A statistic on the symmetric group is called a weighted inversion statistic if there exists an upper triangular matrix such thatwhere is the characteristic function.The inversion count ( for ) defined by Cramer (1750) and the major index (; otherwise) defined by MacMahon (1913) are both weighted inversion statistics (Degenhardt and Milne).
The trimean is defined to bewhere are the hinges and is the statistical median. Press et al. (1992) call this Tukey's trimean. It is an L-estimate.
The mode of a set of observations is the most commonly occurring value. For example, for a data set (3, 7, 3, 9, 9, 3, 5, 1, 8, 5) (left histogram), the unique mode is 3. Similarly, for a data set (2, 4, 9, 6, 4, 6, 6, 2, 8, 2) (right histogram), there are two modes: 2 and 6. A distribution with a single mode is said to be unimodal. A distribution with more than one mode is said to be bimodal, trimodal, etc., or in general, multimodal. The mode of a set of data is implemented in the Wolfram Language as Commonest[data].An interesting empirical relationship between the sample mean, statistical median, and mode which appears to hold for unimodal curves of moderate asymmetry is given by(Kenney and Keeping 1962, p. 53), which is the basis for the definition of thePearson mode skewness.
The median of a statistical distribution with distribution function is the value such . For a symmetric distribution, it is therefore equal to the mean.Given order statistics , , ..., , , the statistical median of the random sample is defined by(1)(Hogg and Craig 1995, p. 152) and commonly denoted or . The median of a list of data is implemented as Median[list].For a normal population, the mean is the most efficient (in the sense that no other unbiased statistic for estimating can have smaller variance) estimate (Kenney and Keeping 1962, p. 211). The efficiency of the median, measured as the ratio of the variance of the mean to the variance of the median, depends on the sample size as(2)which tends to the value as becomes large (Kenney and Keeping 1962, p. 211). Although, the median is less efficient than the mean, it is less sensitive to outliers than the meanFor large samples with population median ,(3)(4)The median is an L-estimate..
The Spearman rank correlation coefficient, also known as Spearman's rho, is a nonparametric (distribution-free) rank statistic proposed by Spearman in 1904 as a measure of the strength of the associations between two variables (Lehmann and D'Abrera 1998). The Spearman rank correlation coefficient can be used to give an R-estimate, and is a measure of monotone association that is used when the distribution of the data make Pearson's correlation coefficient undesirable or misleading.The Spearman rank correlation coefficient is defined by(1)where is the difference in statistical rank of corresponding variables, and is an approximation to the exact correlation coefficient(2)computed from the original data. Because it uses ranks, the Spearman rank correlation coefficient is much easier to compute.The variance, kurtosisexcess, and higher-order moments are(3)(4)(5)Student was the first to obtain the variance...
Given a sample of variates , ..., , reorder them so that . Then is called the th order statistic (Hogg and Craig 1970, p. 146), sometimes also denoted . Special cases include the minimum(1)and maximum(2)Important functions of order statistics include the statisticalrange(3)midrange(4)and statistical median(5)(Hogg and Craig 1970, p. 152).If has probability density function and distribution function , then the probability function of is given by(6)for , ..., (Rose and Smith 2002, pp. 311 and 454).A robust estimation technique based on linearcombinations of order statistics is called an L-estimate.
The word "median" has several different meanings in mathematics all related to the "middle" of mathematical objects.The statistical median is an order statistic that gives the "middle" value of a sample. More specifically, it is the value such that an equal number of samples are less than and greater than the value (for an odd sample size), or the average of the two central values (for an even sample size). The Wolfram Language function Median[list] can be used to find the statistical median of the elements in a list.A triangle median is the Cevian from one of its vertices to the midpoint of the opposite side. The medians intersect in a point known as the triangle centroid that is sometimes also called the median point.Similarly, a tetrahedron median is a line joining a vertex of a tetrahedron to the geometric centroid of the opposite face.The median of a trapezoid is the line segment determinedby the midpoints of..