A typical distance from the mean
Standard deviation answers: how far do values typically sit from their average? It does so in the data's own units (minutes, marks, grams), which is why it survives when fancier measures are forgotten. Build it in four moves: find the mean; subtract it from every value (the deviations); square those, average them (the variance); take the square root to come home to the original units.
Dataset: 2, 4, 4, 4, 5, 5, 7, 9. Mean 5. Deviations −3, −1, −1, −1, 0, 0, 2, 4. Squares: 9, 1, 1, 1, 0, 0, 4, 16, summing to 32. As a population: variance 32 / 8 = 4, SD = 2. Read it as a sentence: values typically sit about 2 units from the mean of 5. Most of this dataset sits within 3 to 7, and the rule holds.
Why the squaring and the root?
Deviations sum to zero always (the mean is the balance point), so their plain average is useless. Absolute values would work, and the mean absolute deviation exists, but squares have better mathematics: they weight large misses more (usually what you want), they connect SD to the normal curve's geometry, and variances of independent quantities add. The square root at the end is what returns the answer to minutes and marks instead of minutes-squared.
Reading SD as the average absolute distance. It is close in spirit but not equal: the root-mean-square construction makes SD a bit larger than the mean absolute deviation whenever values vary in size (here 2.0 against an MAD of 1.5). Quote it as a typical distance, not a literal average of distances.
SD and the bell curve
For roughly normal data, SD unlocks the empirical rule: about 68% of values fall within one SD of the mean, 95% within two, 99.7% within three. Commute times (mean 28.15, SD 4.04) obey it closely: 65% and 95% in the actual data. On skewed data the rule frays, and quartiles take over the description. Either way, "mean ± SD" is the two-number summary of a dataset, and this is the statistic doing the second half of that job.
Frequently asked questions
What does a standard deviation of 0 mean?
All values are identical: no spread, nothing to measure. It happens with constant data and with a single observation (where the sample SD is undefined rather than 0, having no deviations to learn from).
Can standard deviation be negative?
No. It is a square root of an average of squares. The smallest possible value is 0, for constant data.
What is a large standard deviation?
Only relative to the mean and the field: the coefficient of variation (SD / mean) makes it comparable. An SD of 4 minutes on a 28-minute commute (CV 0.14) is tight; the same SD on a 6-minute trip (CV 0.67) is chaos.
Why do reports write mean plus-minus SD?
It compresses centre and spread into one phrase: 28.15 plus-minus 4.04 minutes tells you the typical value and the typical wobble around it. For skewed data, median with quartiles is the better compression.