Buhrman, J.M.

Kaas, R.

English

CWI's Institutional Repository

stichting mathematisch centrum AFDELING MATHEMATISCHE STATISTIEK SW 59/78 JULI (DEPARTMENT OF MATHEMATICAL STATISTICS) R. KAAS & J.M. BUHRMAN A NOTE ON THE MEDIAN OF THE BINOMIAL DISTRIBUTION Preprint ~ MC 2e boerhaavestraat 49 amsterdam P.tu.nte.d a.:t :the. Ma:thema.;t,i,c..a.l Ce.ntll.e., 49, 2e. BoeJ1.haave1>.tna.a.:t, Amti:teAdam. The. Ma:thema.;t,i,c..a.l Ce.ntll.e., 6ou.nde.d :the. 11-:th 06 Fe.bnuaJl.y 1946, ,u.:, a. non-pno6U ,<..YU,;t,Ll:u.,tlon cwning a:t .the. pnomo:Uon o(J pu.11.e. ma:thema.;t,i,c.1.> a.nd --i..:t6 a.ppUc_a.;t,i,ono. I:t ,u.:, .6poYU,one.d by the. Ne.theMa.nd.6 GoveJl.nme.nt thnou..gh the. Ne.theMa.nd.6 Onga.Mza.;t,i,on 6on the. Adva.nc..eme.nt o 6 Pu.11.e. Rel> e.a.nc.h ( Z. W. 0) • AMS(MOS) subject classification scheme (1970): 62EOI d . f h b" . 1 d" "b . *) A note on the me ian o t e inomia istri ution by IL Kaas & J.M. BUHRMAN SUMMARY The location of the median of the binomial distribution is studied. An integer k is shown to be a median of the binomial-(n,p) distribution if it does not differ from np by more than min{p,1-p}. Neither the inequality "mean< median< mode 11 nor its reverse holds if median and mode differ (see [4 ]) . KEY WORDS & PHRASES: BinomiaZ distribution, median, mode. *) This report will be submitted for publication elsewhere. 1 • INTRODUCTION In two papers ([2], [3]) Neumann gives different proofs for the fact that the median of a binomial-(n,p) distribution equals np if np is an integer. We shall show that the median equals an integer k, if lk - np I s min{p,1-p}. First some definitions are given. A weak median of a random variable X will be a number x satisfying P(X ~ x) ~½and P(X s x) ~½·A strong median is a number x for which P(X ~ x - d) >½and P(X s x + d) > ½ for all d > 0. Clearly, a strong median is unique. If a random variable has no strong median, its set of weak medians is a closed interval. Let, fork integer, (1) b(k;n,p) = (n)pk (1-p) n-k k k (2) B(k;n,p) = I b(i;n,p) i=O UHU1ANN [5] gives the following inequalities, cited in [1]. (3) (4) Obviously (5) k B (k;n,--1) n-k+l B (k;n,--1) n+ 2. THE MEDIAN 1 > - > 2 1 > - > 2 k+l B (k;n,--1) n+ k B (k;n,--1) n-I for k < 2(n- l) 1 fork> zCn-1) if n 1.s odd THEOREM 1. If X has a binomial-(n,p) distribution and if the integer k satisfies (6) lk-npl s min{p,1-p} then k is a median of the distribution of X. The value k ~s a strong median 1 unless p = 2. 2 PROOF. First the k satisfying (6) both k ,;. lcn-1) 2 I • case p = 2 1.s . I h. h l.S 2 n, W l.C considered. If n is even, the only integer is obviously the strong median. If n is odd, I and k = -(n+l) 2 satisfy (6) and the set of weak medians is I l the interval in between. Now let p < 2 . We shall show P(X ~ k) > 2 and l P(X ~ k) > 2 , which implies that k is the strong median. Note that B(h;n,p) is decreasing in p. For p <½condition (6) becomes (7a) (n-l)p ~ k ~ (n+l)p or, equivalently, (7b) Now k n+l ~ <~ p - n-1 P(X ~ k) = I - B(k-l;n,p) ~ l - B(k-1 ·n ~) ' 'n+ l l > -2 by the right k ~ (n+ I )p < I hand inequality of (3), since k - l < 2 (n-l), because l 2(n+l). P(X ~ k) = B(k;n,p) k ~ B(k;n,--1) n-l > -2 l if k < 2(n-1) Moreover, for I have P(X ~ k) l . f either k = ½ n and n is p < 2 we > 2 1. l I or k = -(n-1) and n l.S odd. Since (7a) implies k ~ (n+l)p < 2 (n+l), 2 I completes the proof for the case p < -2· even, this Next p >½is considered. Let Y have a binomial-(n,q) distribution (with q = 1-p), and let k' = n-k. Then (6) becomes (8) \k' - nq I ~ q The preceding part of the proof shows that k' is the strong median of the distribution of Y, so k is the strong median of the distribution of X. REMARK. By taking the sum of the lengths of the p-intervals in which the 3 strong median is found by (7b), it can be shown that the theorem gives the median in about half the cases. 3. THE RELATIVE LOCATION OF THE MEAN, MEDIAN AND MODE Let, for fixed n, the numbers O =Po< p 1 < ... < pn < Pn+l = I be defined by I B (k-1 ;n,pk) = 2 k = 1,2, ... ,n This definition of pk makes, that the strong median mn is the following step-function of p. (9) m (p) = k n mn(pk) is undefined, the binomial-(n,pk) distribution has the interval [k-1,k] as a set of weak medians. By considering (7b) it can be seen that theorem (II) if (I 0) (I I) is equivalent to statement (IO) if n is odd, and to statement n is even. 2 2 P1 < n+I < n-1 < P2 < n+I < n-1 < P3 < ... n-1 I I n+3 ... < pn-1 < < - = Pn+l = - < < Pn+3 < ... 2(n+l) 2 2 2(n+l) 2 2 2 n-3 n-1 n-2 n ... < Pn-2 < < -- < Pn-1 < -- < < pn n-1 n+l n-1 n+I 2 2 P < < -- < P2 < -- < < P3 < ... I n+ I n-1 n+ I n-1 n n+2 ... < pn < 2 (n+ I) < 2(n+I) < p < . .. !!. + I 2 2 n-3 n-1 n-2 n ... < pn-2 < -- < < pn-1 < < -- < Pn n-1 n+l n-1 n+I 4 5 4 3 / np n _J xn(p) P 1 0 .129 ' mn(p) Pz 0.314 P3 0.500 P4 0 .686 P5 0. 371 0. 1 0.2 0. 3 0.4 0,5 0.6 0.7 0.8 0.9 0 P1 P2 P3 P4 P5 1 5 ~ n-1 n+l 3 n-1 ~ n+l=~ n+l 4 P1 0. 109 ~- np n = 6 p2 0. 264 _J P3 0.421 "n(p) P4 0. 5 79 2 mn(p) P5 0. 736 p6 0.891 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 P1 Pz P3 P4 P5 p6 2 3 5 4 6 n+l n=-r n=T n-1 4 n-T n+T n-T n+T p·• 01 n+1 Figure I Mean, median and mode as functions of p for n = 5 and for n = 6. Notice, that (12) (13) P + p = 1 i n+ 1-i for all i = 1,2, ... ,n The mode x of a binomial-(n,p) distribution equals [(n+l)p]. Therefore n the mode is given by the following step-function in p. (14) x (p) = k n k k+l if p E (-- --) (k integer) n+ I ' n+ I If p = _!_ both k and k-1 can be considered as the mode. The functions n+l' 5 m (p), x (p) and np are given in figure I for n = 5 and n = 6. The pictures n n suggest the following theorem. i THEOREM 2. Let p f p., p f --1 for whoZe i., i n+ I then for p < 2 (a) m (p) = x (p) or m (p) = x (p) + I n n n n (b) m (p) = x (p) + I => x (p) < np < m (p) n n n n and for p > I 2 (c) m (p) = x (p) n n or m (p) = x (p) - I n n (d) m (p) = x (p) - I =>m (p) < np < x (p). n n n n Propositions (a) and (c) can be summarized as I In words the mode is prone to be further from 2 n than the median. It is well-known, that "mean< median< mode" or "mean> median> mode" holds for a large class of continuous distributions. RUNNENBURG [4] gives suf-ficient conditions. In the binomial case however, if median and mode differ (which they do if e.g. pk< p < k/(n+l)), the mean lies in between, as stated in propositions (b) and (d). 6 PROOF OF THEOREM 2. Notice, that the first condition provides unique medians and modes. We shall now consider m (p), x (p) and np as functions of p on n 1 n . h h+l the interval (n+ 1 , n+ 1) with similarly. Let Os h s 2 n. Greater values of h can be dealt I - 2, then x (p) n by (9) I I By (IO) and (11) we have for Os h s 2 n 2 (15) s __E__ < _!!_ < s h+l Ph n+l n-1 - Ph+l n+l So, h <=> x (p) m (p) h p E: (n+l ,Ph+!) = = n n and h+l <=¢ x n(p) h Am (p) p E: (ph+l 'n+l) = = h + I n By (15) h+l h h+l p E: (ph+l 'n+l) ~ h < n -- < nph+l < np < n-- < h + I n-1 - n+l If h I then by ( I I ) = 2 n, About the difference between median and mean the following can be said. THEOREM 3. For any median m (weak or strong) lnp-ml s max{p,1-p}. 1 considered. The case p > PROOF. Only the case p < - is 21 . syrmnetry and the case p = - 18 t obvious. (16) If m (p) = n m-1 n-1 1 the strong m < - n 18 - 2 2 m+l < p < p < p < ---m m+l - n+l median, m- np < J - p and np -m < I - p • lm-npl s 1-p = max{p,1-p} then 1 then follows by 2 by (IO) or (11) 1 1 Notice, that the equality sign in (16) holds if and only if m = 2 n- 2. If mn(p) = m = ½ n is the strong median, then by (11) and by p < ½ (17) m-1 1 n-1 < Pm < P < 2 • m - np < 1 - p and 1 np < z n = m• 0 < m - np < 1 - p = max{p, 1-p} If [k-1,k] is the set of weak medians of the binomial-(n,p) distribution, then by (10) or (11) (18) k-1 n-1 < p = pk k - np < I - p k < ---n+l and p < k - np < 1 - p np - k < -p • So, for any m E [k-1,k:i we have p - I < m- np < I - p, which completes the proof. D 7 REFERENCES [1] JOHNSON, N.L. & S. KOTZ, (1969), Distributions in Statistics, Discrete Distributions, Chapter 3, p. 53. 8 [2] NEUMANN, P., (1966), Ueber den Median der Binomial-und Poissonverteilung, Wissenschaftliche Zeitschrift der Technischen Universitat Dresden 15, 233-226. [3] NEUMANN, P., (1970), Ueber den Median einiger diskreter Verteilungen und eine damit zusarrtmenhangende monotone Konvergenz, Wissen-schaftliche Zeitschrift der Technischen Universitat Dresden 19, 29-33. [4] RUNNENBURG, J. T., (1978), Mean, median, mode, Statistica Neerlandica 32, 73-79. [5] UHLMANN, w., (1963), RanggroSen ais Schatzfunktionen, Metrika 7, 23-40. 

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