linguistic:voice_change
Derivation of the asymptotic distribution

We take the sample size to be an odd number N = 2 n + 1 {\displaystyle N=2n+1} and assume our variable continuous; the formula for the case of discrete variables is given below in § Empirical local density. The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities F (v) {\displaystyle F(v)}, f (v) {\displaystyle f(v)} and 1 − F (v) {\displaystyle 1-F(v)}. For a continuous variable, the probability of multiple sample values being exactly equal to the median is 0, so one can calculate the density of at the point v {\displaystyle v} directly from the trinomial distribution: