[ghadaahlawya]

a runner covered a distance of 100 meters in 10 seconds and returned the same distance in 80 seconds
then the average speed comes in two ways:
1) 200/90 = 20/9
2) speed 1+speed2 /2
=10+1.25 /2
=5.625
the question is why does the average speed is different in the same thing?

[DrBob1]

Your first number is the time average speed; the second is the distance average speed. Both are completely valid statistics, but they have different stories to tell.
Distance divided by time tells how fast you got from here to there. It works best for a straightforward trip: leave, drive, arrive. How fast were you going? You get a better picture of your event by breaking it into two separate parts: the race at 10 m/sec opposed to walking back at 1.25 m/sec. But 200 meters in 90 seconds is truly 2.22 m/sec. Time-wise it makes perfect sense.
But what if the runner ran into you? I'd rather be hit when he is going at 2.2 m/sec than when he is going at nearly 6 m/sec. The second average gives a better idea of this sort of result. Note that highway safety statistics are based on miles travelled. (deaths per million miles) not for time on the road.
This is why you have to be careful when someone throws statistics at you -- they may have their agenda to promote.