Empirical rule

The empirical rule describes areas under a normal curve. For a distribution that is normal with mean $\mu$ (mu) and standard deviation $\sigma$ (sigma), about 68 percent of the area lies on $\mu \pm \sigma$, about 95 percent on $\mu \pm 2\sigma$, and about 99.7 percent on $\mu \pm 3\sigma$. Those three headline numbers are rounded. The exact areas are 0.6827, 0.9545, and 0.9973, so the rule is a fast sketch rather than a substitute for a table.

Take heights that are normal with $\mu = 68$ inches and $\sigma = 3$ inches. Two standard deviations reach from $68 - 2(3) = 62$ to $68 + 2(3) = 74$ inches, and the rule says about 95 percent land in there. The exact area between 62 and 74 is 0.9545, so the rule is low by roughly half a percentage point. One standard deviation, 65 to 71 inches, holds 0.6827 of the heights.

Here is the error that shows up most: "95 percent are within 2 standard deviations, so 5 percent are above 74 inches." The leftover is split between two tails, not piled into one. The exact area above 74 inches is 0.0228, about 2.3 percent, and the same amount sits below 62 inches. Every one-sided empirical-rule answer needs that halving step, and skipping it roughly doubles the reported probability.

The rule is a fact about the normal curve, not about data in general. Applied to a strongly skewed variable it can be badly off, and applied to data that are only roughly bell-shaped it is an approximation on top of an approximation. When a problem asks for exactly the middle 95 percent, the multiplier is 1.96 rather than 2: for these heights that runs 62.12 to 73.88 inches, an interval whose area is 0.9500.

The normal distribution the rule describes is topic 2.11 of Unit 2. Use the rule for a sketch and a sanity check, and the z-table when the answer has to be right.