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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.
Definitions of other symbols: ... = standard deviation of differences = Chi-squared statistic ^ = = sample proportion, unless specified otherwise ...
Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics , the definition of variance is either the expected value of the SDM (when considering a theoretical distribution ) or its average value (for actual experimental data).
For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is √ 2.9 ≈ 1.7, slightly larger than the expected absolute deviation of 1.5.
Standard deviation is a widely used measure of the spread or dispersion of a dataset. It quantifies the average amount of variation or deviation of individual data points from the mean of the dataset. It uses squared deviations, and has desirable properties. Standard deviation is sensitive to extreme values, making it not robust. [7]
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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 ...
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).