Identity measurements

In analytical chemistry, a number of identical measurements are taken and then an error is estimated by computing the standard deviation. With computational experiments, repeating the same step should always give exactly the same result, with the exception of Monte Carlo techniques. An error is estimated by comparing a number of similar computations to the experimental answers or much more rigorous computations. 

Repeated measurements of the same measurand on a series of identical measuring samples result in random variations (random errors), even under carefully controlled constant experimental conditions. These should include the same operator, same apparatus, same laboratory, and short interval of the time between measurements. Conditions such as these are called repeatability conditions (Prichard et al. [2001]). The random variations are caused by measurement-related technical facts (e.g., noise of radiation and voltage sources), sample properties (e.g., inhomogeneities), as well as chemical or physical procedure-specific effects. 

The RM of the dried product was measured at 50 °C over P205 or in an oven with circulating air at 50 °C, or in the same oven at 90 °C over silica gel. Identical measurements were made with fresh bones. For NMR measurements, a known amount of D20 was added to the bone in a glass container. After equilibrium between DzO and H20 was reached, a known amount of the product was taken from the solution and studied in a Perkin Elmer NMR-spectrometer. In Fig. 3.23 the water contents of fresh and freeze dried bones are listed measured by NMR and the gravimetric methods at 90 °C. The data show that only a certain amount of the total water can be removed at 90 °C, while another amount is so... 

Deviation. How much each measurement differs from the mean is an important number and is called the deviation. A deviation is associated with each measurement, and if a given deviation is large compared to others in a series of identical measurements, the proverbial red flag is raised. Such a measurement is called an outlier. Mathematically, the deviation is calculated as follows ... 

Fig. 15. Comparison of a water suppressed muscle spectrum and a spectrum from yellow bone marrow containing almost pure fat (triglycerides). Measurement parameters STEAM sequence, TE=10 ms, TM=15 ms, TR = 2 s, 40 acq., VOI (11 X 11 X 20) mm. (a) Spectrum from TA muscle recorded after careful positioning of the VOI, avoiding inclusion of macroscopic fatty septa allows separation of extramyocellular (EMCL, broken lines) and intramyocellular lipid signals (IMCL, dotted lines) based on susceptibility differences. For this reason characteristic signals from fatty acids occur double. Signals of creatine (methyl, Crs, and methylene, Cr2) show triplet and doublet structure, respectively, due to dipolar coupling effects. Further signals of TMA (including carnitine and choline compartments), Taurine (Tau), esters, unsaturated fatty acids (-HC=CH-), and residual water are indicated, (b) Spectrum from yellow fatty bone marrow of the tibia with identical measuring parameters, but different amplitude scale.

Median For this same series of identical measurements on a sample, the "middle" value is sometimes important and is called the median. If the total number of measurements is an even number, there is no single "middle" value. In this case, the median is the average of two "middle" values. For a large number of measurements, the mean and the median should be the same number. 

Mode The value that occurs most frequently in the series is called the mode. Ideally, for a large number of identical measurements, the mean, median, and mode should be the same. However, this rarely occurs in practice. If there is no value that occurs more than once, or if there are two values that equally occur most frequently, then there is no mode. 

A measure of the scatter or variability of data is the variance, as discussed earlier. We have seen that a large variance produces broad-interval estimates of the mean. Conversely, a small variability, as indicated by a small value of variance, produces narrow interval estimates of the mean. In the limiting case, when no random fluctuations occur in the data, we obtain exact identical measurements of the mean. In this case, there is no scatter of data and the variance is zero, so that the interval estimate reduces to an exact point estimate. 

The quantum yields of fluorescence of the different systems have also been determined relative to a single crystal of neodymium-doped YAG for which a quantum yield of unity has been assumed (Heller, 1968a). The quantum yields obtained, even if they are accurate only within a factor of two, follow the same trend as for the lifetimes, with the highest values for the acidic solutions 0.70 and >0.75 in presence of S11CI4 and SbCls, respectively. Neutral and basic solutions are less luminescent and have quantum yields of 0.5 and 0.4, respectively. Identical measurements performed on a sodium-compensated neodymium-doped calcium tungstate crystal lead to a value of 0.5. The high quantum efficiency and the low threshold (between 2 and 40 J) of these Nd3+ SeOCl2 systems clearly demonstrate that liquids are not inherently inferior to solids as laser materials. 

The surface structures of standards and investigated sample may not be identical, measurements based on an external standard may be subject to errors,... 

With the development of the non-Newtonian viscosity theories it is now possible to compare the rotary diffusion coefficient and thereby the calculated length (or diameter) of the rigid particles as obtained from this technique with that from the commonly used flow birefringence method. Since both measurements depend upon the same molecular distribution function (Section III) they should give an identical measure of the rotary diffusion coefficient. Differences, however, will arise if the system under study is heterogeneous. The mean intrinsic viscosity is calculated from Eq. (7) whereas the mean extinction angle, x, for flow birefringence is defined by the Sadron equation (1938) ... 

Each individual measurement of any physical quantity yields a value A. But, independently of any possible observation errors associated with imperfect experimental measurements, the outcomes of identical measurements in identically prepared microsystems are not necessarily the same. The results fluctuate around a central value. It is this collection or Spectrum of values that characterizes the observable A for the ensemble. The fraction of the total number of microsystems leading to a given A value yields the probability of another identical measurement producing that result. Two parameters can be defined the mean value (later to be called the expected value ) and the indeterminacy (also called uncertainty by some authors). The mean value A) is the weighted average of the different results considering the frequency of their occurrence. The indeterminacy AA is the standard deviation of the observable, which is defined as the square root of the dispersion. In turn, the dispersion of the results is the mean value of the squared deviations with respect to the mean (A). Thus,... 

It is useful to note that statistical handling for a relatively small number of systematically planned measurements may yield more information than a large number of repeated identical measurements. Consider triplicate analysis of the same sample using different weights or volumes. This may reveal errors that would not be detected if repeated similar sample sizes were taken. A report entitled Principles of Environmental Analysis [2] which can be applied to all types of analyses, states ... 

Note that the particular activity change process included here tends to lessen the difference between the measured and instantaneous response curves. Theoretically, one could get identical measured and instantaneous response curves in the unlikely event that two transient chemical processes exactly offset each other. However, discrepancies would probably still appear in the mass balances if any accumulation-reaction process were present, and discrepancies between the measured and instantaneous response curves would probably appear in other types of transient response experiments on the same catalyst. 

An identical measurement is made at pH 1.2, when 95% of the anthocyanins present are colored. In a mixture containing 1 ml of wine, 7 ml of HCl (n/10) and 2 ml of water, the optical density is measured at 520 nm on a 1 cm optical path, giving the value d. A second measurement is made, replacing the water with sodium bisulfite as before, giving a value dL ... 

Empirical temperature is a number adjoined to every place and instant of process in the system and may be measured by thermometer (see Appendix A. 1 for further details note that a priori assumptions, like thermal equilibrium, Zeroth law, are necessary here). We use for the same units, namely Kelvins, and because we try to measure generally in nonequilibrium situations, we consider as a reliable that value of i read from thermometer the dimension and relaxation time of which are both much smaller than observer s scales (cf. Rem. 8 and Sects. 1.1, 2.3, AppendixA.l). The right value of d is assured if by repeating of identical measurement with other thermometers with smaller and smaller dimensions and relaxation times the same values (in Kelvins) are obtained. 

The possibility of measuring empirical temperature in nonequilibrium situations is connected with the fact that the different calibrated thermometers noted above may have very different dimensions and very different relaxation times (the time intervals necessary to achieve practically thermal equilibrium). Then, a reliable empirical temperature in (even nonequilibrium) situations may be obtained if we use a thermometer with a dimension and relaxation time much less than the space and time of the observer s scales of this situation (cf. Sects. 1.1,1.2,2.3). Moreover, the right value of d is assured if, by repeating identical measurement with other thermome-... 

Compare the measured radioactivities of the test and comparator samples that have been measured under identical measurement conditions. Make any correction necessary for (o) the rate of decay of the radioactive species, (b) chemical yield if the materials have been chem-... 

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