BBC director general Tony Hall has apologised and said a mistake was made after a news report containing a racial slur was broadcast last month.
The N-word was used in full in a report about a racially aggravated attack in Bristol, broadcast by Points West and the BBC News Channel on 29 July.
Lord Hall said he now accepts the BBC should have taken a different approach.
On Saturday, BBC Radio 1Xtra DJ Sideman – real name David Whitely – quit the station over the BBC’s use of the N-word.
He said “the action and the defence of the action feels like a slap in the face of our community”.
His actions were backed by a number of politicians and BBC staff, who offered support to the DJ.
In his statement, Lord Hall said: “I recognise that we have ended up creating distress amongst many people.”