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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid. More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to ), it can be defined on a convex space, not only a vector space.
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
Notice that the a-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if + + =. In the general case, one can consider instead [] / (+ +), which is called a Muirhead mean. [1] Examples
the arithmetic mean of the first and third quartiles. Quasi-arithmetic mean A generalization of the generalized mean, specified by a continuous injective function. Trimean the weighted arithmetic mean of the median and two quartiles. Winsorized mean an arithmetic mean in which extreme values are replaced by values closer to the median.
The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values. ¯ = = + assuming the values have been ordered, so is simply a specific example of a weighted mean for a specific set of weights.
The weighted arithmetic mean is similar ... An alternative perspective for this problem is that of some arbitrary ... For example, consider the weighted mean of the ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.