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## Defining “Median”

The median of a set of numbers is the middle value. In the set (1,2,3), the median is 2. But how about the set (1,2,3,4)? Most commonly, people define the median as 2.5. That is a good measure of central tendency, I guess, but it isn’t satisfactory because it mixes the ideas of mean and median. Also, then the median isn’t a member of the set.

Perhaps the best definition is that the median is X, where X is the lowest value such that 50% of the values are less than or equal to X.

Wikipedia says:

In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking the middle one (e.g., the median of {3, 5, 9} is 5). If there is an even number of observations, then there is no single middle value; the median is then usually defined to be the mean of the two middle values, which corresponds to interpreting the median as the fully trimmed mid-range. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data is contaminated, the median will not give an arbitrarily large result….In a sample of data, or a finite population, there may be no member of the sample whose value is identical to the median (in the case of an even sample size); if there is such a member, there may be more than one so that the median may not uniquely identify a sample member.

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