8-39. (Sampling Distribution of the Median) Given a random sample of size n, let us arrange the sample random variables X1, i = 1, . . . , n, in order of increasing numerical magnitude. Let the resulting set of n values be denoted as X(1),X(2), . . . ,X(n), where, for 1 ≤ i ≤ n, X(1) = minX(i),X(2) =

2nd smallest Xi, . . . ,X(n) = maxXi. Clearly X(1) ≤ X(2) ≤ · · · ≤ X(n). Here Xi is termed the ith order statistic and is a function of the sample elements Xi. In general, the order statistics of a random sample are simply the sample values placed in ascending order. Although the X(i), i = 1, . . . , n, are indeed random variables, they are not independent since, if Xi ≥ t, then obviously X(i+1) ≥ t.

Consider the order statistic, which is (approximately) the (np)th order statistic X(np) for a random sample of size n, 0< p < 1. Choose pn so that npn is an integer and n |pn − p| is finite. Then X(npn) is the (npn)th order statistic for a random sample of size n. Based upon these considerations, let X1, . . . ,Xn be a set of independent and identically distributed random variables with common probability density function f (x) and strictly monotone cumulative distribution function F(t), 0 < F(t) < 1.

Then for γp the pth population quantile (the quantile of order p, 0 < p < 1, is a value γp such that F(γP) = P(X ≤ γp) = p), the (npn)th order statistic X(npn) is asymptotically normal with E(X(npn)) = γp, V(X(npn)) = p(1 − p)
nf (γp)
2 , where f (·) is the population probability density function.
In particular, if p = 0.5, then γ0.5 is the population median. (In terms of order statistics, if X1, . . . ,Xn denotes a sample of size n, then the sample median is md = ⎧⎨
⎩
X((n+1)/2) if n is odd;
1 2 5X(n/2) + X((n/2)+1)6 if n is even.
Then, in the light of the preceding discussion, the sampling distribution of the sample median is asymptotically normal with mean equal to the population median γ0.5 and variance 1/4nf (γ0.5)2. Moreover, if the population probability density function is normal with mean μ and variance σ2, then the sampling distribution of the sample median is asymptotically normal with mean μ and variance 1/4nf (μ)2 = πσ2/2n.
We may express the standard error of the median as σγ0.5 = √
π/2 σ/
√
n.
(Since the standard error of the mean is only σ/
√
n, we see that, in terms of large samples taken from a normal population, the sample mean is less variable (and thus more reliable) than the sample median as a measure of central tendency.)
Armed with these considerations, if a random sample of size n = 100 is taken from a population that is N (40, 10), determine the probability that the sample median (md) will not differ from the population median (γ0.5 = μ = 40) by move than ±5 units.