The standard normal distribution is a normal distribution with a mean of zero and standard deviation of 1.The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation.or the standard normal distribution, 68% of the observations lie within 1 standard deviation of the mean; 95% lie within two standard deviation of the mean; and 99.9% lie within 3 standard deviations of the mean. To this point, we have been using X to denote the variable of interest (e.g., X=BMI, X=height, X=weight).