In your expression for the variance, you need to take a sum (or integral) across the population
$$\text{variance} = \frac{\sum_i(x_i-\text{mean})^2}{n}$$
If your data is a sample from the population then this expression will give you a biased estimate of the population variance. An unbiased estimate would be as follows (note the change in the denominator from your expression), often called the sample variance
$$\text{Sample variance} = \frac{\sum_i(x_i-\text{mean})^2}{n-1} $$
If on the other hand you were trying to estimate the variance of the sample mean, then you vould have a smaller number, closer to your expression. The square root of this is called the standard error of the mean and a reasonable estimate is
$$\text{Standard error} = \sqrt{\frac{\sum_i (x_i-\text{mean})^2}{n(n-1)}} $$