Variance of Binomial total deviation

Question

I have a Binomial random variable $X \sim B(n,p)$. Is there a closed form expression or an upper bound for the variance of the absolute deviation $|X-\mathbb{E}[X]|$ ?

Answer 1

Let $\mu_X=E(X)=np$. Call the mean deviation, $E|X-\mu|$, "$m_X$"

$\text{Var}(|X-\mu|) = E(|X-\mu|^2)- (E|X-\mu|)^2 = \text{Var}(X) - m_X^2$.

$\text{Var}(X)$ for a binomial$(n,p)$ is easily obtained to be $np(1-p)$.

$m_X$ for a binomial is given here [1] as:

$2{n\choose {\lfloor np\rfloor+1}}\,(1-p)^{n-\lfloor np\rfloor}\,p^{\lfloor np \rfloor +1}\,(\lfloor np\rfloor +1)$

According to Diaconis and Zabell [2], this result was given by de Moivre; they write it in (essentially) the form

$2v(1-p)\,{n\choose v}p^v(1-p)^{n-v}\,,$

where $v=\lfloor np+1\rfloor$, which is the same result, but in a form that's slightly easier to write and remember.

So it looks like the exact variance is directly available in closed form.

[1] Weisstein, Eric W. "Mean Deviation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MeanDeviation.html

[2] Diaconis, P. and Zabell, S. (1991),
"Closed Form Summation for Classical Distributions: Variations on Theme of De Moivre,"
Statistical Science Vol 6, No. 3, 284-302
http://statweb.stanford.edu/~cgates/PERSI/papers/demoivre.pdf