Yes, the mean (and median and mode) is the 50th percentile of any normal distribution.
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The cumulative probability up to the mean plus 1 standard deviation for a Normal distribution - not any distribution - is 84%. The reference is any table (or on-line version) of z-scores for the standard normal distribution.
The mean is 0 and the variance is 1. This need not be the case in any other Normal (Gaussian) distribution.
Actually the normal distribution is the sub form of Gaussian distribution.Gaussian distribution have 2 parameters, mean and variance.When there is zero mean and unit variance the Gaussian distribution becomes normal other wise it is pronounced as Gaussian.Wrong! The standard normal distribution has mean 0 and variance 1, but a normal distribution is the same as the Gaussiand, and can have any mean and variance. Google stackexcange "what-is-the-difference-between-a-normal-and-a-gaussian-distribution"
a mean of 1 and any standard deviation
The mean.The mean.The mean.The mean.