What is the difference between using a probability distribution and using the mean of observations only as a measure to describe the data? I am wondering using the mean only without a specific probability distribution to predict or infer an appropriate measure for the average height of female students.
I ask this question because I usually see a probability distribution with the mean and the variance as the parameters of an assumed probability distribution.
So, I wonder that if I want to calculate and use the mean as an approximation of the mean height of female students,

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*Is it a must to assume a probability distribution?
Can't I just use the mean as an approximation or a predicted value without assuming a specific probability distribution ?

*Can I still call using the mean only to approximate the mean height of femal students a statistical model?

*If I can't call it a stistical model, then what should I call the way I try to approximate the mean height?

 A: The benefit of distributional assumptions is that you can do additional inference rather than just prediction.  Using the mean as a prediction of some population level parameter is often good, the mean has lots of very desirable properties and is well studied.  That being said, if the underlying distribution is long tailed and you choose to ignore that, then the mean may appear to be a poor prediction in the short term as you see the majority of observations near the mode.
To answer your questions in bullets:

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*Is it a must to assume a probability distribution? Can't I just use the mean as an approximation or a predicted value without assuming a specific probability distribution ?

No, you don't have to assume a distribution, but you might have to make other assumptions (e.g. finite variance, independent observations, etc) in order to justify the prediction,

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*Can I still call using the mean only to approximate the mean height of female students a statistical model?

Only in the broadest of senses.  Any function which maps data to a number (or a tuple of numbers) is a statistic, so taking your height data and always returning 0 is a -- again, in the broadest of senses -- a statistical model.  It just isn't a very informative one.

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*If I can't call it a statistical model, then what should I call the way I try to approximate the mean height?

I would not call the mean a statistical model.  I would just say you're using the mean as a point prediction or as an estimator.
