Lakhasly

Let X_{1:n},X_{2:n},\ldots.,X_{n:n} be a random sample of size n from a continuous population having pdf f(x) and cdf F(x). The sample observation can be arranged in ascending order of magnitude such that X_{1:n}\le X_{2:n}\le\ldots..\le X_{n:n},where the numbers i=1,2,\ldots, indicate the rank of the observation in the sample.
Such an ordered set of new random variables constitutes the order statistics.
Where,
X_{1:n}\ = the 1^{st} order statistic
= the smallest observation

= min (X_{1:n},X_{2:n},....,X_{n:n})
{\ X}{n:n}\ = the n^{th} order statistic
= the largest observation
= max (X{1:n},X_{2:n},....,X_{n:n})
and
X_{r:n}= the r^{th} order statistic
= the r^{th} smallest value
Remark 1.1. X_{1:n},X_{2:n},....,X_{n:n} is a random sample. They are therefore independent random variables; but Xⅈ:nʹs are dependent because of the inequality relation among them.
Remark 1.2. A more explicit notation of the order statistics is X_{(1,n)},X_{(2,n)},...,X_{(n,n)}.
Remark 1.3. Since X_{r:n} is a random variable, any function of X_{r:n} is also a random variable.
1.1 Functions of Order Statistics
Range and Mid-Range:
A range is the distance between the smallest observation X_{1:n} and the largest observation X_{n:n} observations. It is a measure of the dispersion in the sample and hence reflect the dispersion in the population.
The statistic
R_n=X_{n:n}-X_{1:n} (1.1)

is known as the sample range of the random sample.
The statistics
M=\frac{1}{2}\left(X_{n:n}+X_{1:n}\right)\ (1.2)
is known as the mid-range of the random sample.
Median:
The median is a measure of location that might be considered an alternative to the sample mean.
If n=2m+1(i.e.\ n\ is\ odd), then the {(m+1)}^{th} observation, which is the middle value, is called sample median of the random sample, given as
\widetilde{X}=X_{m+1:n}=X_{\frac{n+1}{2}:n} (1.3)
For n=2m (i.e. n is even), there is not a single middle observation but rather two middle observations. Thus, the sample median becomes
\widetilde{X}=\frac{1}{2}\left(X_{m:n}+X_{m+1:n}\right)=\frac{1}{2}\left(X_{{\frac{n}{2}}:n}+X_{{\frac{n}{2}+1}:n}\right)\ \ \ \ \ (1.4)