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Applying the formula

will always produce a transformed distribution with a mean of zero and a standard deviation of one. However, the shape of the distribution will not be affected by the transformation. If X is not normal then the transformed distribution will not be normal either. One important use of the standard normal distribution is for converting between scores from a normal distribution and percentile ranks.

Areas under portions of the standard normal distribution are shown below. About 0.68 (0.34 + 0.34) of the distribution is between -1 and 1 while about 0.96 of the distribution is between -2 and 2.

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