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Im having trouble seeing how two random variables with the same median will have on ,average, equal number of positive differences and negative differences. Can somebody elaborate and help provide intuition as to why this must be so?

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  • $\begingroup$ Take the expectation of sign of the difference and apply linearity of expectation. $\endgroup$
    – whuber
    Dec 13, 2021 at 18:33
  • $\begingroup$ @whuber i can only do this if i assume that both variables are symmetric and normally distributed. which would make the expectation of the difference 0 under the null hypothesis and half the distribution will be positive and the other half would be negative. i can't think of a way to generalize this to any distribution. $\endgroup$
    – NoLifeKing
    Dec 14, 2021 at 6:57
  • $\begingroup$ Not so: everything follows from the definition of the median and the (here unstated) assumptions that the variables are continuous (at least in a neighborhood of their median) and have the same distribution (the null hypothesis). This implies the chance that the difference of the variables is positive equals the chance that their difference is negative and both of those chances are $1/2.$ $\endgroup$
    – whuber
    Dec 14, 2021 at 16:12

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