Standard deviation and spread

In short: The standard deviation measures how strongly values typically scatter around their mean. A small standard deviation means values lie close together.

In more detail: Two data sets can have the same mean and differ greatly: 49, 50, 51 versus 0, 50, 100. Only the spread shows the difference.

In Depth

• Calculation: Square each value’s deviation from the mean, average (variance), take the square root. For samples one divides by n−1 instead of n.
• Unit: The standard deviation has the same unit as the data, the variance its square.
• Normal distribution: About 68 % of values lie within one standard deviation of the mean, 95 % within two, 99.7 % within three (normal distribution).
• Other measures: Range (maximum minus minimum), interquartile range (more robust against outliers), coefficient of variation (spread relative to the mean).
• Applications: Measurement precision, quality control (Six Sigma), risk in investing (volatility), measurements in experiments, normalising data in AI.
• Pitfall: With skewed distributions or outliers the standard deviation says little. Then prefer quartiles.

See also: mean, median and mode