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From the little that I know ...

1. If the sample = population (census)  
$\sigma^2 = \displaystyle\frac{1}{N}\sum_{i = 1}^N (x_i - \mu)^2$ where $N$ is the size of the population and $\mu$ is the population mean.
The variance is $\sigma^2$ and the standard deviation then is $\sigma$ 

2. If the sample is smaller than the population (any study except a census)  
$\sigma^2 = \displaystyle\frac{1}{n - 1}\sum_{i = 1}^n(x_i - \bar x)^2$ where $n$ is the size of the sample and $\bar x$ is the sample mean. This $\sigma^2$ is called the *unbiased variance* and the standard deviation is $\sigma$.