Statistical definitions Confidence interval

The minimum detectable level, or detection limit, is defined as that concentration of a particular element which produces an analytical signal equal to twice the square root of the background above the background. It is a statistically defined term, and is a measure of the lower limit of detection for any element in the analytical process. (This definition corresponds to the 95% confidence interval, which is adequate for most purposes, but higher levels, such as 99% can be defined by using a multiplier of three rather than two.) It will vary from element to element, from machine to machine, and from day to day. It should be calculated explicitly for every element each time an analysis is performed. 

Fig. 23. Statistical definition of the limit of determination by the confidence interval in the analytical result98)...

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 ... 

By definition, the experimental unit is the smallest unit randomly allocated to a distinct level of a treatment factor. Note that if there is no randomization, there is no experimental unit and (in nearly all cases) no experiment. Although it is possible to perform experiments without randomization, it is difficult to do well, and risky unless the experimental system is very well understood (7). Randomization is important for several reasons. Randomization changes the sources of bias into sources of variation in general, a noisy assay is better than a biased assay. Further, randomization allows estimates of variation to represent variation in the population this in turn justifies statistical inference (standard errors, confidence intervals, etc.). A common practice in cell-culture bioassay is to rotate among a small collection of layouts rather than use random allocation. Whereas rotation among a collection of layouts is certainly better than a fixed layout, it is both possible and practical to use carefully structured randomization on a routine basis, particularly when using a robot. 

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 ... 

If systematic errors are present but not known, the computed statistical characteristics provide an incorrect information on confidence intervals such as (9.3.56). Observe that for example with (9.5.12), the corresponding covariance matrix equals by definition... 

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