T Test vs Z Test: When to Use Each Test

Hypothesis testing with z and t statistics

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T Test

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Key Differences

What is the difference really between z and t tests? The main thing is that t tests are used when you don’t know the population variance! You use the Student’s t distribution instead of the standard normal distribution. This wikiHow compares the t test to the z test.

Things You Should Know

This article covers one-sample z and t tests, comparing their key differences.
Calculate the z statistic using the formula $z={\frac {M-mu}{sigma_{M}}}$
Calculate the t statistic using $t={\frac {M-mu}{s_{M}}}$
The main difference is that the t test is used when population variance is unknown.

Section 1 of 3:

Z Test

1
Use the z statistic to carry out hypothesis testing. The z statistic uses a sampling distribution. Then, it turns it into a standard normal distribution. To calculate the z statistic, use this formula:
- $z={\frac {M-mu}{sigma_{M}}}$
- $sigma_{M}={\frac {sigma}{\sqrt {n}}}$
- where
  - $M$ is the sample mean
  - $mu$ is the population mean
  - $sigma_{M}$ is the sample standard error
  - $sigma$ is the population standard deviation
  - $n$ is the sample size
2
Make a decision. If the calculated z statistic (also called z score) is greater than the critical z value, you reject the null hypothesis and have significant evidence supporting the alternative hypothesis.
- If you’re testing a proportion, check out our guide on performing hypothesis testing for a proportion.

Section 2 of 3:

T Test

1
Use the t statistic to carry out hypothesis testing. The t statistic uses a sampling distribution. Then, it turns it into the t distribution. To calculate the t statistic, use this formula:
- $t={\frac {M-mu}{s_{M}}}$
- $s_{M}={\frac {SD}{\sqrt {n}}}$
- where
  - $M$ is the sample mean
  - $mu$ is the population mean
  - $s_{M}$ is the estimated standard error
  - $SD$ is the sample standard deviation
  - $n$ is the sample size
2
Make a decision. If the calculated z statistic is greater than the critical z value, you reject the null hypothesis and have significant evidence supporting the alternative hypothesis.
- We cover two-sample t tests in this guide.

Section 3 of 3:

Key Differences

Take a look at these key differences between the z test and t test.
- We don’t know the population variance in t testing, while we do know it in z testing.
- The z test uses a normal distribution while the t test uses the Student's t distribution.
- Degrees of freedom are needed in t testing, not in z testing.
- The z statistic is calculated with the standard error. The t statistic uses the estimated standard error.