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Absolute deviation in statistics is a metric that measures the overall difference between individual data points and a central value, typically the mean or median of a dataset. It is determined by taking the absolute value of the difference between each data point and the central value and then averaging these absolute differences. [4]
In statistics, deviance is a goodness-of-fit statistic for a statistical model; it is often used for statistical hypothesis testing. It is a generalization of the idea of using the sum of squares of residuals (SSR) in ordinary least squares to cases where model-fitting is achieved by maximum likelihood .
It is remarkable that the sum of squares of the residuals and the sample mean can be shown to be independent of each other, using, e.g. Basu's theorem.That fact, and the normal and chi-squared distributions given above form the basis of calculations involving the t-statistic:
The reason for this choice is that not only is the deviance a good measure of the goodness of fit, it is also approximately chi-squared distributed, with the approximation improving as the number of data points (K) increases, becoming exactly chi-square distributed in the limit of an infinite number of data points.
In the situation where data is available for k different treatment groups having size n i where i varies from 1 to k, then it is assumed that the expected mean of each group is E ( μ i ) = μ + T i {\displaystyle \operatorname {E} (\mu _{i})=\mu +T_{i}}
Analogously to how the median generalizes to the geometric median (GM) in multivariate data, MAD can be generalized to the median of distances to GM (MADGM) in n dimensions. This is done by replacing the absolute differences in one dimension by Euclidean distances of the data points to the geometric median in n dimensions. [5]
The distribution above is sometimes referred to as the tau distribution; [2] it was first derived by Thompson in 1935. [3] When ν = 3, the internally studentized residuals are uniformly distributed between and +. If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't ...
A large standard deviation indicates that the data points can spread far from the mean and a small standard deviation indicates that they are clustered closely around the mean. For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. Their standard deviations are 7, 5, and 1, respectively.