# When does the sum of the medians = the median of the sum

I have two random variables (say x1 and x2) defined by empirical probability distributions, and would like to calculate the median of their sum.

Under what circumstances (in terms of the distributions of x1 and x2) can I assume the median of the sum is equal to the sum of the medians i.e.

median(x1) + median(x2). (1)

The alternative approach I've used is to randomly generate large samples of x1 and x2 and then calculate the median as

median(sample of x1 + sample of x2). (2)

Approach (1) is quicker and I need to do this calculation many times. Under what circumstances is approach 1 approximately correct? Are their alternatives to my second approach?

If we have two normally distributed random variables then median of the sum is approximately the sum of the medians

N1 <- rnorm(10000, mean = 1, sd = 0.1)
N2 <- rnorm(10000, mean = 0)

# We expect an answer of 1 and get close

median(N1) + median(N2) #[1] 0.9918688
median(N1 + N2) #[1] 0.9962555


This doesn't work for exponential variables

set.seed(2002)
e1 <- rexp(100000, 1)
e2 <- rexp(100000, 1)

median(e1) + median(e2) # expect 2* log(2) = 1.386 and get 1.374
median(e1 + e2) # expect 1.678 and get 1.668


So, looking at @glen_b's comment, is symmetry the sufficient condition that would allow the assumption that the median of the sum is the sum of the medians?

• @repmat Never? that's demonstrably untrue Jul 29, 2019 at 9:12
• @Tony you should clarify exactly what you mean by "sample of x1 + sample of x2". $\:$ There are a number of sufficient conditions. If you have two independent symmetric random variables, then the median of their sum will be the sum of the medians -- unfortunately empirical cdfs are very rarely symmetric. Another case would where the two variables are comonotonic (because of quantile-additivity) but it's not clear that you're talking about sampling their joint distribution (i.e. sampling pairs of $(x_i,y_i)$ together ... ctd Jul 29, 2019 at 9:21
• ctd... by drawing the index $i$ at random); it sounds like you mean to sample them separately as if they were independent. Jul 29, 2019 at 9:21
• But (1) you won't get exact symmetry with an ecdf unless you're astoundingly lucky (its useful for population disttributions, but only approximate at best with samples from them), and (2) you could get additivity without symmetry Jul 30, 2019 at 7:19
• I was suggesting that there would be other conditions than either of the ones I mentioned. I don't have a characterization for you, but examples that are neither symmetric nor comonotonic are easy to make, so those two are not a complete list. Jul 30, 2019 at 12:08

Actually my comment is not entirely correct, allow me to clear up;

The median of a series of numbers $$X$$ is calculated by ordering all the numbers from smallest to largest, then finding the number in the middle. This means that when you change the numbers in $$X$$ you also change the ordering, hence the median changes. Therefore (in general) you can almost always assume that: $$\text{MED}(X + Y) \neq \text{MED}(X) + \text{MED}(Y)$$ However there is at least one exception, whenever the ordering of $$X$$ (after adding $$Y$$ to $$X$$) does not change neither does the median. For instance if all numbers in $$X$$ and $$Y$$ are the same, see this example (written in R):

set.seed(42)
n <- 100
x <- rnorm(n)
c <- x
y <- rnorm(n)

median(x+y)           # 0.0767433
median(x) + median(y) # 0.02050838
median(x + c)         # 0.1795935
median(x) + median(c) # 0.1795935

• The OP does ask about when approximation is close enough, and clearly it can be seen in your example, that with increasing n, the difference between median(x+y) and median(x)+median(y) approaches 0 (?) at some rate. Same can be seen with other symmetric pdfs, as @Glen_b states in the comments.
– runr
Jul 29, 2019 at 12:53

For continuous variables the following are equivalent

$$\text{M}(X + Y) = \text{M}(X) + \text{M}(Y) \\ \iff \\ \mathbb{P}[(X-\text{M}(X)) > -(Y -\text{M}(Y))] = \mathbb{P}[(X-\text{M}(X)) < -(Y -\text{M}(Y))]$$ You can imagine this geometrically from the joint distribution of X and Y. Half the mass needs to be on either side of the line $$x+y=\text{median}(X)+\text{median}(Y)$$ (or equal masses for discrete variables).

This means that for two random x and y, the probability for x to be further above the median of X than y is below the median of Y equals, the probability for x to be less above the median of X than y is below the median of Y.

Comment: This is parallel with other comments, but it might give you a quick way to check whether one variable increases precisely when the other does.

If Spearman correlation between x and y is $$1,$$ I believe the sum of medians is the median of the sum. In R:

x = rexp(100);  y = sqrt(x)
median(x+y)
[1] 1.598729
median(x)+median(y)
[1] 1.598729
cor(x,y, meth="spearman")
[1] 1


The other situation (approximate) discussed in comments is symmetry:

 u = runif(100); z = rnorm(100)
mean(u+z);  median(u+z)
[1] 0.5401409
[1] 0.5229718
mean(u)+mean(z)
[1] 0.5401409
median(u)+median(z)
[1] 0.5866283

• So basically, whenever y is a positive linear function of x, this is true. Makes sense intuitively as well.. Jul 30, 2019 at 2:52
• @ChinG it can also work with zero correlation. For instance if X and Y are independent normal distributed variables. Oct 11, 2021 at 6:49