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Using words, the standard deviation is the square root of the variance of X. The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation.
Probability theory. In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while ...
In statistics, the 68–95–99.7 rule, also known as the empirical rule, and sometimes abbreviated 3sr, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: approximately 68%, 95%, and 99.7% of the values lie within one, two, and three standard deviations of the mean, respectively.
Full width at half maximum. In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value. In other words, it is the width of a spectrum curve measured between those points on the y -axis which are half the maximum ...
when the probability distribution of the value is known, it can be used to calculate an exact confidence interval; when the probability distribution is unknown, Chebyshev's or the Vysochanskiï–Petunin inequalities can be used to calculate a conservative confidence interval; and
That is, if CEP is n metres, 50% of shots land within n metres of the mean impact, 43.7% between n and 2n, and 6.1% between 2n and 3n metres, and the proportion of shots that land farther than three times the CEP from the mean is only 0.2%. CEP is not a good measure of accuracy when this distribution behavior is not met.
Standard normal table. In statistics, a standard normal table, also called the unit normal table or Z table, [1] is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal ...
The 1/e 2 width is important in the mathematics of Gaussian beams, in which the intensity profile is described by () = (). The American National Standard Z136.1-2007 for Safe Use of Lasers (p. 6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0 ...