Depends on how much handwaving you're willing to tolerate. There's definitely some useful statistics you can do without showing the integral sign. One could si

Depends on how much handwaving you're willing to tolerate. There's definitely some useful statistics you can do without showing the integral sign. One could similarly say "How far can you get in statistics without measure theory? Or do you just take the definition of random variable as given?". It's still hand-waving, just at a different level.