Variability vs Standard Deviation

Variability is the reason statistics exists, since identical values would leave nothing to summarize and nothing to infer. Inside a single data set you measure it with the standard deviation or the IQR, which is how two classes can both average 75 while one runs from 70 to 80 and the other from 45 to 100. Across repeated samples it appears as sampling variability, the sample-to-sample bounce in a statistic such as $\bar{x}$ (x-bar), and that bounce shrinks as the sample size grows. Standard error is the name for the size of that second kind of variability.

The standard deviation summarizes spread as the typical gap between a value and the mean. The sample standard deviation is $s = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}$, where $x_i$ are the values, $\bar{x}$ (x-bar) is the mean, and $n$ is the number of values. For example, a small $s$ means the data cluster tightly around the mean, while a large $s$ means they spread out widely. It is the square root of the variance, which returns the measure to the original units.