When might you want the mean?
Examples from finance:
- Bond returns:
- The median bond return will generally be a few percentage points.
- The mean bond return might be low or high depending on the default rate and recovery in default. The median will ignore all this!
- Good luck explaining to your investors, "I know our fund is down 40% this year because almost half are bonds went bust with no recovery, but our median bond returned 1%!"
- Venture capital returns:
- Same thing in reverse. The median VC or angel investment is a bust, and all the return comes from a few winners! (Side note/warning: estimates of venture capital or private equity returns are highly problematic... be careful!)
When forming a diversified portfolio, deciding what to invest in and how much, the mean and covariance of returns are likely to factor prominently into your optimization problem.