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I see from other posts that the average distance within a circle of radius $r$ to the center is $2r/3$. However, assuming a uniformly distributed PDF, shouldn't the average distance be the radius at which the area inside and outside are the same? For example, in a circle radius 9 (area=254), a circle with radius $R = \sqrt{r^2/2} = 6.36$ would give you a circle with exactly half the area (area=127). So why doesn't this work?

Tesla Daybreak
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rufus100
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    Of course not. Is the average household income the median household income (the income for which half the population earns more, half the population earns less)? Of course not! – David G. Stork Feb 21 '21 at 23:29
  • "Average" is an ambiguous term. The OP found the median, which may be one variety of "average". Probably, if a math text just says "average" it means the "mean" not the "median". – GEdgar Feb 22 '21 at 00:02
  • At that radius you have two equal areas, but the distance to the centre from any point in the inside half is smaller the distance to the centre from any point from the outside half. – N. S. Feb 22 '21 at 00:03
  • Fantastic point David, I think that helps. For context, I'm looking to find the average walk distance to a transit stop (on a rectilinear grid with uniform population density, but a circle made it easier to explain for this post). I think you've cleared it up well enough, in terms of walk distance for this situation, I think mean will be more useful than median. – rufus100 Feb 22 '21 at 03:57

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