Sum of the Median = Median of the Sum?

Question

Analysing data that shows people time spent at stages of the process

Stage 1 Then Stage 2 Then Stage 3 Then Stage 4 Then Stage 5

I found the sum of average of each Stage = the average of the sum

$Avg(Stage1) + Avg(Stage2) + Avg(Stage3) + Avg(Stage4) + Avg(Stage5) = Avg(Stage1 + Stage2 + Stage3 + Stage4 + Stage5)$

Is the same rule apply for median

$Median(Stage1) + Median(Stage2) + Median(Stage3) + Median(Stage4) + Median(Stage5) = Median(Stage1 + Stage2 + Stage3 + Stage4 + Stage5)$

and why? have a proof?

Answer 1

No: If the times spent are as follows:

Stage 1: 1, 3, 4

Stage 2: 3, 4, 1

Stage 3: 4, 1, 3

then each stage's median is 3 so the sum of medians is 9, but each sum is 8 so the median of sums is 8.