Empirical Rule vs Z-Score

Both terms below come up in the same part of the course, and students mix them up. Here is each one defined on its own, side by side, so you can see where they part company.

Empirical rule

Random variables and distributions

The empirical rule says that in a normal distribution, about 68, 95, and 99.7 percent of values fall within 1, 2, and 3 standard deviations of the mean.

The empirical rule, also called the 68-95-99.7 rule, describes how data cluster in a normal distribution. About 68 percent of values fall within 1 standard deviation of the mean, about 95 percent within 2, and about 99.7 percent within 3. For example, if adult heights are normal with mean 68 inches and standard deviation 3 inches, about 95 percent fall between $68 - 2(3) = 62$ and $68 + 2(3) = 74$ inches. It gives quick estimates without needing a z-table.

Full entry for empirical rule

Z-score

Describing data

A z-score tells how many standard deviations a value lies above or below the mean of its distribution.

A z-score standardizes a value by measuring its distance from the mean in standard deviations. You compute it as $z = \frac{x - \mu}{\sigma}$ , where $x$ is the value, $\mu$ (mu) is the mean, and $\sigma$ (sigma) is the standard deviation. For example, a score of 85 in a distribution with mean 70 and standard deviation 5 has $z = (85 - 70)/5 = 3$ , so it sits 3 standard deviations above the mean. A negative z-score means the value is below the mean.

Full entry for z-score

Where each one fits in the course

Empirical ruleRandom variables and distributions Z-scoreDescribing data