Normal distribution, definition

Normal distribution curves showing the definition of detection limit and limit of identification (LOI). The probability of a type 1 error is indicated by the dark shading, and the probability of a type 2 error is indicated by light shading. 

A first approach to the definition of the confidence regions in parameter space follows the linear approximation to the parameter joint distribution that we have already used If the estimates are approximately normally distributed around 9 with dispersion [U. U.] then an approximate 100(1 - a)%... 

A conceptual definition of the follows from consideration of a set of numbers drawn at random from the standard normal distribution, the one with mean zero and standard deviation one. Ordering this set of numbers gives a sequence called order statistics. The are the... 

Long and Winefordner along with several other authors agree on a value of k = 3, which allows a confidence level of 99.86% if the values of xb follow a normal distribution, and 89% if the values of Xb do not follow a normal distribution. A value of k = 2 has also been used by some workers, but this decreases the confidence level in Cl. The definition of LOD was later expanded on by lUPAC in 1995 to include the probabilities of false positives and negatives. 

Assuming that j are normally distributed and uncorrelated, with zero mean and known positive definite covariance matrix the parameter estimation problem can be formulated as minimizing with respect to zj and 0 ... 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

If y1 Y2, and Y3 are normally distributed, the constant probability surfaces are ellipsoids centered at y (Figure 5.12) and the statistical projection y of y will be defined as the point where the plane is tangent to the innermost probability ellipsoid. Points on the same ellipsoid are by definition at the same statistical distance from y. If Sy is the covariance matrix of the vector y, the statistical distance c between y and y is given by... 

Parameter Two distinct definitions for parameter are used. In the first usage (preferred), parameter refers to the constants characterizing the probability density function or cumulative distribution function of a random variable. For example, if the random variable W is known to be normally distributed with mean p and standard deviation o, the constants p and o are called parameters. In the second usage, parameter can be a constant or an independent variable in a mathematical equation or model. For example, in the equation Z = X + 2Y, the independent variables (X, Y) and the constant (2) are all parameters. 

For a log-normal distribution, o-g is still a measure of spread of the distribution, but it has a slightly different definition because In D, rather than D, is assumed to have a normal distribution. For a log-normal distribution o-g is defined by Eq. (E) ... 

A curve with the shape given by Eq. 18-2 is called a normal (or Gaussian) distribution. Usually it is denoted as p (x) where x is the spatial coordinate and a is the standard deviation which characterizes the width of the distribution along the x-axis. The mathematical definition and properties of the normal distribution are presented in Box 18.2. 

W onder of wonders Data that were non-significant are now revealed as significant (P = 0.034). It is usually at about this point that the cynical cry cheat How dare we use this statistical fiddle to convert non-significant results into significant ones Essentially, we need have no qualms about this approach. It is entirely respectable and is definitely superior to the analysis of the original data, because the transformed data are much closer to a normal distribution. The only caveat would be that, if we are... 

Usefulness of the normal distribution curve lies in the fact that from two parameters, the true mean p. and the true standard deviation true mean determines the value on which the bell-shaped curve is centered, and most probability concentrated on values near the mean. It is impossible to find the exact value of the true mean from information provided by a sample. But an interval within which the true mean most likely lies can be found with a definite probability, for example, 0.95 or 0.99. The 95 percent confidence level indicates that while the true mean may or may not lie within the specified interval, the odds are 19 to 1 that it does.f Assuming a normal distribution, the 95 percent limits are x 1.96 where a is the true standard deviation of the sample mean. Thus, if a process gave results that were known to fit a normal distribution curve having a mean of 11.0 and a standard deviation of 0.1, it would be clear firm Fig. 17-1 that there is only a 5 percent chance of a result falling outside the range of 10.804 and 11.196. 

The alpha spectrometry results were also significantly different at a 99% confidence level from the assigned NPL values (which deviations are 0% by definition). Application of the non-parametric Wilcoxon Signed Rank test, which, like the Rank Sum test, does not assume a normal distribution and does not require the removal of outliers, also resulted in a significant difference at a 99% confidence level between the alpha spectrometry results and the assigned NPL values (the absolute z-value being 3.72). 

The agreement between the concentration profiles predicted using the more exact numerical schemes and that obtained by assuming a normal distribution indicates that this definition of resolution is consistent with the other assumptions made in this model. 

The spectral overlap integral J can be expressed in terms of either wavenumbers or wavelengths (Equation 2.36). The area covered by the emission spectrum of D is normalized by definition and the quantities / and lx are the normalized spectral radiant intensities of the donor D expressed in wavenumbers and wavelengths, respectively. Note that the spectral overlap integrals J defined here differ from those relevant for radiative energy transfer (Equation 2.33). Only the spectral distributions of the emission by D /,P and, are normalized, whereas the transition moment for excitation of A enters explicitly by way of the molar absorption coefficient sA. The integrals J" and Jx are equal, because the emission spectrum of D is normalized to unit area and the absorption coefficients sA are equal on both scales. 

Equation 2.62 indicates that 68.3 percent of the total area under the Gaussian is included between m — a and m + a. Another way of expressing this statement is to say that if a series of events follows the normal distribution, then it should be expected that 68.3 percent of the events will be located between m — a and m + O. As discussed later in Sec. 2.13, Eq. 2.62 is the basis for the definition of the standard error. 

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