# T-Test vs. Z-Test: What's the Difference?

Edited by Aimie Carlson || By Janet White || Published on February 20, 2024
T-Test assesses means of two groups with unknown variances, typically for small samples. Z-Test evaluates population means with known variance, usually for large samples.

## Key Differences

T-Test and Z-Test are both statistical methods used to test hypotheses. The T-Test is often used when the data sample is small and the population variance is unknown. In contrast, the Z-Test is applied for large sample sizes when the population variance is known.
The T-Test is particularly useful in situations where the sample size is less than 30. It is more adaptable to smaller samples where the Central Limit Theorem doesn't hold. The Z-Test, however, is suitable for larger samples, typically over 30, where the sample mean tends to normally distribute, making the Z-Test a good fit.
In T-Test, the test statistic follows a T-distribution under the null hypothesis. This is a consequence of the unknown population variance, which requires estimation from the sample. The Z-Test, on the other hand, uses a Z-distribution (normal distribution) for the test statistic, applicable when the population variance is known and the sample size is large.
The T-Test includes various types like the one-sample, two-sample, and paired T-Test, each suited for different experimental designs. The Z-Test, while generally simpler, is mainly used for comparing sample and population means, and for testing proportions in large samples.
For both T-Test and Z-Test, statistical significance is determined by comparing the p-value with a threshold (alpha level). However, due to the different distributions used, the specific critical values and interpretation of results may vary between the tests.

## Comparison Chart

### Sample Size

Typically used for small samples (<30).
Ideal for large samples (≥30).

### Variance

Used when population variance is unknown.
Applied when population variance is known.

### Distribution

Follows a T-distribution.
Follows a Z-distribution (normal distribution).

### Types

Includes one-sample, two-sample, paired.
Mainly for sample vs population mean, proportions.

### Usage Context

More common in academic and research settings.
Often used in quality control, large surveys.

## T-Test and Z-Test Definitions

#### T-Test

T-Test assesses differences in means between two groups.
Researchers used a T-Test to compare the effectiveness of two medications.

#### Z-Test

Z-Test compares the mean of a sample to a known population mean.
A Z-Test was used to determine if the factory's output was different from the industry standard.

#### T-Test

T-Test determines if a sample is significantly different from the population.
A T-Test showed that the sample's average age significantly differed from the national average.

#### Z-Test

Z-Test is utilized for testing the proportion in large samples.
To analyze voter preference, a Z-Test was applied to the survey data.

#### T-Test

T-Test compares the mean values of two related samples.
To evaluate the before-and-after effects of a diet, a paired T-Test was used.

#### Z-Test

Z-Test is ideal for hypothesis testing in large sample sizes.
A Z-Test confirmed that the new process significantly increased production efficiency.

#### T-Test

T-Test evaluates the impact of an intervention in experimental studies.
A T-Test was conducted to analyze the impact of a new teaching method on student performance.

#### Z-Test

Z-Test assesses whether two population means are different when variances are known.
The Z-Test revealed no significant difference in average heights between the two regions.

#### T-Test

T-Test is a parametric test based on mean and standard deviation.
The T-Test indicated significant differences in average incomes between the two cities.

#### Z-Test

Z-Test uses the normal distribution for statistical inference.
The Z-Test showed that the medication's success rate was above average.

#### T-Test

(statistics) Student's t-test

## FAQs

#### Is Z-Test applicable for small samples?

Z-Test is not ideal for small samples as it assumes a known population variance and a normal distribution which might not hold in small samples.

#### Can T-Test be used for large samples?

While T-Test is best for small samples, it can be used for larger samples, but Z-Test is more appropriate in such cases.

#### What is a T-Test?

T-Test is a statistical test used to compare the means of two groups, especially when the sample sizes are small and variances are unknown.

#### What types of T-Tests are there?

There are one-sample, two-sample, and paired T-Tests, each for different experimental setups.

#### How does sample size affect the choice between T-Test and Z-Test?

Sample size is crucial; use T-Test for smaller samples and Z-Test for larger ones.

#### Can Z-Test be used for two independent samples?

Yes, Z-Test can compare two independent population means, provided the sample sizes are large and variances are known.

#### What if population variance is unknown for a large sample?

If the variance is unknown, even in large samples, a T-Test might be more appropriate.

#### How do variances impact the choice of test?

T-Test is used when variances are unknown and estimated from the data, while Z-Test requires known variances.

#### When should I use a Z-Test?

Use a Z-Test for large sample sizes (30 or more) when you know the population variance.

#### Is T-Test only for normal distributions?

T-Test assumes a normal distribution, but it's robust to some deviations, especially in larger samples.

#### How are T-Test and Z-Test related to hypothesis testing?

Both are used in hypothesis testing to determine if there are significant differences between groups or between a sample and a population.

#### Why is Z-Test more common in quality control?

Z-Test is used in quality control due to large sample sizes and known population parameters.

#### Are there online tools to perform T-Test and Z-Test?

Yes, there are various online calculators and statistical software that can perform these tests.

#### What is the significance level in T-Test and Z-Test?

The significance level, typically set at 0.05, is the threshold for determining statistical significance in both tests.

#### What does the ‘T’ in T-Test stand for?

The 'T' in T-Test refers to the T-distribution, which the test statistic follows.

#### Does Z-Test require normality?

Z-Test assumes a normal distribution, which is generally met in large sample sizes due to the Central Limit Theorem.

#### Can T-Test and Z-Test results differ?

Yes, especially in small samples, due to the difference in distribution assumptions.

#### Can T-Test be used for paired data?

Yes, the paired T-Test is specifically designed for comparing two related samples.

#### Can I use T-Test for non-parametric data?

No, T-Test is a parametric test; for non-parametric data, consider other tests like the Mann-Whitney U test.

#### What is the main difference in the distribution between T-Test and Z-Test?

T-Test uses the T-distribution, which adjusts for smaller samples, while Z-Test uses the normal distribution, suitable for larger samples.