Hallo! Basically, the evidences for Piceni slingers are limited to literature (Polibio and Pilinio, "Historiae"), and in a lot of lead glans - a very huge amount of them can be seen in our museums, and can still be found in the fields around the most important town of the ancient region - like Ascoli Piceno (Ausclum) and Fermo (Firmum Picaena).
Greetings,
Mauro.

The figure is from Pliny NH 3.13.

The sling bullets from Ascoli may come from the battle of Asculum (89 BC), during the Social War. In other words, the profusion of lead bullets does not reflect any Picene prowess with the sling, but slingers in the ranks of the Roman army that fought against the Italian rebels. Not sure if this works for Firmum ?

I think that there must have been a very significant number of specialist cohorts of slingers in the Roman army because of the massive abundancy of evidence, archaeological and historical.

I will deal the second century CE, at the height of Roman power. The Roman army comprised roughly 300,000 men, of which approximately half were auxiliary troops. The vast majority of these auxilia were heavy infantry very like the legions, but the auxilia also comprised cavalry, specialist light infantry and missiles - let's estimate these specialist supporting units to make up maybe a third of the auxiliary army - about fifty thousand men. This would make them about a sixth of the whole Roman army, which is roughly what you would expect from most contemporary battle descriptions: the heavy infantry was by far the most important and numerous with about a tenth of the army being cavalry support, so it is probably reasonable to suggest another tenth to have comprised missile and light infantry.

If the cavalry made up about a tenth of the Roman army, then of our fifty thousand specialist auxilia only twenty thousand are probably foot-soldiers. This force included all sorts - Syrian and Greek archers, Thracian falx-wielders, Caucasian club-men, etc etc - but I'm prepared to bet that slingers made up the majority, because the Romans demonstrated such a preference for them and they feature in so many battles. Maybe 10,000 men then - a thirtieth of the entire Roman army. Twenty cohorts. It doesn't seem too unrealistic.

That's overall. On a practical level, if a standard Roman force comprised about twelve thousand men (one legion plus auxilia and various detachments from all over, with none of the units quite up to their full numbers as seems to have been common), then it probably had about a cohort of slingers in support - Balearics, Rhodians, Spaniards, Britons, Anatolians, or just local shepherds.

Sorry I didn't cite any sources, I don't remember where most of my knowledge comes from.

Hallo there!
I've provoked a lot of questions... let see if I can even answer to them  
Thearos, evidences of Picenian slingers come from all of those glans - a great part of them have been thrown from Picenian slingers, be they enlisted in the Roman Army or on the wall of Asculum, protecting their capital against it. A smaller number of those glans have been used by Venetian auxiliares that helped the Roman repressing the Social War of 91b.C.
I'll post a picture of a sculpture now at the Archaeological Museum in Ascoli:

(courtesy of www.archart.it); this sculpture has been made to celebrate the victory over the city; those represented are probably Picenian slingers serving the Roman Army, analyzing their poor clothes and the absence of any other weapon, and, most important, the fact that this sculpture has been exposed in front of the city gate just after the Romans conquered it, as a memento.
For the 360'000 Picenian at the beginning of IIIrd Century: Plinius talks about this (Naturalis Historiae, III 18, 110). You've done a very good operation: 360'000:2=180'000 males. Of those, it's easy to find just 10'000 capable of handing a sling or some other weapon.

Howewer, you will never find some ancient historician saying clearly that "Piceni are famous slingers". You have to deduct this by a lot of other evidences, the most important being literature, founding of hundreds of glans, the fact that Picenian's way of warmaking has always been guerrilla-style (in which the sling fits well), and the fact that Piceni have been the last Italic population to fall into the Roman orbit, in the center of Italy - they were tough people: after conquering them, the Romans deported about 1/3 of the population in South West Italy, and another 1/3 in South East.
Howewer, we still find dozens of glans, such as this from my collection:

found in the shores of the Tronto river, were the Piceni have been involved in their last important battle.

This has been a long off-topic, howewer, but I hope to have been clear enough  
For Picenian slingers serving in the Roman Army during the IIIrd Century, check out the Trasimeno and Cannae Battles  
Greetings,
Mauro.

While I'm at it:

Asculum I (Pisa, 1975), has a study by U. Laffi on ancient Ascoli Piceno. p. xxvvii "Come e noto, la grande  magioranza della glandes ascolane e stata rinvenuta nell ' alvea del Castallano, il torrente che cinge la citta a S. e a E e confluisce nel Tronto a NE della citta stessa. (other findspots listed). Questa distribuzione dei reperti fornisce, a mio avviso, utili indizi per la riconstruzione della topografia dell'assedio...

But what if they weren't any full timers ? You'd have the archer units always there. And slingers ? Ad hoc stuff. At least worth thinking about.

Hallo everyone!
Thearos, I agree with your point of view: probably, Piceni did use slings, but were not skillful as Balears, Rhodians, etc...
I don't remember where I read of Piceni involved in the Trasimeno Battle, but I'm 100% sure that they helped Romans during the Cannae Battle, with slingers - I'm gonna ask my teachers as soon as possible 
Oh and I forgot another important battle, the Sentinum one (295 b.C.): here, the Romans fought Etrurians, Umbri, Sanniti and Galli (but the first two populations did not partecipate), helped by the Piceni. Note that, in that time, Piceni were not yet conquered by Rome, against which they began a war in 269 b.C. Even in the ancient Sentinum (now Sassoferrato), we have foundings of lead glans, but they may have been thrown by everyone - it's a fact that Piceni did partecipate.
Still, a lot of evidence that Piceni did use the sling, but much less that they were skilled slingsmen: I had misunderstood the topic's title, which is "How many slingers were in the Roman Army", and not "How good were Roman Army's slingers?" 

A small note about the relief: it is conserved in Rome, but it comes from Ascoli Piceno, and doesn't show dancers, because at least the two in the left wear helmets - I have a very detailed pic of that particular in my "Archeologia nelle Marche", I'll try to post it as soon as possible!
Greetings,
Mauro.