Mathematics
Empirical Rule (68-95-99.7 Rule)
Quick fact
The 68-95-99.7 rule, also called the empirical rule, applies only to data that follows a normal (bell-shaped) distribution—not all datasets follow this pattern.
Why this is interesting
You've probably heard that most people have an IQ between 85 and 115, but have you ever wondered why that precise range?
Read the full explanation
Understanding Empirical Rule (68-95-99.7 Rule)
Imagine you have a large set of data that forms a bell curve when plotted—most values cluster near the center, and fewer appear as you move outward. The center is the mean (average). The spread is measured by the standard deviation. The empirical rule tells you that about 68% of all data points lie within one standard deviation on either side of the mean. This means if you know the average and how spread out the data is, you can quickly estimate where the bulk of observations will fall. For IQ tests, the mean is 100 and standard deviation is 15, so 68% of people score between 85 and 115.
A deeper explanation
The rule works because the normal distribution has a precise mathematical shape defined by its probability density function. The percentages (68%, 95%, 99.7%) are the integrated areas under the curve between the mean and ±1, ±2, and ±3 standard deviations. They follow from the properties of the normal distribution's formula. This rule is crucial in quality control (e.g., setting process limits), psychology (interpreting test scores), and finance (assessing risk). It also underpins the concept of z-scores, as each standard deviation from the mean corresponds to a specific percentile.