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In statistics, the mid-range or mid-extreme is a measure of central tendency of a sample defined as the arithmetic mean of the maximum and minimum values of the data set: [1] M = max x + min x 2 . {\displaystyle M={\frac {\max x+\min x}{2}}.}
L-estimators are often much more robust than maximally efficient conventional methods – the median is maximally statistically resistant, having a 50% breakdown point, and the X% trimmed mid-range has an X% breakdown point, while the sample mean (which is maximally efficient) is minimally robust, breaking down for a single outlier.
In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
In some circumstances, mathematicians may calculate a mean of an infinite (or even an uncountable) set of values. This can happen when calculating the mean value of a function (). Intuitively, a mean of a function can be thought of as calculating the area under a section of a curve, and then dividing by the length of that section.
The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed.
The range, T, has the cumulative distribution function [3] [4] = [(+) ()].Gumbel notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express G(x + t) by G(x), and that the numerical integration is lengthy and tiresome."
L ∞ norm statistics: the mid-range minimizes the maximum absolute deviation trimmed L ∞ norm statistics: for example, the midhinge (average of first and third quartiles ) which minimizes the median absolute deviation of the whole distribution, also minimizes the maximum absolute deviation of the distribution after the top and bottom 25% ...
In both cases, the resulting formula reduces to dividing the total distance by the total time.) However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the inverse of speed.