Explore how substituting population standard deviation ($\sigma$) with sample standard deviation ($s$) changes the test statistic distribution from Normal to Student's t.
When we use the sample standard deviation ($s$) instead of the true population standard deviation ($\sigma$), we introduce extra uncertainty. Because $s$ varies from sample to sample, sometimes it underestimates $\sigma$. When a small $s$ is in the denominator of our test statistic, it inflates the resulting $t$-value, pushing it further into the tails. This extra variability is why the t-distribution has "fatter" tails than the normal distribution, especially at small sample sizes.
Degrees of Freedom (df) = 4
Expected Sample SD ≈ 0.94
Z-Statistic: $Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$ (Uses known $\sigma$)
t-Statistic: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$ (Uses sample $s$)
Distribution of raw sample means ($\bar{x}$). Narrows as sample size ($n$) increases due to a smaller standard error.
Always standard normal $N(0,1)$, regardless of sample size $n$. This assumes we know the true population variance.
Heavier tails than normal, depends on $df = n - 1$. Approaches normal as $n$ increases.
The histograms show the empirical distribution of calculated test statistics from repeated random sampling. Notice how the simulated t-statistics match the theoretical t-curve, not the normal curve, especially at small sample sizes.