Statistical definitions Standard deviation

Sample Statistics Many types of sample statistics will be defined. Two very special types are the sample mean, designated as X, and the sample standard deviation, designated as s. These are, by definition, random variables. Parameters like [L and O are not random variables they are fixed constants. 

Mean aud standard deviation The statistical normal curve shows a definite relationship among the mean, the standard deviation, and normal curve. The normal curve is fully defined by the mean, that locates the normal curve, and the standard deviation that describes the shape of the normal curve. A relationship exists between the standard deviation and the area under the curve. 

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... 

Note that in data analysis we divide by n in the definition of standard deviation rather than by the factor n - 1 which is customary in statistical inference. Likewise we can relate the product-moment (or Pearson) coefficient of correlation r (Section 8.3.1) to the scalar product of the vectors (x - x) and (y - y) ... 

Quantification of the limits of detection (LOD), or minimum detectable levels (MDL statistically defined in Section 13.4), is an important part of any analysis. They are used to describe the smallest concentration of each element which can be determined, and will vary from element to element, from matrix to matrix, and from day to day. Any element in a sample which has a value below, or similar to, the limits of detection should be excluded from subsequent interpretation. A generally accepted definition of detection limit is the concentration equal to a signal of twice (95% confidence level) or three times (99% confidence) the standard deviation of the signal produced by the background noise at the position of the peak. In practice, detection limits in ICP-MS are usually based on ten runs of a matrix matched blank and a standard. In this case ... 

Consistent with these definitions, there are two methods for IDL determination. The first method consists of multiple analyses of a reagent blank, followed by the determination of the standard deviation of the responses at the wavelength of the target analyte. The standard deviation multiplied by a factor of three is the IDL. This calculation defines the IDL as an analyte signal that is statistically greater than the noise. 

Unlike Lc and LoD, LoQ is not related to statistical probabilities (a- and /3-error). LoQ is the minimum signal (concentration or amount) that can be quantified. The ability to quantify in analytical chemistry is related to a signal (concentration or amount) that can be readily reproduced. Therefore, the residual standard deviation (RSDq) is included in the definition /q = (1 /RSDq)oq. The IUPAC default value for RSDq is 0.1. Using the default value, one obtains Lq = IOctq. 

Note that the term (n - 1) is used in the denominator of this equation to ensure that 5 is an unbiased estimate of the population standard deviation, a. (There is a general convention in statistics that English letters are used to describe the properties of samples, Greek letters to describe populations). The term (n - 1) is the number of degrees of freedom of the estimate, s. This is because if x is known, it is only necessary to know the values of (n — 1) of the individual measurements, as by definition S(.x,- — x) = 0. The square of the standard deviation, is known as the variance, and is a very important statistic when two or more sources of error are being considered, because of its additivity properties. 

There are several concepts and terms that are essential to discussions about QA, even the concept itself. While at a very detailed level any definition can be challenged as being too narrow or too broad, the definitions presented below are useful (20). It should be noted that these terms are quite recent in definition and are not usually given in statistics books. Other terms like pollution have evolved over hundreds of years (21). Some key terms used in the field like reproducibility, standard deviation, standard error, replicate analysis, blanks, spiked samples and blind samples are self-explanatory. 

There is nothing definitive about the selected number of 20. Quite generally, the estimate of the imprecision improves the more observations that are available. Exact confidence limits for the standard deviation can be derived from the distribution. Estimates of the variance, SD, are distributed according to the distribution (tabulated in most statistics textbooks) (N-l)SDVa X(v-i)j where (N-1) is the degrees of freedom. Then the two-sided 95% confidence interval (Cl) (95% Cl) is derived from the relation ... 

Equation (1) describes the idealized distribution function, obtained from an infinite number of sample measurements, the so-called parent population distribution. In practice we are limited to some finite number, n, of samples taken from the population being examined and the statistics, or estimates, of mean, variance, and standard deviation are denoted then by x, and s respectively. The mathematical definitions for these parameters are given by equations (2H4),... 

Hyperuricemia may be an asymptomatic condition with an increased serum uric acid level as the only apparent abnormality. Statistically, hyperuricemia is defined as serum urate concentrations greater than two standard deviations above the population mean. However, for determination of the risk for gout, hyperuricemia is defined as a supersaturated urate concentration. By this definition, a urate concentration greater than 7.0 mg/dL is abnormal and is associated with an increased risk for gout. This corresponds to a measured value greater than 7.5 mg/dL by most autoanalyzers. 

In the homo-octahedral family, the three M sites are by definition identical in content and size. Any difference in one of the M sites violates the H centering, lowering the symmetry of the O sheet to that of the meso-octahedral family. A difference between the other two M sites destroys also the inversion center and lowers the symmetry of the O sheet to that of the hetero-octahedral family. From the practical viewpoint, differences among the M sites are often small and must be evaluated on statistical grounds. As discussed by Bailey (1984c) for the specific case of micas [cf. an application in Amisano-Canesi et al. (1994)], if o/ is taken as the estimated standard deviation (esd) of an individual quantity and o = Oiln is the esd of the mean of n values, the esd of a difference (A) between two mean values is given by (3 = Two quantities are... 

Instead of trying to use the covariance itself as a standard for comparing the degree of statistical association of different pairs of variables, we apply a scale factor to it, dividing each individual deviation from the average by the standard deviation of the corresponding variable. This results in a sort of normalized covariance, which is called the correlation coefficient of the two variables (Eq. (2.9)). This definition forces the correlation coefficient of any pair of random variables to always be restricted to the [—1,+1] interval. The correlations of different pairs of variables are then measured on the same scale (which is dimensionless, as can be deduced from Eq. (2.9)) and can be compared directly. 

By definition, the standard deviatirai is the root-mean-square deviation about the mean value. It does not provide an indicator of the statistical error about the mean of multiple measurements. If the distribution is unimodal and not too skewed, then the standard deviation will give a reasonable indication of dispersity.. 

A more rigorous definition of uncertainty (Type A) relies on the statistical notion of confidence intervals and the Central Limit Theorem. The confidence interval is based on the calculation of the standard error of the mean, Sx, which is derived from a random sample of the population. The entire population has a mean /x and a variance a. A sample with a random distribution has a sample mean and a sample standard deviation of x and s, respectively. The Central Limit Theorem holds that the standard error of the mean equals the sample standard deviation divided by the square root of the number of samples ... 

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