sqrtVar

sqrtVar is the square root of the variance of a random variable, and is commonly known as the standard deviation. It provides a measure of dispersion in the same units as the original data and is widely used to summarize variability in a dataset or model.

Mathematically, if X has mean mu and variance Var(X), then sqrtVar(X) = sqrt(Var(X)) = sigma. The variance is defined as Var(X) = E[(X - mu)^2] for a population, or an appropriate estimator of this quantity from a sample.

For a finite data sample x1, ..., xn, the usual estimator is the sample standard deviation s = sqrt( (1/(n-1)) sum (xi - x_bar)^2 ), where x_bar is the sample mean. If the population variance is known, one can compute sigma using the same formula with the population mean and the population size in the denominator, but this is rarely available in practice.

Key properties include non-negativity (sigma >= 0) and the scaling behavior under affine transformations: sd(aX + b) = |a| sd(X). The standard deviation is not additive; in general sd(X + Y) ≠ sd(X) + sd(Y). It also depends on the scale of the data and can be sensitive to outliers.

Interpretation and usage: sqrtVar quantifies dispersion around the mean and is fundamental for assessing variability, risk, and reliability in statistics and data analysis. In a normal distribution, it is part of the 68-95-99.7 rule, where a large portion of observations fall within a few standard deviations of the mean.

Notes: The term sqrtVar is not standard in formal literature; standard deviation is the conventional term. The sample standard deviation is a biased estimator of the population standard deviation for small samples but becomes unbiased in expectation only in specific asymptotic contexts.