standard deviation

Meaning: SD is roughly the typical distance of scores from the mean. You square each deviation from the mean, average the squares (giving the variance), then take the square root to return to the original units. A small SD means scores cluster near the mean; a large SD means they are spread widely. Two groups can have the same mean and different SDs.

Equation: σ = √[Σ(x − μ)² / N], where x is each score (units of the measurement), μ is the mean (same units), and N is the number of scores (unitless). σ has the same units as the data, such as IQ points. For a sample, divide by n − 1 and call it s.

On the MCAT: Passages give means and SDs for groups and ask which group is more variable, or whether a gap between means is large compared with the spread. Know the normal curve: about 68%, 95% and 99.7% of scores lie within 1, 2 and 3 SD. For IQ with mean 100 and SD 15, about 68% score between 85 and 115. Adding a constant to every score shifts the mean but leaves the SD unchanged.

Common confusion: SD is not the standard error of the mean. SD describes the spread of individual scores; standard error (SD/√n) describes how precisely a sample mean estimates the population mean, and it shrinks as sample size grows. SD also is not the range, and the 68-95-99.7 rule applies to roughly normal data.