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A normal distribution simply enables you to convert your values, which are in some measurement unit, to normal deviates. Normal deviates (i.e. z-scores) allow you to use the table of normal values to compute probabilities under the normal curve.
The arithmetic mean is more commonly known as the average. It is the sum of the values divided by the number of values.
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The data from a normal distribution are symmetric about its mean, not about zero. There is, therefore nothing strange about all the values being negative.