Statistics and Probability Letters 83 (2013) 1254–1259 Contents lists available at SciVerse ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro A sharp estimate of the binomial mean absolute deviation with applications Daniel Berend a,b , Aryeh Kontorovich a,∗ a Department of Computer Science, Ben-Gurion University, Beer Sheva, 84105, Israel b Department of Mathematics, Ben-Gurion University, Beer Sheva, 84105, Israel article info Article history: Received 1 October 2012 Received in revised form 17 January 2013 Accepted 17 January 2013 Available online 23 January 2013 Keywords: Binomial Mean absolute deviation Density estimation Total variation abstract We give simple, sharp non-asymptotic bounds on the mean absolute deviation (MAD) of a Bin(n, p) random variable. Although MAD is known to behave asymptotically as the stan- dard deviation, the convergence is not uniform over the range of p and fails at the endpoints. Our estimates hold for all p ∈[0, 1] and illustrate a simple transition from the ‘‘linear’’ regime near the endpoints to the ‘‘square root’’ regime elsewhere. As an application, we provide asymptotically optimal tail estimates of the total variation distance between the empirical and the true distributions over countable sets. © 2013 Elsevier B.V. All rights reserved. 1. Introduction The mean absolute deviation (MAD) of a random variable Y is defined by E |Y − EY |. For Y ∼ Bin(n, p), a closed-form expression for MAD is known, apparently having been first discovered by De Moivre: D n (p) := 2(1 − p) n−⌊np⌋ p ⌊np⌋+1 (⌊np⌋+ 1)  n ⌊np⌋+ 1  , (1) where the floor function is defined by ⌊x⌋= max {n ∈ N : n ≤ x}. Diaconis and Zabell (1991) give a fascinating account of the history of (1) and provide generalizations to other distribution families. Since MAD is a measure of dispersion of the random variable, 1 it is natural to compare it with more familiar quantities, such as the standard deviation: S n (p) :=  E(Y − EY ) 2 =  np(1 − p). Blyth (1980) discusses this comparison in some detail, observing the obvious relation D n (p) ≤ S n (p) and also showing that lim n→∞ S n (p) D n (p) =  π/2, 0 < p < 1. (2) ∗ Corresponding author. E-mail addresses: berend@cs.bgu.ac.il (D. Berend), karyeh@cs.bgu.ac.il, lkontor@gmail.com (A. Kontorovich). 1 In some sense, MAD is more natural than standard deviation, but the latter is usually preferred for reasons of analytic tractability. 0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2013.01.023