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Since the arithmetic mean is not always appropriate for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles: Convert all angles to corresponding points on the unit circle , e.g., α {\displaystyle \alpha } to ( cos α , sin α ) {\displaystyle (\cos \alpha ,\sin \alpha )} .
The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean. When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data ...
There are associated concepts, such as the DRMS (distance root mean square), which is the square root of the average squared distance error, a form of the standard deviation. Another is the R95, which is the radius of the circle where 95% of the values would fall, a 95% confidence interval .
The first pass determines the circular mean of those values, while the second pass determines the variance. This double-pass method requires access to all values. There is also a single-pass method for calculating the standard deviation, but this method is unsuitable for angular data such as wind direction. Trying to calculate angular moments ...
In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy , S. Ramanujan , and J. E. Littlewood , who developed it in a series of papers on Waring's problem .
In probability and statistics, a circular distribution or polar distribution is a probability distribution of a random variable whose values are angles, usually taken to be in the range [0, 2π). [1] A circular distribution is often a continuous probability distribution , and hence has a probability density , but such distributions can also be ...
The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant ...
A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.