# What's the difference between variance and standard deviation?

I was wondering what's the difference between variance and standard deviation?

If you calculate it its is clear that you get the standard deviation out of the variance...

But what does that mean in terms of the distribution you are looking at?

Why do you really need a standard deviation?

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You probably got the answer by now. Still this link has the simplest and best explanation. mathsisfun.com/data/standard-deviation.html –  user20726 Feb 11 at 13:09

The standard deviation is the square root of the variance.

The standard deviation is expressed in the same units as the mean is, whereas the variance is expressed in squared units, but for looking at a distribution, you can use either just so long as you are clear about what you are using. For example, a Normal distribution with mean = 10 and sd = 3 is exactly the same thing as a Normal distribution with mean = 10 and variance = 9.

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yeah thats the mathematical way to explain these two parameters, BUT whats the logical explenation? Why do I really ned two parameters to show the same thing(the deviation around the arithmetical mean)... –  Le Max Aug 26 '12 at 12:40
You don't really need both. If you report one, you don't need to report the other –  Peter Flom Aug 26 '12 at 12:47

You don't need both. They each have different purposes. The SD is usually more useful to describe the variability of the data while the variance is usually much more useful mathematically. For example, the sum of uncorrelated distributions (random variables) also has a variance that is the sum of the variances of those distributions. This wouldn't be true of the SD. On the other hand, the SD has the convenience of being expressed in units of the original variable.

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If John means independent random variables when he says unrelated distributions then he is right. However, to answer your question there are several things that can be said.

1. The mean and variance are the natural parameters for a normal distribution.

2. The Chebyshev inequality bounds the probability of a observed random variable being within k standard deviations of the mean.

3. The standard deviation is used to normalize statistics for statistical tests (e.g. the known standard deviation is used to normalize a sample mean for the z test that the mean differs from 0 or the sample standard deviation is used to normalize the sample mean when the true standard deviation is unknown, resulting in the t test).

4. For a normal distribution 68% percent of the distribution is within 1 standard deviation. 95.4% within 2 standard deviations and over 99% within 3 standard deviations.

5. The margin of error is express as a multiple of the standard deviation of the estimate.

6. Variance and bias are measures of uncertainty in a random quantity. The mean square error for an estimate equals the variance + the squared bias.

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You should probably not say "natural parameter", which are mean divided by variance, and 1 divided by variance: en.wikipedia.org/wiki/Natural_parameter –  Neil G Feb 2 at 6:31

In terms of the distribution they're equivalent (yet obviously not interchangeable), but beware that in terms of estimators they're not: the square root of an estimate of the variance is NOT an (unbiased) estimator of the standard deviation. Only for a moderately large number of samples (and depending on the estimators) the two approach each other. For small sample sizes you need to know the parametric form of the distribution to convert among the two, which can become slightly circular.

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while calculating the variance, we squared the deviations.. means, if the given data (observations) is in meters, it will become meter square... hope its not correct representation abt the deviations..so we square root again(S.D)that is nothing but S.D.

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