Question

For a normal distribution, find a) the percentage of data that is greater than 3 standard deviations above the mean and b) the percentage of data that is less than 2 standard deviations below the mean.

Answer 1

Let, X follows normal distribution with mean m and standard deviation s

X ~ N( m, s^2 )

a) Percentage of data that is greater than 3 standard deviations above the mean = P[ X > m + 3*s ] = P[ ( X - m )/s > ( m + 3*s - m )/s ] = P[ Z > 3 ] = 1 - P[ Z < 3 ] = 1 - .9987 = 0.0013 = 0.13%

b) Percentage of data that is less than 2 standard deviations below the mean = P[ X < m - 2*s ] = P[ ( X - m )/s < ( m - 2*s - m )/s ] = P[ Z < -2 ] = 0.0228 = 2.28%