Standard deviation measures dispersion in the identical physical units as the original data observations.
It is defined mathematically as the positive square root of the variance.
By taking the square root, the squared units introduced during variance calculation return to baseline measurements.
For example, if variance equals sixteen squared seconds, the standard deviation is exactly four seconds.
In mathematical notation, the Greek letter sigma denotes population standard deviation, while the letter s denotes sample standard deviation.
A small standard deviation indicates that data points cluster tightly around the arithmetic mean.
A large standard deviation indicates that data points scatter widely across a broad range of values.
In a symmetric bell-shaped normal distribution, standard deviation provides precise probabilistic coverage rules.
Approximately sixty-eight percent of all observations fall within one standard deviation of the mean.
Roughly 95 percent fall within 2 standard deviations, and 99.7 percent fall within 3.
This empirical rule allows analysts to quickly flag observations that sit more than three standard deviations away as potential anomalies.
Standard deviation is also used to compute z-scores, which express individual measurements as standardised distances from the mean.
However, standard deviation remains sensitive to extreme outliers because it derives directly from squared deviations.
When distributions are severely skewed, standard deviation can mislead, and robust alternatives like interquartile range are preferred.
A common error is comparing raw standard deviations across variables with different scales without adjusting for their average magnitudes.