Standard deviation

Standard deviation is a measure of dispersion: it summarizes the extent to which numerical observations vary around their arithmetic mean. It is the positive square root of variance, which is calculated from squared distances between observations and the mean. Taking the square root restores the measure to the original unit of the data—for example, centimetres rather than square centimetres. A small standard deviation indicates values concentrated relatively close to the mean; a larger one indicates greater spread. It is a root-mean-square measure of distance from the mean, not the average absolute distance.

For a population of N values x₁, …, xₙ with mean μ, the population standard deviation is σ = √[Σ(xᵢ − μ)² / N]. Each value's difference from the mean is squared, the squared differences are averaged, and the square root of that average is taken. For example, the values 2, 4 and 6 have mean 4 and squared deviations 4, 0 and 4. If these three values are the entire population, the standard deviation is √(8/3), or about 1.63. The calculation makes a dataset's overall spread easier to describe with one number, but it does not show where individual values fall.

The denominator depends on what the data represent. When the observations are the whole population of interest, the population variance divides the sum of squared deviations by N; its square root is the population standard deviation. When n observations are treated as a sample from a larger population, the commonly used sample variance divides the sum of squared deviations from the sample mean by n − 1 before taking the square root. The n − 1 divisor accounts for estimating the mean from the sample. For the example 2, 4, 6, treating the numbers as a sample gives a sample standard deviation of √(8/2) = 2.

Standard deviation is widely used in descriptive statistics and in methods that compare data with a mean or quantify uncertainty. In a normal distribution, it helps describe how widely values are spread around the centre; the mean and standard deviation together determine the location and scale of that distribution. This interpretation should not be applied automatically to every dataset: standard deviation alone does not establish that data are normally distributed, nor does it describe skewness or the shape of the tails. Because deviations are squared, distant observations can have a strong effect. For skewed data or distributions with long tails, the median absolute deviation or interquartile range can provide a less outlier-sensitive complement. The National Institute of Standards and Technology recommends considering the distribution's shape when choosing a measure of scale.