Hence, it’s a two sample t test. If you have the same people or items in both groups, you can use the paired t-test.

Related post: Independent and Dependent Samples

Groups can have equal or unequal variances but use the correct form of the test

Variance, and the closely related standard deviation, are measures of variability. Because the two sample t test uses two independent samples, each sample has its own variance. Consequently, the independent samples t test has two methods. One method assumes that the two groups have equal variances while the other does not assume they are equal. The form that does not assume equal variances is known as Welch’s t-test.

When the sample sizes for both groups are roughly equal, and you have a moderate sample size, t-tests are robust to unequal variances. If one group has twice the standard deviation of another group, it’s time to use Welch’s t-test! However, you don’t need to worry about smaller differences.

If you have unequal variances and unequal sample sizes, it’s vital to use the unequal variances version of the two sample t test!

Related post: Standard Deviations

Let’s run an example independent sample t test! Our hypothetical scenario is that we are comparing scores from two teaching methods. We drew two random samples of students. Students in one group learned using Method A while the other group used Method B. These samples contain entirely separate students.

Now, we want to determine whether the two means are different. Download the CSV file that contains the independent samples t test example data: t-TestExamples.

Here is what the data look like in the datasheet.

Let’s assume that the variances are equal and use the Assuming Equal Variances version.

Interpreting the Results

Here’s how to read and report the results for an independent samples t test.

The output indicates that the mean for Method A is 71.50 and for Method B it is 84.74. Looking in the Standard Deviation column, we can see that they are not exactly equal, but they are close enough to assume equal variances.

Because the p-value (0.000) for our independent samples t test is less than the standard significance level of 0.05, we can reject the null hypothesis. If the p-value is low, the null must go! Our sample data support the claim that the population means are different. Specifically, Method B’s mean is greater than Method A’s mean. If high scores are better, then Method B is significantly better than Method A.

The two sample t test estimates that the mean difference is -13.24. However, that estimate is based on 30 observations split between the two groups and it is unlikely to equal the population difference. The confidence interval indicates that the mean difference between these two methods for the entire population is likely between -19.89 and -6.59. Learn more about confidence intervals.

The negative values reflect the fact that Method A has a lower mean than Method B (i.e., Method A – Method B < 0). Because the confidence interval excludes zero (no difference), we can conclude that the population means are different.

To learn more about performing t-tests and how they work, read the following posts:

• Running T Tests in Excel
• T-Values and T-Distributions