Google

The standard deviation ($$SD$$) for describing the variability of a sample is computed by finding the difference between each score ($$X$$) and the mean ($$M$$), squaring each difference, finding the mean of these squared values, and taking the square root. The formula is: $$SD = \sqrt{\frac{\sum (X-M)^2}{N}}$$, where $$N$$ is the total number of scores.

Statistical Standard Deviation Formula

To compute the standard deviation for a set of scores (e.g., $$3, 5, 4, 2, 7, 6, 5$$, and $$8$$, which have a mean of $$5$$), first determine the difference between each score and the mean. Square each difference to make all values positive, then find the mean of these squared differences (the variance), which is $$28 / 8 = 3.50$$. Finally, the square root of the variance provides the standard deviation: $$\sqrt{3.50} = 1.87$$.

Example of Computing the Standard Deviation

When researchers compute the standard deviation of a sample to estimate the variability of a larger population, they divide the sum of squared differences by $$N - 1$$ rather than by $$N$$. Because a sample's standard deviation naturally tends to underestimate the true standard deviation of the population, applying this $$N - 1$$ correction provides a more accurate estimate.

Learn Before

Related