standard deviation
Variance answers 'how spread out?' but in awkward squared units — squared dollars, squared seconds — which are hard to feel. Standard deviation fixes this by taking the square root, returning a spread measure in the same units as the data itself, so it can sit right next to the mean and be read as a typical distance from it.
Formally, the standard deviation sigma = sqrt(Var(X)). It is, roughly, the typical amount by which an outcome differs from the mean. If heights have mean 170 cm and standard deviation 8 cm, then 'most' people are within a band of about 8 cm of the average — and for the familiar bell-shaped normal distribution this intuition is sharp: about 68 percent of values fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three (the empirical rule).
Standard deviation is the everyday language of variability: error bars, volatility in finance, measurement uncertainty, and the z-score (how many standard deviations a value sits from the mean) all speak in units of sigma. Two cautions: the 68-95-99.7 figures are exact only for the normal distribution and merely a rough guide elsewhere; and since it descends from variance, a distribution with no finite variance has no standard deviation either.
Exam scores have mean 72 and standard deviation 10. A score of 92 is (92 - 72)/10 = 2 standard deviations above the mean — its z-score is 2. If scores are roughly normal, only about 2.5 percent of students score that high or higher.
Standard deviation puts spread back in the data's own units — a typical distance from the mean.
The 68-95-99.7 rule is exact only for the normal distribution; for skewed or heavy-tailed data it can be badly off, and a distribution with infinite variance has no standard deviation.