No, they are rarely the same.
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nop its not
A standard normal distribution has a mean of zero and a standard deviation of 1. A normal distribution can have any real number as a mean and the standard deviation must be greater than zero.
Because the standard deviation is one of the two parameters (the other being the mean) which define the Normal curve. The mean defines the location and the standard deviation defines its shape.
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If the Z-Score corresponds to the standard deviation, then the distribution is "normal", or Gaussian.