Weighted median of the set $\{1,2,3,4,5\}$ with weights $\{0.1, 0.1, 0.1, 0.1, 0.6\}$?

Question

The weighted median is described in Wikipedia in the following way: link.

What is the weighted median of the following set of numbers $\{1,2,3,4,5\}$ with each number having weights $\{0.1, 0.1, 0.1, 0.1, 0.6\}$, respectively?

Note that, the sum of weights is $\displaystyle \sum_{i=1}^{5}w_{i}=1$, but there is no element $x_{k}$, such that:

$$\sum_{i=1}^{k-1}w_{i}\leq 0.5\qquad\text{and}\qquad \sum_{i=k+1}^{5}w_{i}\leq 0.5.$$

Namely:

• For $x_{2}=2$, we have

$$\sum_{i=1}^{1}w_{i}=0.1\leq 0.5\qquad\text{but}\qquad \sum_{i=3}^{5}w_{i}=0.8.$$

• For $x_{3}=3$, we have

$$\sum_{i=1}^{2}w_{i}=0.2\leq 0.5\qquad\text{but}\qquad \sum_{i=4}^{5}w_{i}=0.7.$$

• For $x_{4}=4$, we have

$$\sum_{i=1}^{3}w_{i}=0.3\leq 0.5\qquad\text{but}\qquad \sum_{i=5}^{5}w_{i}=0.6.$$

Answer 1

I choose $x_5$. Note $$ \sum_{i=1}^4 w_i = 0.4,\qquad \sum_{i=6}^5 w_i = 0, $$ the last is an empty sum.

Answer 2

1	2	3	4	5
0.1	0.1	0.1	0.1	0.6
0.1	0.2	0.3	0.4	1

The last row is the cumulated weighs.

Then you can see that the median is $5$