Some person says that the conservative estimate would be 7 rounds per minute, meaning that slingers can fire more than that.

However, I need sources to confirm this. Does anybody have one? What about the rate of fire for 400 meters?

Thank you

given that only two modern slingers have recorded officially witnessed throws over 400 metres, that's a tough one. And an unlikely distance for ancient slingers to shoot regularly.
200 metres - yeah with the right ammo seven shots a minute would be reasonable slinging.
I can do that easily enough. Probably 10 distance shots a minute with the right setup.

bear in mind every 3 shots you need to go to your ammo pouch to reload (I can hold 3 2 oz glandes in my slinging hand easily enough) that's pretty much the only bit that slows you down.

A good distance throw is pretty smooth and generally only one movement - we're not talking helicopter here ;-)

When dealing with glandes I would second what Curious Aardvark has said, seven is a prudent choice,  maybe even a little more.   And furthermore, a slinger using glandes could cary several minutes worth of ammo on him with little inconvience, even if it was he only got six or seven shots per killogram of weight. (lead shot is what I am thinking of)

A slinger using rocks on the otherhand would be equaly incumbered, if not more so, just carrying one minutes worth of ammuntion.  Even if the projectiles are the same weight, that stone is going to be much bigger, and likely irregular in shape.  Even if the slinger could manage to get 7 stones out in a minute, he probably wont have many more stones to throw without searching for some on the ground.   And so his rate of fire drops rapidly.

I suppose all the above would not matter if the slingers were defending something where they had ammo stockpiled and availble.  Then I would say seven stones a minute for a working range, or slightly more,  would be a good estamate, no matter what the ammo is, stones, clay, or lead.

One shot every six seconds isnt hard to do. Thats ten per minute.

This is how I do it - take the pouch with the left hand and load it with the right hand:

http://www.zinfinium.net/index.php?id=157

This summer I was trying some continous firing to a bush at ~150 or 200 meters, but I had a pile between my legs and picked rocks 1 by 1 (makes great exercise for back).
So after some slinging I was able to do a shot in 4secs, best was ~20 shots a minute.
I don't think having a bag of rocks on your shoulder could do much better than that, because you still got to catch the tail of sling that I was doing when bending down to reach for rock, and came up with a rock already in the pouch and half swing done. IMHO the bag or 3 rocks ready will just make your back feel much easier after half an hour of slinging ;)
And I bet it's possible to make it 30 shots/minute, even with this "bendy" technique.