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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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Standard deviation is very useful in metrology and when testing analog to digital converters (ADCs) input noise.

Given that a ADC input has noise, to measure its noise one can plot an histogram of values x ocurrences and then calculate the standard deviation of this data. The standard deviation value representes de rms noise of the ADC input.

Using only this statistic data you can check if an ADC is faulty.

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