T-Distribution vs Standard Normal Distribution

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.

The standard normal distribution is a normal curve with mean $\mu = 0$ (mu) and standard deviation $\sigma = 1$ (sigma). Any normal value converts to it through the z-score $z = \frac{x - \mu}{\sigma}$, which rescales the data onto this common curve. For example, a value 2 standard deviations above its mean maps to $z = 2$ on the standard normal. A z-table or calculator then gives the area, and therefore the probability, to the left of that z-score.