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https://en.wikipedia.org/wiki/Median https://en.wikipedia.org/wiki/Median has this definition. It's also the definition that I was taught at school. Oxford La

https://en.wikipedia.org/wiki/Median https://en.wikipedia.org/wiki/Median has this definition. It's also the definition that I was taught at school. Oxford Languages defines it as: "denoting or relating to a value or quantity lying at the midpoint of a frequency distribution of observed values or quantities, such that there is an equal probability of falling above or below it", which also supports the idea of taking the midpoint between the two middle values. Merriam Webster define it explicitly in the way OP does (arithmetic mean of the two middle values): https://www.merriam-webster.com/dictionary/median https://www.merriam-webster.com/dictionary/median I'm not sure why you think it's a 'completely useless definition'. It would perhaps be a logical argument that data sets of even length do not have a median, but I think this would be even more useless. To the layperson (which Mary Jones undoubtedly is), providing the 'median' age as an age half of the employees are older than/younger than, is what would be expected. Your first bullet point makes sense here. For example here: 18, 20, 25, 30, [35, 40], 45, 50, 55, 60 Most people would expect the median to be calculated as 37.5. However, as you mention, any real number higher than 35 and lower than 40 would equally fulfill the definition of 'equal probability of falling above or below' assuming this is the full population. Would you then argue that there are infinitely many medians in this data set, or none? There are other issues with calculating 37.5, such as the fact it implies a higher degree of precision than it ought to. But ultimately, as a practical and usable insight into data, especially data with extreme outliers or high levels of skew, I strongly disagree that this definition is completely useless.

qsort 4y ago
> Would you then argue that there are infinitely many medians in this data set By the definition I usually go by (the one with the inequalities), that dataset has infinitely many medians. There is no reason to say the midpoint is privileged. This is a better definition because it also works on sets where an order is defined but the arithmetic mean is not. If asked to report a single value with no qualifications, I would report either extreme of the interval.
- d1sxeyes 4y ago
  > There is no reason to say the midpoint is privileged. This is a matter of opinion. I agree it's not 'privileged', but it sacrifices a minimal amount of accuracy for a usable answer. > If asked to report a single value with no qualifications, I would report either extreme of the interval. That's a very interesting answer - so for my contrived example of 18, 20, 25, 30, [35, 40], 45, 50, 55, 60, you would report 35 or 40? How about if I change the dataset a bit (so it's even more contrived): 18, 18, 18, 18, [18, 60], 60, 60, 60, 60 - would you still report either 18 or 60 (again, asking without qualifications)? I think most people would agree that the median calculated as the arithmetic mean of the two middle values (39) is misleading, but that it is less misleading than ignoring half of the data set. How would you respond to that?
  - qsort 4y ago
    > you would report 35 or 40? Generally I report the lower end because it more easily generalizes to different quantiles. > How about if I change the dataset a bit would you still report either 18 or 60 (again, asking without qualifications)? Yes. I don't see how answering 39 is any better in that case. > How would you respond to that? I would respond that you should look at the distribution. Reporting the median is particularly misleading in the case of bimodal distributions. I would try to explain that summary statistics are to be taken with caution, possibly showing a picture.
    - hgsgm 4y ago
    - d1sxeyes 4y ago
      > Yes. I don't see how answering 39 is any better in that case. Fair enough. I think 39 would show there's no bias towards younger or older employees. I think 39 is better but I can see both sides of the argument, and can ultimately see that it boils down to a question of personal preference as to which of two wrong answers is 'less wrong'.

qsort 4y ago

> Would you then argue that there are infinitely many medians in this data set By the definition I usually go by (the one with the inequalities), that dataset has infinitely many medians. There is no reason to say the midpoint is privileged. This is a better definition because it also works on sets where an order is defined but the arithmetic mean is not. If asked to report a single value with no qualifications, I would report either extreme of the interval.

d1sxeyes 4y ago
> There is no reason to say the midpoint is privileged. This is a matter of opinion. I agree it's not 'privileged', but it sacrifices a minimal amount of accuracy for a usable answer. > If asked to report a single value with no qualifications, I would report either extreme of the interval. That's a very interesting answer - so for my contrived example of 18, 20, 25, 30, [35, 40], 45, 50, 55, 60, you would report 35 or 40? How about if I change the dataset a bit (so it's even more contrived): 18, 18, 18, 18, [18, 60], 60, 60, 60, 60 - would you still report either 18 or 60 (again, asking without qualifications)? I think most people would agree that the median calculated as the arithmetic mean of the two middle values (39) is misleading, but that it is less misleading than ignoring half of the data set. How would you respond to that?
- qsort 4y ago
  > you would report 35 or 40? Generally I report the lower end because it more easily generalizes to different quantiles. > How about if I change the dataset a bit would you still report either 18 or 60 (again, asking without qualifications)? Yes. I don't see how answering 39 is any better in that case. > How would you respond to that? I would respond that you should look at the distribution. Reporting the median is particularly misleading in the case of bimodal distributions. I would try to explain that summary statistics are to be taken with caution, possibly showing a picture.
  - hgsgm 4y ago
  - d1sxeyes 4y ago
    > Yes. I don't see how answering 39 is any better in that case. Fair enough. I think 39 would show there's no bias towards younger or older employees. I think 39 is better but I can see both sides of the argument, and can ultimately see that it boils down to a question of personal preference as to which of two wrong answers is 'less wrong'.