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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?

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I believe standard deviation is used rather than variance to make units consistent (variance is in square units, by virtue of how it is calculated). –  Daniel Littlewood Nov 18 '12 at 17:07

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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.

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