Those related to 48 sexual assaults, 201 dangerous or negligent acts, 27 abduction/harassment cases, 27 public order offences and 33 assaults.
To protect the privacy of individuals, where there are fewer than three people in one category it is rounded up to three.
Data for February showed three of the children were given formal police warnings for theft.
The other three children were dealt with by "non-court action not further defined" in November.
All the recorded offenders were female, with the February incidents taking place in Waikato and the November incidents in the Central police district.
Those numbers were down from nine in the year ending June 30, 2015 when six preschoolers were dealt with for theft and six for property damage.
Victoria University crime and justice lecturer Dr Sarah Wright said that children under 10 cannot be charged under New Zealand law.
However, the incidents must have been serious enough for police to take the actions recorded.
"I can't comment about the individual cases, but there must have been good reason why the police [took action]."
Criminologist Dr Jarrod Gilbert said intervention was needed from a number of agencies, not just police, to tackle crime at both ends of the spectrum.
"Punishment at those extreme ends is only one element we should be considering. It's not just a police issue, because it requires a community response, and not only addressing individual crimes but the problems as a whole and reducing future victims."
He said it was especially important to address young people's offending as soon as possible.
"When you're dealing with really young people like that, it is critical that crime is seen more than just a police issue.
"If these young people, or their problems, are left unattended then they will continually pop up in the crime data time and time again: A current indicator of future offending is previous offending when really young."
Dr Wright said overall, youth offending was trending down. With declines for all age groups under 19..
"Young people have always done things, if you look at the long term trend of people being apprehended . . . it's very steadily on the decline."
When it came to the elderly offenders, Dr Gilbert said it was a probable side effect of the country's ageing population.
"You'd expect to see greater crime as that demographic increases."
A police spokeswoman said very young and very old offenders comprised a very small number of total offenders.
"While any offending, particularly that of very young children is of concern, the number of offenders at either end of the age spectrum represents a very small fraction of the total number of offenders nationally.
"The combined total of offenders between the ages of 0-4 and over 80 were less than 400 out of the national total 113,553 offenders, and any overall impact of these crimes are reflective of this fraction. "
She said that criminal activity was prioritised on a case-by-case basis regardless of the age of an offender.
With youth offending, non-court approaches such as family conferences, alternative actions and warning were used in about 85 per cent of cases.
Crime statistics: Young and old
Up to six children aged under 4 received warnings from police for theft or burglary
348 offences were committed by people ages 80 and over, including:
• 5 homicides
• 33 acts intending to cause injury
• 48 sexual assaults
• 201 dangerous or negligent acts
• 27 abduction / harassment
• 27 public order offences
(For the year June 30. Source: NZ Police)