# Two different definitions of a median

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?

• 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. Mar 14, 2015 at 3:36
• 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". Mar 14, 2015 at 3:41

• 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.