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I read from the book "Introduction to the Practice of Statistics", The International Seventh Edition, page 31 that "The $\textbf{median $M$}$ is the midpoint of a distribution. Half the observations are smaller than the median and the other half are larger than the median."

But I also read from the book "Statistical Inference", second edition, international student edition, page 78 problem 2.17 that "A $\textit{median}$ of a distribution $X$ is a value $m$ such that $P(X\leq m)\geq 1/2$ and $P(X\geq m)\geq 1/2$."

Are there differences between those definitions?

For example, is the median of a given distribution necessarily a unique number in the latter definition?

Have I understood correctly that if we have a distribution with only two values, $0$ and $1$ with equal probabilities, then the median of this distribution is $\frac{1}{2}$ by the first definition and an arbitrary real number $x\in [0,1]$ in the second definition?

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    $\begingroup$ Please give an exact quote, in context, for the first definition, along with a proper reference (including edition and page number). (in fact, please also give a proper reference for the other book, for completeness sake); do they explain what that actually means? Also be careful you don't conflate the sample median with the median of a distribution; make sure both definitions relate to the same thing. $\endgroup$ – Glen_b Mar 14 '15 at 3:36
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    $\begingroup$ For example, does the first reference go on to explain something like "such that half of the data have values that are lower than the median and half have values that are higher than the median"? In that case, notice that it's referring to a sample ('half the data'). The definitions of median in samples and in distributions of random variables (i.e. in populations) are related, but the distribution of a random variable doesn't have "data". $\endgroup$ – Glen_b Mar 14 '15 at 3:41
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The Wikipedia page has a good definition of "median". Regarding the first reference, "midpoint" should be defined more precisely. Your second reference gives more precise, formal definition, though leaves out some aspects. m is only included in the discrete distribution of the number of elements in the distribution is odd. "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". Thus, in the case you described with discrete distribution of only two values -- 0 and 1, with equal probability -- then the median is 1/2. However, the majority of situations where median is used as a descriptive statistic of central tendency, there are many more than two values in the support.

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Note that the first definition refers to the median, while the second refers to a median. This is not a minor distinction. According to some uses of the term "median" you can have more than one -- a median is any value that satisfies certain requirements (typically either dividing the set in half, or minimizing some distance/loss function based on absolute distances between points) and so in your example, any real number in [0,1] is indeed a median, while convention dictates that we refer to 1/2 as the median. The distinctions between the various common definitions (averaging "middle" values, finding any point that divides the set, or finding points that minimize the distance according to the L1 norm) won't actually make a difference in most circumstances, and so most commonly people will refer to the median and ignore the others. Think of it analogously to how people will often casually refer to 2 as the square root of 4, even though -2 is also a square root of 4. The less commonly encountered cases are simply ignored.

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    $\begingroup$ I am troubled by the fact that many datasets have no median at all according to the first definition. For instance, in the dataset $(1,2,3)$ there is no median because the only possible candidate is $2,$ but fewer than half exceed it and fewer than half are less than it. As far as square roots go, the choice of a nonnegative root for nonnegative real numbers is a convention and when taking roots of other numbers (or matrices etc) one has to be careful to note there are multiple solutions. $\endgroup$ – whuber May 1 at 21:07
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    $\begingroup$ Good point regarding the first definition. I suppose it should really be "No more than half the observations are smaller than the median and no more than half are larger than the median" (or equivalently but more verbosely: "At least half the observations are smaller than or the same size as the median, and at least half the observations are larger than or the same size as the median") $\endgroup$ – Bill Clark May 1 at 21:37
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    $\begingroup$ Also an excellent expansion on the square root example -- that's in line with the point I was trying to make. When you come across references to the median, it's typically the one given by the convention of averaging the middle/two-middle values. But in other contexts you may be looking for a more general sense of median, and in those cases you'll often see reference to a median. $\endgroup$ – Bill Clark May 1 at 21:42
  • $\begingroup$ If you would like to modify your post to incorporate some of your comments, please feel welcome to do so: many more people may read and benefit from them in that fashion. $\endgroup$ – whuber May 1 at 21:49

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