To understand the probability of a value being more than one standard deviation away from its mean, we need to dive into the concept of the normal distribution, often depicted as a bell curve.
In a normal distribution:
- The mean is the center of the distribution.
- About 68% of the data falls within one standard deviation (σ) from the mean (μ).
- This implies that roughly 32% of the data lies outside this range.
Specifically:
- Approximately 16% of the data is more than one standard deviation above the mean (i.e., μ + σ).
- Similarly, about 16% of the data is more than one standard deviation below the mean (i.e., μ – σ).
Putting these together, the total probability of a value being more than one standard deviation away from its mean (either above or below) is:
16% + 16% = 32%
Thus, in a normally distributed dataset, there is a 32% chance that a random value will fall more than one standard deviation away from the mean. This insight is crucial for understanding data dispersion and the behavior of distributions in probability and statistics.