Standard Deviation
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 vs Sample
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.
In the Normal Distribution
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.
Use in Effect Sizes and Reporting
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.
References
- Field, A. (2018). Discovering Statistics Using IBM SPSS Statistics (5th ed.). London: Sage.
- Larson-Hall, J. (2016). A Guide to Doing Statistics in Second Language Research Using SPSS and R (2nd ed.). New York: Routledge.
- Plonsky, L., & Oswald, F. L. (2014). How big is "big"? Interpreting effect sizes in L2 research. Language Learning, 64(4), 878–912.