What is the variance of x = (1, -1, 0)? Using the standard formula, I always arrive at 2/3, but shouldn't the answer be 1?
 A: You are thinking of range intuitively, and equating it with variance. But if you think of the mathematical formulation of the concept, you'll see that what leads you to some mistaken conclusion is that the zero point in your data is identical to the mean (zero), and consequently, one of the squared distances is zero, bringing down the calculation of the variance.
In other words, your data is not so spread on either side of the mean as to be $1$, because one of the data points lies precisely on the mean.
A: The correct answer is $2/3$.
$\mu = 0$ (average of x).
$Var(x) = \frac{(1-0)^2 + (0-0)^2 + (-1-0)^2}{3} = \frac{2}{3}$
A: You can think of the variance as the average squared deviation from the mean.  Your mean is clearly $0$.  With three data, you have three deviations (and thus three squared deviations):
\begin{align}
&\ (1-0)^2  &=& &1^2&  &=& &1  \\
&(-1-0)^2 &=& &-1^2& &=& &1  \\
&\ (0-0)^2  &=& &0^2&  &=& &0
\end{align}
The mean of $\{1, 1, 0\}$ is clearly $^2/_3$.  
I think your intuition is neglecting the fact that there is a $0$ deviation in the set.  
