Z-Test vs T-Test

Maths: Statistics for machine learning

2 min read

Published Oct 22 2025, updated Oct 23 2025

Machine LearningMathsNumPyPandasPythonStatistics

Both Z-tests and T-tests are statistical hypothesis tests used to determine whether there’s a significant difference between sample data and a known or assumed population parameter.

In simple terms:

“Both tests check if your sample mean is far enough from what you expect —
the Z-test uses the population standard deviation, while the T-test uses the sample standard deviation.”

The Z-Test

When to Use:

The population standard deviation (σ) is known
The sample size is large (n ≥ 30)
The data are normally distributed (or approximately normal via CLT)

Example:

“Is the average test score of 50 students different from the known population mean of 75 (σ = 10)?”

Use Z-test because σ is known and sample is large.

The T-Test

When to Use:

The population standard deviation (σ) is unknown
You have to estimate σ using the sample standard deviation (s)
The sample size is small (n < 30)
Data are approximately normally distributed

The t-distribution has heavier tails than the normal distribution, accounting for extra uncertainty from estimating σ.

Distribution Used

Test	Distribution	Description
Z-test	Standard Normal (Z)	Bell-shaped, fixed shape
T-test	Student’s t	Similar to normal, but heavier tails; shape changes with sample size (degrees of freedom)

As n increases, the t-distribution approaches the normal distribution, meaning the T-test ≈ Z-test for large samples.

Typical Use Cases

Scenario	Use	Why
Population σ is known	Z-test	We can use exact population variability
Population σ is unknown	T-test	Must estimate variability from sample
Small sample (n < 30)	T-test	Accounts for higher uncertainty
Large sample (n ≥ 30)	Z-test (or T-test, both valid)	CLT ensures normal approximation
Comparing two means	Two-sample T-test	Usually σ unknown
Paired data (before/after)	Paired T-test	Works on differences
Comparing proportions	Z-test	Uses population or large sample proportions