Back to Inference: Hypothesis Testing and Beyond

T-Tests, Z-Tests, and When to Use Each

The most-used tests in practice. The choice of which is more about sample size and known variance than anything else. FIND_VIDEO: search 't-test z-test paired tutorial' — recommended channel: StatQuest. Aim for 10 min or under.

5 minutesVideo LessonPDF notes

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Key moments

T-Test Categories — T-tests are broadly categorized into paired and unpaired types.
Paired T-Test Use — A paired T-test is used for before-and-after measurements taken from the same test subject.
Unpaired T-Test Use — An unpaired T-test compares two separate, independent groups, such as Group A and Group B.
Variance Assumptions — Unpaired tests require deciding whether to assume equal variation or not assume equal variation between the groups.
Conservative Variance Choice — It is generally recommended to use the test that does not assume equal variation because it is slightly more conservative.
One vs Two-Tailed — The second major choice is between using a one-tailed or a two-tailed T-test.
Two-Tailed Test — A two-tailed test checks both directions, testing if Group A is significantly higher or smaller than Group B.
Conservative Tail Choice — Always use two-sided T-tests because they are more conservative and allow the data to speak for itself.
Final Recommendation — For unpaired data, use the unequal variance and two-tailed T-test to ensure rock-solid results.

PDF notes

Bottom line: You will be able to select the appropriate T-test (paired vs. unpaired, equal vs. unequal variance, one-tailed vs. two-tailed) based on your experimental design. You will understand why choosing more conservative test parameters leads to more robust and reliable statistical conclusions.

This lesson on T-Tests, Z-Tests, and When to Use Each is hands-on and example-driven. You will be able to select the appropriate T-test (paired vs. unpaired, equal vs. unequal variance, one-tailed vs. two-tailed) based on your experimental design. You will understand why choosing more conservative test parameters leads to more robust and reliable statistical conclusions.

What You'll Be Able To Do

Distinguish between paired and unpaired data structures in experimental design.
Select the appropriate T-test type based on whether data is paired or unpaired.
Explain the difference between assuming equal variance and not assuming equal variance.
Justify the recommendation to use T-tests that do not assume equal variance.
Contrast the purpose and application of one-tailed versus two-tailed T-tests.
Recommend the most conservative T-test parameters for robust data analysis.

Topics Covered in T-Tests, Z-Tests, and When to Use Each

T-Test Categories (0:00 - 0:15) — T-tests are broadly categorized into paired and unpaired types.
Paired T-Test Use (0:15 - 0:55) — A paired T-test is used for before-and-after measurements taken from the same test subject.
Unpaired T-Test Use (0:55 - 1:20) — An unpaired T-test compares two separate, independent groups, such as Group A and Group B.
Variance Assumptions (1:20 - 2:10) — Unpaired tests require deciding whether to assume equal variation or not assume equal variation between the groups.
Conservative Variance Choice (2:10 - 2:45) — It is generally recommended to use the test that does not assume equal variation because it is slightly more conservative.
One vs Two-Tailed (2:45 - 3:10) — The second major choice is between using a one-tailed or a two-tailed T-test.
Two-Tailed Test (3:10 - 3:55) — A two-tailed test checks both directions, testing if Group A is significantly higher or smaller than Group B.
Conservative Tail Choice (3:55 - 4:30) — Always use two-sided T-tests because they are more conservative and allow the data to speak for itself.
Final Recommendation (4:30 - 5:00) — For unpaired data, use the unequal variance and two-tailed T-test to ensure rock-solid results.

SQL Cheat Sheet

Paired T-test — Compares before and after measurements taken from the same subject
```
SELECT AVG(after_bp - before_bp) FROM blood_pressure_study;
```
Unpaired T-test — Compares measurements between two separate, independent groups (A and B)
```
SELECT AVG(height) FROM group_a;
```
Equal Variance Assumption — Assumes variation within Group A is the same as variation within Group B
```
SELECT VARIANCE(height) FROM group_a;
```
Unequal Variance T-test — Recommended test; does not assume variation is the same in both groups
```
SELECT VARIANCE(height) FROM group_b;
```
Two-Sided T-test — Tests if Group A is significantly higher OR significantly smaller than Group B
```
SELECT * FROM results WHERE p_value < 0.05;
```
One-Sided T-test — Tests for significance in only one predefined direction (e.g., A > B)
```
SELECT * FROM results WHERE p_value < 0.05;
```

Comparison Table

Test Type	Data Structure	Recommendation
Paired T-test	Before/after measurements on same subject.	Use when data is linked by individual.
Unpaired T-test	Two separate, independent groups (A and B).	Use when groups are distinct entities.
Two-Tailed Test	Tests both directions (A>B and A<B).	Recommended; more conservative and agnostic.
One-Tailed Test	Tests only one predefined direction (e.g., A>B).	Avoid; less conservative, requires pre-notion.

Common Pitfalls

Mistake: Using a paired T-test when comparing two distinct, separate groups. Avoid: Use an unpaired T-test if subjects in Group A are different from Group B.
Mistake: Assuming equal variance between two unpaired groups. Avoid: Select the T-test that does not assume equal variance for conservatism.
Mistake: Using a one-tailed test to prove a hypothesis directionally. Avoid: Always use a two-sided T-test to let the data speak for itself.

FAQs

Why is the unequal variance test recommended? It is slightly more conservative, meaning the data must be stronger to pass the significance threshold. If your data passes this test, it is considered rock solid.
What does 'conservative' mean in statistical testing? A conservative test requires stronger evidence (a larger effect or smaller variance) to reject the null hypothesis, leading to more reliable results.
When would I use a one-tailed test? Rarely, as it requires a strong, pre-existing theoretical reason to predict the direction of the effect, which is often discouraged in academic journals.
Does the source material cover Z-tests? No, this lesson focuses exclusively on the selection criteria for various types of T-tests.

Common pitfalls

Mistake: Using a paired T-test when comparing two distinct, separate groups. Avoid: Use an unpaired T-test if subjects in Group A are different from Group B.
Mistake: Assuming equal variance between two unpaired groups. Avoid: Select the T-test that does not assume equal variance for conservatism.
Mistake: Using a one-tailed test to prove a hypothesis directionally. Avoid: Always use a two-sided T-test to let the data speak for itself.

Hello, and welcome to a stat quickie on t-tests. I work in a large genetics lab and people often ask me questions about t-tests. The first question they ask me is what type of t-test they should use. There are two main categories for t-tests, paired and unpaired. Paired t-tests are useful when you have before and after measurements taken from the same test subject. For example, if you have a drug for, uh, blood pressure, uh, say like you take a group of people and you measure their blood pressure, and then you give them this drug, and then after they've taken the drug, you test their, you test their blood pressure again. So for each person in the study, each individual, you have a pair of measurements. You have a before measurement and then you have an after measurement. And so, when you have data like that that's paired, use a paired t-test. When you don't have paired data, for example, say like you have a, uh, one group of, of people, uh, and you measure their height, and we call that group A, and then we have another group of people and we measure their height, and we, when we call that group B, then we have unpaired data. And so that's when you use an unpaired t-test. And there are two subcategories of unpaired t-tests. Uh, one assumes that the variance within each group, so the variation around the height measurements in group A, is the exact same as the variation around the height measurements in group B. So, uh, they call that a, well, actually, I don't know what they call that. They call that a t-test that assumes equal variation or equal variance. There's another type of t-test that does not assume equal variation. So you could have a different measurement of variance in group A as opposed to group, uh, B. Uh, now, I often recommend that people select the test that does not assume that the variation is equal in both groups, simply because that test is slightly more conservative. And if your data can pass this more conservative t-test, then you know your data are rock solid. So I think that's my general recommendation for using t-tests. Uh, you go with, if, if you have unpaired data, go with the test that does not assume equal variance in both groups. Okay, now the second question people ask me is should they use a one-tailed or one-sided t-test or a two-tailed or two-sided t-test? Now, a two-sided t-test, uh, say for example, we'll go back to our height measurements. Uh, so we have group A and group B, and we measured the heights in both groups. A two-tailed t-test would test to see if group A is higher than group B, and it would also test to see if it's significantly smaller than group B. So it tests both sides. It tests two conditions. So you've got a tail up here and you've got a tail down here. Um, it's, it's agnostic to the data. It doesn't have some preconceived notion that group A should be taller or group, uh, A should be smaller. It says, "I'm going to test both sides and just see what group A is." Uh, a one-tailed t-test is a lot less conservative because it, uh, it requires you to say, "Oh, I know which direction A is going to be. A is has to be higher than B." And generally speaking, in academic journals, that's not a good way to go. Generally speaking, you want the data to speak for itself, and that's why I always recommend people use two-sided t-tests, because it's slightly more conservative, and it lets the data speak for itself. And again, the conservative thing means you've got rock-solid rock-solid data. So, in summary, if you've got paired data, meaning you've got a single, you've got subjects, and for each subject, you've got two measurements, you've got a before and an after measurement. Oh, well, then use a paired t-test. But if you don't have paired, uh, data, so you've got two groups, two separate groups of people, you've and you've measured everything, and there's, um, and there's not overlap between who's in which group. So if you've got unpaired data, use an unpaired t-test, and try not to assume equal variance, because that'll make the test more conservative, and use a two two-tailed t-test, because that makes the data speak for itself, and again, it makes it a little more conservative. So if your data can pass that test, you know it's rock solid. All right, that's all there is to it. Thanks so much for paying attention to this stat quickie on t-tests.

Frequently asked questions

Why is the unequal variance test recommended?

It is slightly more conservative, meaning the data must be stronger to pass the significance threshold. If your data passes this test, it is considered rock solid.

What does 'conservative' mean in statistical testing?

A conservative test requires stronger evidence (a larger effect or smaller variance) to reject the null hypothesis, leading to more reliable results.

When would I use a one-tailed test?

Rarely, as it requires a strong, pre-existing theoretical reason to predict the direction of the effect, which is often discouraged in academic journals.

Does the source material cover Z-tests?

No, this lesson focuses exclusively on the selection criteria for various types of T-tests.