you calculate it by using a supercomputer that has over 100 petabytes of data it is a search engine it searches throughout the world looking for information it calculates it in the so called brain of the computer which is a mother board and hardware wall linked to the main computer than calculates sample data to answer a question that once never had an answer to it. So that means that it brings the closest correct information that it can find throughout the web and makes the no answered question answerable and thats how it calculates the variance of sample data.
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Because it is in same units as the original data. For example, if you have a sample of lengths, all in centimetres, the sample variance will be in units of centrimetres2 which might be more difficult to interpret but the sample standard deviation with be in units of centimetres, which would be relatively easy to intepret with reference to the data.
The variance is: 1.6709957376e+13
You cannot prove it because it is not true.The expected value of the sample variance is the population variance but that is not the same as the two measures being the same.
The sample mean is distributed with the same mean as the popualtion mean. If the popolation variance is s2 then the sample mean has a variance is s2/n. As n increases, the distribution of the sample mean gets closer to a Gaussian - ie Normal - distribution. This is the basis of the Central Limit Theorem which is important for hypothesis testing.
The mean, by itself, does not provide sufficient information to make any assessment of the sample variance.