Population Standard Deviation So we know that the population standard deviation $\sigma$ is just the square root of the variance $\sigma ^2$. But what are the uses of the standard deviation? At this juncture it just seems are random variable to me, but what are the uses of the standard deviation?

The variance (and therefore the standard deviation) measures the variability of a random variable. It has a large number of uses, mostly connected to estimating the reliability of a result obtained from an experiment.

For example, think of precision scale for measuring weights. If we weigh something several times, typically a precision scale will give 9slightly) different answers.

Two scales A and B might both on average give the right answer. But if scale A has smaller variance than scale B, then a single weighing on scale A is more likely to give a result close to the truth than a single weighing on scale B.