Standard deviation
The values above the mean and below it cancel when you add their distances. How do you still get one honest number for how spread out the data are?
Where you are 3 inside
What this area is about
The standard deviation is the typical distance of a value from the mean, in the data’s own units. Signed distances from the mean always sum to 0, so they are squared before averaging, which gives the variance; its square root brings the units back. For a sample, \displaystyle s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}.
How these fit together
The definition. Variance of a data set is the average squared distance from the mean, and explains why squaring is the fix for the cancelling.
Procedures. Computing a standard deviation works through distances, squares, average and root, and says when to divide by n (a whole population) and when by n - 1 (a sample). Interpreting a standard deviation turns the number into a sentence in the setting’s units: what a typical value looks like, which group varies more, and when one value is far out.
A path through it
Variance first, then the computation, then interpretation.
The standard deviation is the yardstick of most later statistics: it sets the scale of the normal distribution, shows how shifting and rescaling affect spread, and is built into correlation.