What is the probability that x is more than 1 standard deviation from its mean value?

Solution:

The probability that x is more than one standard deviation is 0.3413.

From the normal standard table, the area within one standard deviation is 0.3413. Hence the area beyond one standard deviation is 0.5 - 0.3413 = 0.1587 from the center of the normal distribution. Similarly, the relevant area beyond one standard deviation on the left-hand side of the centre of the normal distribution curve is also 0.3413.

Hence the probability of the value of x being beyond one standard deviation is 0.1587 x 2 = 0.3174

The probability is depicted in the diagram by the shaded area. The areas add up to 0.3174 and hence the probability of the mean of x being beyond one standard deviation will be 0.3174

What is the probability that x is more than 1 standard deviation from its mean value?

Summary:

The probability of the mean of x being beyond one σ (z =1) is given by the areas under the normal distribution curve beyond 1𝜎 from the mean value of x. The probability of this happening is 0.3174 of 31.74%.