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I have been learning discrete probability and know that variance and sd are both measures of spread. I also know that sd is just the variance squared. But I don't know the difference between them. Do they measure different types of spread?

Also, I know that the expected value is used in terms of a full census so the mean is called the expected value. In a sample, however, the mean is just called the mean and is only an approximation of $\mu$ (E(x)). Similarly, $S$ is an approximation of $\sigma$. But is there such thing as an approximation of $Var(x)$ in a sample?

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  • $\begingroup$ What's the difference between a number $x$ and the number $\sqrt{x}$? They contain essentially the same information. $\endgroup$ Oct 16, 2020 at 10:36
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    $\begingroup$ ahh ok. That's a nice insight. So why bother with sd and the square if x contains the same information for both? $\endgroup$
    – user71207
    Oct 16, 2020 at 11:15
  • $\begingroup$ Good point, after all we don't go squaring the mean $\endgroup$
    – develarist
    Oct 16, 2020 at 11:24
  • $\begingroup$ Yes, exactly. Any other discussions online are very vague about the answer to this question and I'm yet to find a suitable explanation $\endgroup$
    – user71207
    Oct 16, 2020 at 11:40

1 Answer 1

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The standard deviation will have the same units as the mean, while the variance has the units squared. This makes the standard deviation easier to think about intuitively and define rules that are easy to remember such as: For a normal distribution, 68% of the probability mass is within 1 standard deviation of the mean.

There's no such rule for variance: consider the case of standard normal as before. In this case, the variance is equal to the standard deviation (1), so we could use the same rule. But if the distribution has a variance of 100, then almost all the probability mass will be within 100 from the mean.

The sample variance is what you're looking for in the second part.

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