Normal Distribution vs T-Distribution

The normal distribution is a bell-shaped curve centered at its mean $\mu$ (mu) and spread out by its standard deviation $\sigma$ (sigma). Many natural measurements, such as heights or measurement errors, are approximately normal. For example, adult heights cluster near an average, with fewer people far above or below it. The empirical rule says about 68, 95, and 99.7 percent of the data fall within 1, 2, and 3 standard deviations of the mean.

The $t$-distribution accounts for the extra uncertainty of estimating the population standard deviation with the sample standard deviation $s$. Its exact shape depends on the degrees of freedom: fewer degrees of freedom give fatter tails, and as they grow the curve approaches the standard normal. For example, a one-sample $t$ test with $n = 15$ uses the $t$ model with $15 - 1 = 14$ degrees of freedom. The test statistic is $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$, where $\bar{x}$ is the sample mean and $\mu_0$ the hypothesized mean.