t-Distribution — When σ is Unknown
Foundations of Statistics
The Real-World Workhorse for Means
The t-distribution accounts for the extra uncertainty when estimating σ with s, making it the standard for real-world mean comparisons. Its heavier tails provide more conservative inference than the normal distribution.
- Quality Control — Comparing process means when population variance is unknown
- Clinical Research — Testing treatment effects with small sample sizes
- Business Analytics — A/B testing with limited data to make faster decisions
When σ is unknown, the t-distribution is your trusted companion.
Core Concepts
The t-distribution arises when we estimate the population standard deviation with the sample standard deviation . It has heavier tails than the normal, reflecting additional uncertainty from estimating .
Interactive Visualization
Derivation: Why the t-Distribution Appears
Degrees of Freedom and Tail Behavior
Critical Values
Worked Example
A biochemist measures enzyme reaction rates (in μmol/min) for samples: , . Test vs at .
Step 1. Compute the t-statistic:
Step 2. With degrees of freedom, the critical values are .
Step 3. Since , we fail to reject . The observed difference is not statistically significant at the 5% level.
Step 4. For comparison, if we had used the normal approximation: with critical value . We would still fail to reject, but the normal approximation underestimates the tail probability. The exact p-value from is 0.133, while the normal approximation gives 0.113.