title: Pseudomedian
text: # Pseudomedian
  
  In statistics, the pseudomedian is a measure of centrality for data-sets and populations. It agrees with the median for symmetric data-sets or populations. In mathematical statistics, the pseudomedian is also a location parameter for probability distributions.
  
  ## Description
  
  The pseudomedian of a distribution $F$ is defined to be a median of the distribution of $(Z_{1}+Z_{2})/2$, where $Z_{1}$ and $Z_{2}$ are independent, each with the same distribution $F$.
  
  When $F$ is a symmetric distribution, the pseudomedian coincides with the median; otherwise this is not generally the case.
  
  The Hodges–Lehmann statistic, defined as the median of all of the midpoints of pairs of observations, is a consistent estimator of the pseudomedian.
  
  Like the set of medians, the pseudomedian is well defined for all probability distributions, even for the many distributions that lack modes or means.
  
  ## Pseudomedian filter in signal processing
  
  In signal processing there is another definition of pseudomedian filter for discrete signals.
  
  For a time series of length 2N + 1, the pseudomedian is defined as follows. Construct N + 1 sliding windows each of length N + 1. For each window, compute the minimum and maximum. Across all N + 1 windows, find the maximum minimum and the minimum maximum. The pseudomedian is the average of these two quantities.
  
  