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If a data distribution is approximately normal then about 68 percent of the data values are within one standard deviation of the mean (mathematically, μ ± σ, where μ is the arithmetic mean), about 95 percent are within two standard deviations (μ ± 2σ), and about 99.7 percent lie within three standard deviations (μ ± 3σ).
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.
About 68% of values drawn from a normal distribution are within one standard deviation σ from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. [6] This fact is known as the 68–95–99.7 (empirical) rule, or the 3-sigma rule.
The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix is positive definite. In this case the distribution has density [5] where is a real k -dimensional column vector and is the determinant of , also known as the generalized variance.
Normal curve equivalent (NCE): A normalized standardized score with a mean of 50 and a standard deviation of 21.06 resulting in a near equal interval scale from 0 to 99. The NCE was developed by RMC Research Corporation in 1976 to measure the effectiveness of the Title I Program across the United States and is often used to measure gains over time.
Under the assumption of normality of returns, an active risk of x per cent would mean that approximately 2/3 of the portfolio's active returns (one standard deviation from the mean) can be expected to fall between +x and -x per cent of the mean excess return and about 95% of the portfolio's active returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per ...
Unbiased estimation of standard deviation. In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.