Used when extending inference to the population mean $\mu$ when the population standard deviation $\sigma$ is unknown and substituted with the sample standard deviation $s$.
Key Properties
Heavier Tails: More probability in the extreme tails compared to standard normal, accounting for the extra uncertainty of using $s$.
Convergence: As sample size $n \to \infty$ ($df \to \infty$), the t-distribution becomes exactly the standard normal $z$-distribution.
Multiplier Penalty: Because of the heavier tails, $t^* > z^*$ for any given confidence level, making our intervals safely wider.
Tags
t-distributionnormalz-scoredegrees of freedomconfidence
Critical Values ($C\%$)
Confidence Level ($C$):
Sample Size ($n$):
Degrees of Freedom ($df$):
Standard Normal $z^*$
Student's $t^*$
Penalty Width Ratio ($t^*/z^*$):
Target Confidence Level
Sample Size ($n$)
Watch the purple t-curve. At $n=2$ ($df=1$), its tails drop very slowly, meaning extreme outliers are much more common. As $n \to 100$, it perfectly masks the cyan normal curve.