Can anyone explain the concept of median rank? http://reliawiki.org/index.php/Parameter_Estimation I found some things here but it does not make me understand it. 
For example, if I have data:
1 piece 10.91 MR
2 pieces 26.55 MR
3 pieces 42.18 MR
4 pieces 57.32 MR
5 pieces 73.75 MR
6 pieces 89.09 MR

How would you interpret it?
 A: The following three questions have similar answers
Question 1. Suppose you have any random variable $Y$ with a continuous cumulative distribution function $F(y)=\mathbb P(Y \le y)$, you sample $6$ observations, order these, and consider the distribution of the $n$th ordered observation counting from the lowest; let's call this $Y_{(n)}$.  The distribution of $Y_{(n)}$ will have a median, say $m_n$.  What is $F(m_n)$?
Question 2. Suppose you have a random variable $U$ with a uniform distribution on $[0,1]$, you sample $6$ observations, order these, and consider the distribution of the $n$th ordered observation counting from the lowest; let's call this $U_{(n)}$.  The distribution of $U_{(n)}$ will have a median, say $u_n$.  What is $u_n$?   
Question 3. Suppose you have a random variable $B$ with a Beta distribution with parameters $\alpha=n$ and $\beta=7-n$.  The distribution of $B$ will have a median, say $b_n$.  What is $b_n$?  
The answer to all three questions is the value $k$ which makes $$\int_{x=0}^k \frac{(n-1)!(6-n)!}{6!} x^{n-1}(1-x)^{6-n}\,dx = \frac12$$
In your question, the suggested interpretation is that of question 1, i.e. how far up the probability distribution is a sample ordered value likely to be centred on, in the sense of being equally likely to be above or below that point in the original distribution, described the median rank.  
In my view, question 2 gives a simpler way of conceptualising this, while question 3 gives a way of easily calculating the six answers for different $n \in \{1,2,3,4,5,6\}$, for example in R:
qbeta(1/2 , 1:6, 6:1)

giving
# [1] 0.1091013 0.2644500 0.4214072 0.5785928 0.7355500 0.8908987 

and you can see that these are close to the percentages in your question, and even closer to the values in the page you link to. 
