Solved – Can mean be less than half of median if all numbers are non-negative

I feel like I might be missing something dumb, but in a set of non-negative numbers, is it possible for the mean to be less than half of the median?

Example: there are 999 numbers and we are told that the median is 10. This means that the sum of the numbers cannot be any less than 5,000. The configuration to achieve that sum is: 500 numbers are equal to 10, 499 are equal to 0. This makes the mean 5,000 / 999 = (slightly more than) 5. Since 5,000 is the lowest possible sum, (slightly more than) 5 is the lowest possible mean. Therefore the lowest possible mean is more than one half of the median.

Am I wrong?

You are correct. This is one example of a general result called Markov's inequality, which says that for a non-negative random variable $X$ and number $a$, $$P(Xgeq a)leq frac{E[X]}{a}$$ If you plug in the median of $X$ for $a$ you get $$P(Xgeq text{median})leq frac{E[X]}{text{median}}$$ so $$0.5leq frac{E[X]}{text{median}}$$ and $$ 0.5timestext{median}leq text{mean}$$

Your argument is also roughly how Markov's inequality is proved.

Similar Posts:

Rate this post

Leave a Comment