Standard deviation measures how spread out values are
Standard deviation gauges how tightly values hug their arithmetic mean. Take the square root of average squared gaps from that mean, and you recover a spread figure in the same units as the raw data, unlike variance, which squares those units.
In statistics, standard deviation measures how far values spread from their arithmetic mean—abbreviated SD or σ. It is the square root of variance, the average of squared deviations from the mean, but unlike variance it shares the same units as the data. For a normal bell curve, roughly 68.3 percent of values lie within one standard deviation, 95.4 percent within two, and 99.7 percent within three—the empirical 68–95–99.7 rule.
Consider eight students scoring 2, 4, 4, 4, 5, 5, 7, and 9. The mean is 5. Squaring deviations (9, 1, 1, 1, 0, 0, 4, 16) and averaging yields variance 4, so population standard deviation is 2. When data come from a sample rather than the full population, a modified unbiased estimator may replace the simple formula. Adult US men average about 69 inches tall with a standard deviation near 3 inches—meaning about 68 percent stand between 66 and 72 inches.
Standard deviation differs from standard error: the latter is the spread of sample means if you repeated sampling infinitely, equal to population σ divided by the square root of sample size. Poll "margin of error" reports this expected variability. Scientists often publish both SD (data spread) and SE (estimate uncertainty); effects beyond two standard errors from a null expectation are conventionally deemed statistically significant.
Not every distribution has a finite standard deviation. The normal distribution does despite infinite tails; the Cauchy distribution has neither mean nor standard deviation, and some Pareto distributions have a mean but infinite σ. For discrete weighted data, σ equals the square root of the probability-weighted sum of squared deviations from the expected value.
Source: Standard deviation
