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In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f ( x ) over the interval ( a , b ) is defined by: [ 1 ]
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.
For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year is +60%, then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R that makes ...
A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation. [6] For example, the arithmetic mean of 3 {\displaystyle 3} and 5 {\displaystyle 5} is 3 + 5 2 = 4 {\displaystyle {\frac {3+5}{2}}=4} , or equivalently 3 ⋅ 1 2 + 5 ⋅ 1 2 = 4 ...
For p = 0 and p = ∞ these functions are defined by taking limits, respectively as p → 0 and p → ∞. For p = 0 the limiting values are 0 0 = 0 and a 0 = 0 or a ≠ 0, so the difference becomes simply equality, so the 0-norm counts the number of unequal points. For p = ∞ the largest number dominates, and thus the ∞-norm is the maximum ...
The arithmetic mean of a population, or population mean, is often denoted μ. [2] The sample mean ¯ (the arithmetic mean of a sample of values drawn from the population) makes a good estimator of the population mean, as its expected value is equal to the population mean (that is, it is an unbiased estimator).
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Smoothing of a noisy sine (blue curve) with a moving average (red curve). In statistics, a moving average (rolling average or running average or moving mean [1] or rolling mean) is a calculation to analyze data points by creating a series of averages of different selections of the full data set.