when is it appropriate to use arithmetic mean as opposed to median
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It is then the [arithmetic] mean of the two number. The sam applies if you have 6 numbers. The median is the mean of the 3rd and 4th numbers (in order).
Median of the set = arithmetic mean of 8 and 9 = 8.5
It is not: it may be better in some circumstances. The arithmetic mean and median are totally useless if the data are categoric. Think about a class of pupils recording what their favourite movie is and - unless they all choose the same film - try and figure out what the arithmetic mean or median could mean! There are also situations - particularly with highly skewed data - where the median is a superior indicator of the central tendency compared with the mean. Having said that, the arithmetic mean has some very important statistical properties and it has been studied extensively so there is a lot of information about it. It is beyond the scope of this site but if you are really interested you might want to read about the Central Limit Theorem. Warning: it is basic university level stuff.
You take the arithmetic mean of the two middle numbers.
Take the middle value (arithmetic mean) of those two.