The probability that a random variable x is more than one standard deviation away from its mean can be understood by referencing the properties of the normal distribution, which is often used in statistics.
In a normal distribution, approximately 68% of the data falls within one standard deviation (σ) from the mean (μ). This means that about 32% of the data lies outside of this range. More specifically, about 16% of the data will be more than one standard deviation above the mean, and another 16% will be more than one standard deviation below the mean.
Thus, the total probability that x is more than 1 standard deviation away from its mean (either above or below) is approximately 32%. This concept is crucial in statistics as it helps to quantify variations and understand the spread of data in relation to the mean.