In statistics, a weighted median of a sample is the 50% weighted percentile. It was first proposed by F. Y. Edgeworth in 1888. Like the median, it is useful as an estimator of central tendency, robust against outliers. It allows for non-uniform statistical weights related to, e.g., varying precision measurements in the sample.
Definition
General case For n {\displaystyle n} distinct ordered elements x 1 , x 2 , . . . , x n {\displaystyle x_{1},x_{2},...,x_{n}} with positive weights w 1 , w 2 , . . . , w n {\displaystyle w_{1},w_{2},...,w_{n}} such that ∑ i = 1 n w i = 1 {\displaystyle \sum _{i=1}^{n}w_{i}=1} , the weighted median is the element x k {\displaystyle x_{k}} satisfying
∑ i = 1 k − 1 w i ≤ 1 / 2 {\displaystyle \sum _{i=1}^{k-1}w_{i}\leq 1/2} and ∑ i = k + 1 n w i ≤ 1 / 2 {\displaystyle \sum _{i=k+1}^{n}w_{i}\leq 1/2}
Special case Consider a set of elements in which two of the elements satisfy the general case. This occurs when both element's respective weights border the midpoint of the set of weights without encapsulating it; Rather, each element defines a partition equal to 1 / 2 {\displaystyle 1/2} . These elements are referred to as the lower weighted median and upper weighted median. Their conditions are satisfied as follows: Lower Weighted Median
∑ i = 1 k − 1 w i < 1 / 2 {\displaystyle \sum _{i=1}^{k-1}w_{i}<1/2} and ∑ i = k + 1 n w i = 1 / 2 {\displaystyle \sum _{i=k+1}^{n}w_{i}=1/2}
Upper Weighted Median
∑ i = 1 k − 1 w i = 1 / 2 {\displaystyle \sum _{i=1}^{k-1}w_{i}=1/2} and ∑ i = k + 1 n w i < 1 / 2 {\displaystyle \sum _{i=k+1}^{n}w_{i}<1/2}
Ideally, a new element would be created using the mean of the upper and lower weighted medians and assigned a weight of zero. This method is similar to finding the median of an even set. The new element would be a true median since the sum of the weights to either side of this partition point would be equal. Depending on the application, it may not be possible or wise to create new data. In this case, the weighted median should be chosen based on which element keeps the partitions most equal. This will always be the weighted median with the lowest weight. In the event that the upper and lower weighted medians are equal, the lower weighted median is generally accepted as originally proposed by Edgeworth.
Properties The sum of weights in each of the two partitions should be as equal as possible. If the weights of all numbers in the set are equal, then the weighted median reduces down to the median.
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