# The value range for std of movie ratings?

For a movie rating, we can have its distribution like this:

Each rating would have a percentage. Consequently, we can obtain the mean, std from this distribution.

The question is what is the value range for the mean - std for the movie ratings?

Obivously, its bounded, and the std would be 0 for mean rating at 10. For 5.5 rating, the upper limit of std would be around 4? (given the extreme case that 50% for 10 and 50% for 1). And it should be centered around 5.5.

My questions are that:

1. Is there any distribution for the mean - std for such distribution?

2. How can I obtain the value range, in Python? (I can probably obtain it through generating random values and then calculate the mean and std. but it's a little bit hard to do it from scratch.)

UPDATE:

My simulation result:

Let's let $X$ be a random variable for the movie rating. Let's represent the probability distribution over $X$ as a vector $p$ s.t. $p_x = P(X = x)$. Let's let $f(p)$ be mean - std as a function of $p$. This gives us $f(p) = \sum_{x=1}^{10}xp_x - \sqrt{\sum_{x=1}^{10}(x - \sum_{x=j}^{10}xp_x)^2}$. We're looking for the range of values that mean - std can take for different probability distributions. This is equivalent to finding the minimum and maximum value of $f(p)$ under the constraint that $p$ represents a valid probability distribution. More precisely, we want:
$\min_p f(p)$ s.t. $\sum_{x=1}^{10}p_x = 1$ and $\forall x$, $p_x \geq 0$, which is the minimum value that mean - std can take, and we want $\min_p -f(p)$ s.t. $\sum_{x=1}^{10}p_x = 1$ and $\forall x$, $p_x \geq 0$, which is the max value that mean - std can take.
These are both examples of constrained optimization problems and can be solved in python using minimize from scipy.optimize. However, this may not give the correct minimum if $f(p)$ is not convex and may not give the correct maximum if $-f(p)$ is not convex.
• I now thinks that I can use Dirichlet distribution to simulute the rating distribution, and then calculate the mean - std relationship. – cqcn1991 May 24 '17 at 4:16