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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.
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 unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n − 1.5 yields an almost unbiased estimator. The unbiased sample variance is a U-statistic for the function ƒ ( y 1 , y 2 ) = ( y 1 − y 2 ) 2 /2, meaning that it is obtained by averaging a 2-sample statistic ...
The demonstration of the t and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models , including linear regression and analysis of variance .
If is a standard normal deviate, then = + will have a normal distribution with expected value and standard deviation . This is equivalent to saying that the standard normal distribution Z {\textstyle Z} can be scaled/stretched by a factor of σ {\textstyle \sigma } and shifted by μ {\textstyle \mu } to yield a different normal distribution ...
where X is a random variable which we have sampled N times, m is the sample mean, k is a constant and s is the sample standard deviation. This inequality holds even when the population moments do not exist, and when the sample is only weakly exchangeably distributed; this criterion is met for randomised sampling.
For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases. If the uncertainties are correlated then covariance must be taken into account ...
Important examples include the sample variance and sample standard deviation. Without Bessel's correction (that is, when using the sample size n {\displaystyle n} instead of the degrees of freedom n − 1 {\displaystyle n-1} ), these are both negatively biased but consistent estimators.