Test results for the expected median of the counts in uniformly randomly filled bins; see http://math.stackexchange.com/questions/1749770.

Question1749770.java

import java.util.Arrays;
import java.util.Random;

public class Question1749770 {
	final static int ntrials = 10000000;
	final static int k = 1;
	final static int nbins = 2 * k + 1;
	
	final static Random random = new Random ();
	
	public static void main (String [] args) {
		int [] bins = new int [nbins];
		
		for (int n = nbins;;n += nbins) {
			double sum = 0;
			double sum2 = 0;
			for (int i = 0;i < ntrials;i++) {
				Arrays.fill (bins,0);
				for (int j = 0;j < n;j++)
					bins [random.nextInt (nbins)]++;
				Arrays.sort (bins);
				sum += bins [k];
				sum2 += bins [k] * bins [k];
			}
			double mean = sum / ntrials;
			double mean2 = sum2 / ntrials;
			System.out.println (n + " : " + mean + " ± " + Math.sqrt ((mean2 - mean * mean) / ntrials) + " / " + (n / (double) nbins - 1 / (2 * Math.PI * Math.sqrt (3))));
		}
	}
}