Yes
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Mean = 0 Standard Deviation = 1
Mean 0, standard deviation 1.
There are two points of infection (the points where the curvature changes its direction) which lie at a distance of one standard deviation above mean and one standard deviation below mean.
Standard error of the mean (SEM) and standard deviation of the mean is the same thing. However, standard deviation is not the same as the SEM. To obtain SEM from the standard deviation, divide the standard deviation by the square root of the sample size.
Mean and standard deviation are not related in any way.