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An inequality on location and ... - Wikipedia, the free encyclopedia
For probability distributions having an expected value and a median, the mean (i.e., the expected value) and the median can never differ from each other by more than one standard deviation.
en.wikipedia.org/wiki/An_inequality_on_location_and_sca... en.wikipedia.org/wiki/An_inequality_on_location_and_scale_parameters |
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The median of an exponential distribution with parameter λ is the scale parameter times the natural log of 2, λln 2. See an inequality on location and scale
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The article "An inequality on location and scale parameters" is part of the Wikipedia encyclopedia. It is licensed under the terms of the GNU FDL.
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But this inequality is trivially true if the variance is infinite.) This proof uses Jensen's inequality twice. We have...
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In probability theory and statistics, a median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value...
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, the location parameter. The median of an exponential distribution with parameter <math>\lambda</math> is the natural log of 2 divided by the scale parameter: <math>\frac{\ln 2}{\lambda}</math>.
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Whereas the mean is a way to describe the location of a distribution, the variance is a way to capture its scale or degree of being spread out.
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In plain language, it can be expressed as "The average of the square of the distances from every data points to the mean". In practice, when dealing with large populations, it is almost never possible to find the exact value of the population variance, due to time, cost, and other resource constraints.
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variance, Variance, Variance - Definition, Variance - Generalizations, Variance - History, Variance - Moment of inertia, Variance - Population variance and sample variance, Variance - Properties, Variance - An unbiased estimator, expected value, standard deviation, skewness, kurtosis, statistical dispersion, an inequality...
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Chebyshev's inequality proves that in any data set, nearly all of the values will be nearer to the mean value, where the meaning of "close to" is specified by the standard deviation. Chebyshev's inequality entails that for (nearly) all random distributions, not just normal ones, we have the following weaker bounds:
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