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Bias in standard deviation for autocorrelated data. The figure shows the ratio of the estimated standard deviation to its known value (which can be calculated analytically for this digital filter), for several settings of α as a function of sample size n. Changing α alters the variance reduction ratio of the filter, which is known to be
The mean and the standard deviation of a set of data are descriptive statistics usually reported together. In a certain sense, the standard deviation is a "natural" measure of statistical dispersion if the center of the data is measured about the mean. This is because the standard deviation from the mean is smaller than from any other point.
The standard deviation and the expected absolute deviation can both be used as an indicator of the "spread" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics ...
This correction is so common that the term "sample variance" and "sample standard deviation" are frequently used to mean the corrected estimators (unbiased sample variation, less biased sample standard deviation), using n − 1. However caution is needed: some calculators and software packages may provide for both or only the more unusual ...
If we repeat this procedure many times, the average value of (y − y ′) 2 is equal to twice the Allan variance (or Allan deviation squared) for observation time τ. The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers.
This algorithm can easily be adapted to compute the variance of a finite population: simply divide by n instead of n − 1 on the last line.. Because SumSq and (Sum×Sum)/n can be very similar numbers, cancellation can lead to the precision of the result to be much less than the inherent precision of the floating-point arithmetic used to perform the computation.
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
Variance (the square of the standard deviation) – location-invariant but not linear in scale. Variance-to-mean ratio – mostly used for count data when the term coefficient of dispersion is used and when this ratio is dimensionless, as count data are themselves dimensionless, not otherwise. Some measures of dispersion have specialized purposes.