A measure of the spread of a set of values around their mean, expressed in the same units as the data. The standard deviation is the square root of the variance — the average squared deviation from the mean. A class with mean reading score 70 and SD 5 is much more homogeneous than a class with the same mean and SD 15.
Population standard deviation, σ, is computed by dividing the sum of squared deviations by N. Sample standard deviation, s, divides by N − 1, applying Bessel's correction — the −1 corrects the upward bias that arises when the sample mean is used in place of the unknown population mean. Statistical software defaults to the sample formula; SPSS, R, and Excel all use N − 1 unless explicitly told otherwise. The distinction matters at small N and disappears at large N.
Under a Normal Distribution, roughly 68% of values fall within one SD of the mean, 95% within two SDs, and 99.7% within three. This 68–95–99.7 rule supports the conventional 95% Confidence Interval (mean ± 1.96 × SE), z-score interpretation, and standardised test scaling. Many language tests — TOEFL iBT section scores, IELTS-aligned scoring scales, school placement batteries — are constructed on a standardised metric where one SD has a fixed score-point value.
Standard deviation is the denominator in standardised mean-difference effect sizes such as Cohen's d and Hedges' g. Reporting an SD alongside every mean is a baseline expectation in quantitative SLA reporting: without it, readers cannot evaluate the practical meaning of group differences, recompute effect sizes, or include the study in a meta-analysis. Plonsky and Oswald (2014) flag missing SDs as the single most common reason published L2 studies cannot be incorporated into syntheses.