Precision and standard deviation

Measuring the Precision and Standard Deviation of the Methods (Y ouden/Steiner)... 

Note that for the calculations of precision and standard deviation (equations 38-1 through 38-4), the numerator expression is given as 2(n — 1). This expression is used due to the 2 times error contribution from independent errors found in each independent set (i.e., X and Y) of results. 

Ensure that quantitation yields accurate and precise results by monitoring the background, recoveries and standard deviations. 

The accuracy and precision of the analytical methods were determined by the average and standard deviation of individual method recoveries of the fortitied-control samples in 50 different matrices (see Tables 1 and 2). These methods were also demonstrated to be very rugged based on the results of accuracy and precision for a variety of crop and animal matrices. 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

Quantification Precision can be quantified by suitable dispersion characteristics. It is proposed to characterize precision by standard deviation, see Eqs. (4.12)-(4.14) and relative standard deviation, see Eq. (4.15) (Fleming et al. [1996b] Prichard et al. [2001]). Because of some uncertainty, the characterization of analytical proceedings in their hierarchy (see Fig. 7.1) and of analytical results, respectively, will be considered in detail. 

By making replicate analytical measurements one may estimate the certainty of the analyte concentration using a computation of the confidence limits. As an example, given five replicate measurement results as 5.30%, 5.44%, 5.78%, 5.00%, and 5.30%. The precision (or standard deviation) is computed using equation 73-1,... 

The reproducibilities of both the peak heights and peak areas are shown for replicate injections of a 100 ng mC solution in Fig. 5.9. The results reveal that the injection volume has no significant effect on precision for the standard solution, providing the volume does not exceed 200 pi. Peak area generally offers greater precision (relative standard deviation 2%) at all injection volumes. As the injection volume is decreased, so the width at halfheight decreases. 

The insectivorous bird assessment can be compared to a more traditional probabilistic assessment based on precise distribution functions and particular dependence assumptions. For comparison purposes, we conducted such a simulation. The variable BW was modeled by the same normal distribution with mean 14.5 g and standard deviation 3 g. The variable FIR, on the other hand, was modeled by a log-normal distribution with mean 5.23 and variance 2.3 g per day. The choice of... 

Why is any of this of interest If it is known that some data are normally distributed and one can estimate p and a, then it is possible to state, for example, the probability of finding any particular result (value and uncertainty range) the probability that future measurements on the same system would give results above a certain value and whether the precision of the measurement is lit for purpose. Data are normally distributed if the only effects that cause variation in the result are random. Random processes are so ubiquitous that they can never be eliminated. However, an analyst might aspire to reducing the standard deviation to a minimum, and by knowing the mean and standard deviation predict their effects on the results. 

In the case where duplicate measurements are made and there is no other estimate of the measurement precision, the standard deviation of the effect is calculated from the differences of the pairs of measurements (dj)... 

Precision The average and standard deviation for the individual and total... 

Figure 21.1—Graphical illustration of the data presented in Table 21.1.To illustrate accuracy and precision, the standard deviations are calculated according to equation (21.8). Variations are indicated on the graph for arithmetic averages by a vertical bar and for the corresponding median values by arrows. For the results from chemist 3, the deviation between the mean and median is large. Chemist 1 has probably committed a systematic error. On the right is the classical target illustration of precision and accuracy. This image appears simpler than it is because there are uncertainties in x and y.

The repeatability (precision) of the technique was determined by analyzing 139 urine samples in duplicate. Values ranged from 0.06 to 13.86 mg/24 hr. The mean and standard deviation were 1.597 0.022 mg/24 hr. 

In 1959, four independent and simultaneous reports defined the problem, van Bekkum et al. (1959), not distinguishing among the ordinary and electrophilic side-chain reactions, showed the o--constants to be broadly variable. They tabulated the a-parameters for twelve groups as derived from each reaction of an extended series. Just as illustrated in Fig. 50 for the p-methoxy, p-methyl, and p-chloro substituents, they detected extensive variations in the apparent values of the constants. The authors concluded that the postulation of two or three sets of -constants could not be regarded as sound. By eliminating all those reactions involving important resonance contributions Wepster and his associates (van Bekkum et al., 1959) obtained a series of reactions similar to those presented in Table 23, and hence derived a series of normal, an, parameters for meta and para substituents. These normal values, presumably representing the reactions for which the Hammett relationship is most precise, exhibit standard deviations of 10-30%. 

Thus, control charts measure both the precision and accuracy of the test method. A control chart is prepared by spiking a known amount of the analyte of interest into 4 to 6 portions of reagent grade water. The recoveries are measured and the average recovery and standard deviation are calculated. In routine analysis, one sample in a batch is spiked with a known concentration of a standard and the percent spike recovery is measured. An average of 10 to 20 such recoveries are calculated and the standard deviation about this mean value is determined. The spike recoveries are plotted against the frequency of analysis or the number of days. A typical control chart is shown below in Figure 1.2.2. 

The stability of the values of bond order is even more striking. In the described data set the values of the bond orders span the range from 0.929 to 0.992 with an average of 0.974 and standard deviation of 0.017. This corresponds to the precision of 1.7%. Of course the high stability is explained by the validity of the above limit, which in its turn is due to the fact that the difference between one- and two-center electron-electron repulsion integrals (Aym) at interatomic separations characteristic of chemical bonding is much smaller than the resonance interaction at the same distance. The most important reason for the stability (i.e. of transferability) of the bond orders is that they deviate from the ideally transferable value in the second order in two small parameters < and i. 

Detection limits at or below 1 ppt (1 pg/mL) are routinely attainable for many elements by ICP-MS as long as sources of contamination and reagent purity are carefully controlled. Detection limits as low as 10 ppq (10 fg/mL) are attainable in some cases. A linear dynamic range of up to 108 can be provided by ICP-MS. Short-term precision (relative standard deviation) of 1% to 3% is typical for clean samples. Long-term precision (relative standard deviation) of 5% or better over 8 hours is common for clean samples. Spectral overlaps, discussed previously, can... 

If A is larger than 20, provides a satisfactory estimate of a. Note that the estimated standard deviation 5is a finite sample standard deviation and the so-called standard deviation a is more precisely the standard deviation of an infinite sample set. 

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