There have been many Andertons who have been granted Coats of Arms throughout the ages and the thing you have to remember is that the grant is to the individual, NOT the family. Certainly the Arms can and usually are passed on as a legacy within a family. I will always be proud to say that these are the arms of Andertons of bygone ages but it would be untrue of me to say that they are MY family arms ... unless of course I could prove a descent from someone who HAD been granted arms .
If you want to be specific, you need to ask yourself which branch of the Anderton family you are interested in. In Lancashire you have Anderton of ... ANDERTON, BIRCHLEY, BURGH HALL, CLAYTON, ELDERTON, EUXTON, HAIGHTON HALL, LANCASTER, LOSTOCK ... all of whom have granted subtly different arms. From Yorkshire I know of two Grants of Arms not to mention Welsh-Andertons who funnily enough come from Wales (Welsh in this case is a family name as well).
You are right in that the root form of the Anderton Arms - the recurring them that probably harks back to the ancestral arms of Anderton of Anderton is Sable[black field], three shackbolts argent [silver].
The earliest Arms I have been able to find are from "Ballards Roll of Arms, c. 1485" - Harl. MS 2076. They are of Thurstan Anderton of Anderton, c.1473 - 6 May 1516, who quartered the Anderton Arms with those of de Kenyon ...
"No.42 … Arms - Quarterly: 1 and 4, Sable, three shacklebolts [sic] argent; 2 and 3, vert, three hunting horns, argent, stringed or. Wreath - Uncoloured. In M.3. argent and vert Crest - A curlew proper. In the margin the bird is marked grey, but Wall, No. 10, says it is a hilpe, otherwise called a curlew, and the wreath should be silver and vert."
Thurston was the grandson of Oliver de Anderton and his wife and alleged poisoner, Ellen de Kenyon, mustn't have held any hard feelings eh?
If you want see an example of the ancestral Anderton Arms on the Net see ... ...Contact me off list if you want full references.