Jimmy Savile may have abused as many as 25 victims over a 40-year period on a "national scale", police said.

Scotland Yard has formally recorded eight criminal allegations against Savile so far in its investigation, named Operation Yewtree.

The alleged abuse involves teenage girls as young as 13 and includes two complaints of rape and six of indecent assault, with officers looking into up to 120 lines of inquiry.

The scale of the alleged abuse by the former Top of the Pops presenter was disclosed as Savile's family planned to remove the headstone from his grave to ensure the "dignity and sanctity" of the cemetery.

A family spokesman said in a statement: "The family members are deeply aware of the impact that the stone remaining there could have on the dignity and sanctity of the cemetery.

"Out of respect to public opinion, to those who are buried there, and to those who tend their graves and visit there, we have decided to remove it."

Commander Peter Spindler, head of specialist crime investigations at Scotland Yard, said the allegations – the earliest of which dates back to 1959 – span four decades, with information suggesting abuse was on a "national scale".

He said: "The reality is this really has captured the public's mind. We are getting calls from victims, from witnesses and third parties who believe they know something about it.

"We have formally recorded eight criminal allegations against Savile. Two of those are rape, six of indecent assault.

"These are primarily against girls in their mid-teens, so between 13 and 16, and it spans four decades of abuse.

"The pattern of his offending behaviour does appear to be on a national scale."

Mr Spindler said the first allegation dates back to about 1959 but most claims seemed to be in the 1970s and 80s.

Scotland Yard has contacted ITV and the BBC, who in turn are contacting alleged victims to see if they will co-operate, he said. Mr Spindler added that there may be 20 or more victims in total. "It is too early for us to give an accurate picture of what 120 lines of inquiry will distil down to but we believe we will come up with between 20 to 25 victims."

Prime Minister David Cameron addressed the issue of the prospect that Savile could be posthumously stripped of his knighthood. But the move would require a change in the law as technically, when the former DJ died, the honour ceased to exist.