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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}}.}
The mean (L 2 center) and midrange (L ∞ center) are unique (when they exist), while the median (L 1 center) and mode (L 0 center) are not in general unique. This can be understood in terms of convexity of the associated functions (coercive functions).
The two are complementary in sense that if one knows the midhinge and the IQR, one can find the first and third quartiles. The use of the term hinge for the lower or upper quartiles derives from John Tukey 's work on exploratory data analysis in the late 1970s, [ 1 ] and midhinge is a fairly modern term dating from around that time.
For n independent and identically distributed discrete random variables X 1, X 2, ..., X n with cumulative distribution function G(x) and probability mass function g(x) the range of the X i is the range of a sample of size n from a population with distribution function G(x).
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.
Robust measures of scale can be used as estimators of properties of the population, either for parameter estimation or as estimators of their own expected value.. For example, robust estimators of scale are used to estimate the population standard deviation, generally by multiplying by a scale factor to make it an unbiased consistent estimator; see scale parameter: estimation.
The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If the data set has an odd number of observations, the middle one is selected (after arranging in ascending order).
Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal N(0,σ 2) Population. In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. [1]