The survey also found a quarter were worried about cyberbulling on social networking sites and other shared websites.
Respectme, Scotland’s anti-bullying service, which conducted the survey, yesterday called for parents and carers to monitor more closely the online activities of their children.
Brian Donnelly, director of respectme, said: “When a child asks if they can go into town with their friends you would want to know who they were going with, how long for and where exactly they would be.
“We need to have the same concerns and a similar response if they are going online, because they are still going somewhere.”
Eileen Prior, executive director of the Scottish Parent Teacher Council, backed the call.
“The internet is another avenue used by bullies to pursue their target,” she said.
“The advice to parents is out there -- the first step is to talk to your children about what they are doing and who they are meeting, whether in person or in cyberspace.”
The survey of around 4000 young Scots in October also looked at where children go online, how long they spend surfing the web, what devices they use to access the internet and the impact of cyberbullying.
Of those young people who had experienced cyberbullying, 63% said they knew the identity of the perpetrator.
In terms of general internet habits, 55% of children and young people use the internet every day, 53% are online for up to two hours a day and 8% said they are online for more than five hours a day.
The most common devices used to access the internet are mobile phones and laptops -- both used by 59% -- with games consoles also common.
The most common places young people go to online are Facebook, popular with 68% of the survey, followed by the 28% who use BlackBerry Messenger.
Children and young people who have been cyberbullied were most likely to tell their parent or carer and their friends, followed by a teacher.
The 16% who said they had been bullied is in line with previous surveys which indicate the problem impacts on a significant minority, but does not seem to be growing.
However, there are specific concerns about the nature of cyberbullying which can make it more difficult to combat.
Unlike physical bullying, electronic bullies can remain virtually anonymous using temporary email accounts, pseudonyms in chat rooms, instant messaging programmes and mobile phone text messaging.
Additionally, electronic forums often lack supervision and, even where chat rooms are policed, personal messages sent between users are viewable only by the sender and the recipient.
Anti-bullying charities have previously called for social networking websites to stop the capacity for users to send messages anonymously, saying that it is particularly damaging to children and teenagers.
The practice, known as “trolling”, has been linked to self-harm among victims and even, in the most extreme cases, suicide.