Standard deviation
lastmod 2023-10-26created 2023-10-265 edits
The standard deviation (SD) is a number that indicates the degree of dispersion of a statistical population or the spread of data, defined as the non-negative square root of the variance, that is, the square root of the variance. The smaller the standard deviation, the closer the values are to the mean. In statistics and probability, it is mainly applied to probability distributions, random variables, or measured populations or multisets. By convention, populations are written with Greek letters and samples with Latin letters, so the population standard deviation is denoted \(\sigma\) (sigma) and the sample standard deviation \(s\).
The deviation is an observed value minus the mean or median.
The variance is obtained by squaring each observed value minus the mean, summing them all, and dividing by the total count. In other words, it is the mean of the squared differences. Since the deviations (observed values minus the mean) sum to 0, they are squared before summing.
The standard deviation is the square root of the variance. Starting from the variance, which is the mean of the sum of squares (SS, sum of square) of the deviations, it is obtained by taking the square root again to return to the original units.
The population standard deviation \(\sigma\) is the standard deviation of a population. It is obtained by taking the square root of the population variance \(\sigma^{2}\).
The sample standard deviation \(s\) is the standard deviation of a sample. It is obtained by taking the square root of the sample variance \(s^{2}\).
Meaning of the value
Because the variance is too large a number, it is scaled down appropriately with a square root, and this is called the standard deviation. (That is, standard deviation^2 = variance)
- A large standard deviation means the values are scattered all over the place.
- A small standard deviation means the values are mostly similar and close to one another.