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If I have an order statistic, I can rank them from smallest to largest like

$$X_{(1)}, X_{(2)},...,X_{(n)}$$

and there will be some median $X_{m}$.

But I can also have some distribution $f(x)$ and define an integral

$$\int_{0}^m f(x)dx = \frac{1}{2}$$ and, if I solve for $m$, this will be the median for the distribution.

My Question

What is the difference between these two types of medians?

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Your order statistic $X_m$ is the median of your data. This is your sample median and can change depending on what sample you pull. For example, if you gather a sample of 50 observations on one day and a sample of 50 observations on the next day, it is very likely that $X_m$ calculated from sample 1 will be different from $X_m$ calculated from sample 2.

Using your integral above to solve for $m$ will find you the population median - perhaps denoted $M$. This will not change regardless of what data you pull or the size of your sample. $M$ is a parameter, not a statistic.

You may use $X_m$ to estimate $M$. If you were interested in finding the median height of American adults, you may gather a sample of $n$ people and calculate $X_m$. Of course it will be logistically impossible to measure the height of every adult and calculate $M$ directly, so you use $X_m$ to estimate $M$.

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  • $\begingroup$ I thought a population statistic was like the maximum possible number of data points for sample statistic and that then, once you had all the data, it becomes a population statistic and I can find a population median. Is that wrong? if not, how is this kind of population statistic different from simply finding the median of a distribution? $\endgroup$ – Stan Shunpike Mar 14 '16 at 22:04
  • $\begingroup$ A statistic is a function of data. Generally data comes in the form of a sample and thus most often a statistic is based off of a sample, not a population. Thus, $X_m$ will be a sample statistic. If there is some case where you are able to gather all of your data, then $X_m$ will still be a statistic but will be based on the entire population and will equal your true median $M$. Since $f(x)$ is your probability density function, $f(x)$ defines your population and thus finding the median $M$ by integrating $f(x)$ from 0 to $M$ will get you the population median. $\endgroup$ – Matt Brems Mar 14 '16 at 22:11
  • $\begingroup$ In most cases, however, it is not possible to gather 100% of the population's data so we use sample statistics to estimate population parameters. If you can gather 100% of the population's data, then there is no need to conduct any statistical inference because you already know the true population values (parameters). In this case, you can technically have population statistics but I think this term is more confusing than using statistic to refer to functions of data from a sample and parameters to refer to true population values. $\endgroup$ – Matt Brems Mar 14 '16 at 22:17
  • $\begingroup$ Similar logic for sample mode vs population mode I assume then. $\endgroup$ – Stan Shunpike Mar 15 '16 at 2:26
  • $\begingroup$ That is correct. $\endgroup$ – Matt Brems Mar 15 '16 at 9:17

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