Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance?

I understand The variance is calculated as:

$$\text{Var} = \frac{1}{N}(x_i-\mu)^2$$

and the sample variance is computed as

$$\text{Var}_s = \frac{1}{N-1}(x_i-\mu)^2$$

In real world data sets would you use the sample variance most of the time? What if the population is also a sample? How would you know? Also does the mean change when you are looking at a population or a sample?

• For a population mean does not change. If the sample is also the population the mean is same for both. mean change for sample in general. for example if a population consists of 100 elements and three different people take samples of say, 10 elements each .in general no two people will select same elements. hence their means will be different. – SA-255525 Jun 6 '14 at 17:55
• The sample variance is computed as $$\text{Var}_s = \frac{1}{N-1}(x_i-\bar{x})^2$$ and it is the possible difference between $\mu$ and $\bar{x}$ which leads to using the changed denominator to make the sample variance an unbiased estimator of the population variance – Henry Oct 26 '15 at 15:43

What you wrote $Var_s$ is an unbiased estimator of the variance. Sample variance usually is defined in the same way as population variance. It is a best estimator of real variance in certain sense, however it is biased. $Var_s$ is not the best but unbiased. I would suggest you to read specila literature to understand the notions of "best estimator" and "biased, unbiased estimator".