Standard Deviation

Definition

The standard deviation measures how far values typically lie from their average, in the same units as the data. A small value means the data cluster tightly around the mean; a large one means they are spread out.

In plain terms

The mean says where the data are centred; the standard deviation says how widely they scatter around it. It is computed by squaring each value’s distance from the mean, averaging those (dividing by n − 1 for a sample), and taking the square root.

For roughly bell-shaped data, about two thirds of values fall within one standard deviation of the mean and about 95% within two, but that rule is a property of the bell shape, not of standard deviations in general. It is the building block of Z-scores and Confidence Intervals.

Example

Weekly sales of two products, both averaging 100:

The averages are identical, but A is almost constant and B swings by about a third either way. A plan that works for A’s steady demand could leave B out of stock or overstocked.

Common misunderstandings

• “The standard deviation is the error of the data.” It describes natural spread, not a mistake.
• “68–95–99.7 always holds.” Only for roughly bell-shaped data.
• “Dividing by n or n − 1 makes no difference.” For small samples it does; use n − 1 when estimating from a sample.