Variance of discrete distribution exceeds variance of discrete uniform distribution

Question

I am not a mathematician, so I don't quite understand how comes that a variance of some discrete probability distribution could exceed the variance of the discrete uniform distribution. I thought that variance of any distribution will be capped by the variance of the flat distribution.

My case:
I have the following probability distribution for the discrete random variable on the [0,4] interval:
0.66$\hspace{0.5cm}$0.16$\hspace{0.5cm}$0.01$\hspace{0.5cm}$0.01$\hspace{0.5cm}$0.16

I compute the variance of this distribution using:
$$ \text{Var}(X) = \sum_{i=1}^n p_i\,\cdot\,(x_i - \mu)^2 $$ and I get 2.1275, while the variance for the discrete uniform distribution, according to this formula $$ \text{Var}(X) = \frac{n^2 - 1}{12}, \quad \text{where }n=b-a+1 $$ should be 2.0 on the same interval, i.e., for $b-a = 4$.

This bugs me a lot. Please tell me what I am missing. Thank you for your time.

Answer 1

Popoviciu's inequality states that the sharp upper bound on the variance $\sigma^2$ of any bounded probability distribution with support on $[a, b]$ is: $$ \sigma^2\leq \frac{1}{4}(b - a)^2 $$ In your case, $a= 0, b = 4$ so we have $\sigma^2\leq 4$. This is achieved precisely when half of the probability is concentrated at each of the two bounds.