No. Very often the mean will not be part of the data set.
In most countries, the average number of children per family is some fractional number. For example, in 2006, the mean number of children in a UK family was 1.8. However, I have yet to come across 0.8 of a child!
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No.
mean
A set of data will not always have a mode because some data sets will not have a number that occurs more than once.
No.
The mean of a set of data is the sum of that data divided by the number of items of data.