What's the difference between variance and standard deviation? I was wondering what the difference between the variance and the standard deviation is. 
If you calculate the two values, it is clear that you get the standard deviation out of the variance, but what does that mean in terms of the distribution you are observing?
Furthermore, why do you really need a standard deviation?
 A: 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.
A: 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.
A: While calculating the variance, we squared the deviations.
It mean that if the given data (observations) is in meters, it will become meter square.
Hope it's not correct representation about the deviations. So, we square root again (SD) that is nothing but SD.
A: If John refers to independent random variables when he says "unrelated distributions," then his response is correct. However, to answer your question, there are several points that can be added: 


*

*The mean and variance are the two parameters that determine a normal distribution.

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

*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).

*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.

*The margin of error is expressed as a multiple of the standard deviation of the estimate.

*Variance and bias are measures of uncertainty in a random quantity.  The mean square error for an estimate equals the variance + the squared bias.
A: The variance of a data set measures the mathematical dispersion of the data relative to the mean. However, though this value is theoretically correct, it is difficult to apply in a real-world sense because the values used to calculate it were squared. The standard deviation, as the square root of the variance gives a value that is in the same units as the original values, which makes it much easier to work with and easier to interpret in conjunction with the concept of the normal curve.
A: 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.
A: In adition to Hassan's response, you need to be careful on interpreting standard deviation. Some people define it as the mean distance between every observation and its mean, but this is the definition of mean absolute deviation (MAD), thus wrong.
For a better understanding of both concepts, variance and SD, I highly recommend Taleb's video series on statistics (first lesson is about SD).
