SFA chief executive Stewart Regan says they take full responsibility for child protection failings.
Three-Year-Olds Linked To Offences
Children as young as three have been recorded by police for committing offences.
Five three-year-olds and 20 four-year-olds are among the thousands of children recorded by Police Scotland in the past two years, according to a freedom of information request.
In total 376 children aged under eight - the age of criminal responsibility - are included in the figures.
Chief Inspector Hilary Sloan said officers have to record all offences.
"The police record crimes because we have to have an understanding of what crimes are occurring within the communities and because of the Scottish Crime Recording Standard we have a responsibility to do that," she said.
Offences linked to the young children include shoplifting and vandalism.
Among all children under 16, there were 44,341 offences recorded.
In 2013-14 alone there were 5,154 violent crimes and 409 sexual crimes recorded among under-16s.
Although the age of criminal responsibility remains eight, the age at which children can be tried in a court of law is 12.
A Scottish Government spokeswoman said youth crime has fallen by 52% over the last five years and continues to fall, dropping 22% since 2011-12.
"The Scottish Government's early intervention initiative, the whole system approach, tackles all aspects of youth offending from low level crime to the most serious and harmful offences and aims to stop young people following the wrong path into a life of crime by identifying at the earliest opportunity when they are in trouble," the spokeswoman said.
"Police, courts, education and social services work together to address minor offending behaviour before it becomes a major problem. This approach was first piloted in Aberdeen in 2010 to 2011 and saw marked improvement in youth offending with youth crime down 20% and offence referrals to the Children's Reporter falling by 40% on the previous year.
"Recorded crimes and offences are not proven offences. Police have a requirement to record all crimes under the Scottish Crime Recording Standard. This ensures consistency and a victim-centred approach to crime recording across Scotland.
"The Scottish Government will consider whether the age of criminal responsibility should be raised within the lifetime of this Parliament. To this end, we have set up a working group to look at the practical challenges of doing so."
The 55-year-old died at the scene of the incident in Dryden Street at about 9.50am on Monday.
She called on both sides of the Brexit divide to treat each other with respect and stop accusations of "racism''.
Police found the body of Sharon Greenop at a house in Aldersyde Avenue, Troon, South Ayrshire, on November 10.
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