Some parts of your question are unclear. It might help to give the context of what you’re trying to achieve, rather than what are the specific steps you’re taking.
1) + 3) In a Normal distribution – fitting the distribution, and estimating the mean and standard deviation – are basically the same thing. The mean and standard deviation completely determine the distribution.
mu, std = norm.fit(data)
is tantamount to saying “find the mean and standard deviation which best fit the distribution”.
4) Calculating the Z score – you’ll have to explain what you’re trying to do. This usually means how much above (or below) the mean a data point is, in units of standard deviation. Is this what you need here? If so, then it is simply
(np.array(data) - mu) / std
2) Mixture of normal distribution – this is completely unclear. It usually means that the distribution is actually generated by more than a single Normal distribution. What do you mean by this?
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solved Python – Statistical distribution