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The skewness value can be positive, zero, negative, or undefined. For a unimodal distribution (a distribution with a single peak), negative skew commonly indicates that the tail is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness ...
Real skew-symmetric matrices are normal matrices (they commute with their adjoints) and are thus subject to the spectral theorem, which states that any real skew-symmetric matrix can be diagonalized by a unitary matrix. Since the eigenvalues of a real skew-symmetric matrix are imaginary, it is not possible to diagonalize one by a real matrix.
For a large class of unimodal distributions that are positively skewed the mode, median and mean fall in that order. [41] Conversely for a large class of unimodal distributions that are negatively skewed the mean is less than the median which in turn is less than the mode. In symbols for these positively skewed unimodal distributions
Thus the skewed generalized t distribution can be highly skewed as well as symmetric. If − 1 < λ < 0 {\displaystyle -1<\lambda <0} , then the distribution is negatively skewed. If 0 < λ < 1 {\displaystyle 0<\lambda <1} , then the distribution is positively skewed.
It is customary to transform data logarithmically to fit symmetrical distributions (like the normal and logistic) to data obeying a distribution that is positively skewed (i.e. skew to the right, with mean > mode, and with a right hand tail that is longer than the left hand tail), see lognormal distribution and the loglogistic distribution. A ...
Roughly speaking, a distribution has positive skew (right-skewed) if the higher tail is longer, and negative skew (left-skewed) if the lower tail is longer. Perfectly symmetrical distributions always have zero skewness, though zero skewness does not necessarily imply a symmetrical distribution.
The accompanying plot of skewness as a function of variance and mean shows that maximum variance (1/4) is coupled with zero skewness and the symmetry condition (μ = 1/2), and that maximum skewness (positive or negative infinity) occurs when the mean is located at one end or the other, so that the "mass" of the probability distribution is ...
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [ 1 ] represented in an orthonormal basis over a real inner product space .