Measurement of Distribution

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

Errors affecting the distribution of measurements around a central value are called indeterminate and are characterized by a random variation in both magnitude and direction. Indeterminate errors need not affect the accuracy of an analysis. Since indeterminate errors are randomly scattered around a central value, positive and negative errors tend to cancel, provided that enough measurements are made. In such situations the mean or median is largely unaffected by the precision of the analysis. 

The distribution of measurements subject to indeterminate errors is often a normal distribution. 

Most environmental sampling studies are not amenable to classical statistical techniques. Correlation among samples, non-normal distributions of measurements, and multivariate requirements are typical In environmental studies. The effective use of statistics In an environmental study thus depends on meaningful Interaction between statisticians and other environmental scientists. 

Fig. 4.3. Schematic frequency distribution of measured values y (a), GAUSsian normal distributions of measured values y (b) as well as of analytical values x(c), and standard normal distribution (d)...

Fig. 4.9. Distributions of measured values which belong to the content at the limit of specification xLSp and at the screening limit Xscr dis is the limit of discrimination...

In this way, deviations can be characterized between an experimentally found distribution of measured values, p(x), and an a priori distribution po(x)y e.g., corresponding to an expected normal range of values. There are situations, especially with some spectroscopic methods, in which relations of the signal position, experimentally recorded on the one hand and theoretically expected on the other hand, may contain essential chemical information on the species (chemical shifts). 

Dispersion parameter for the distribution of measured values, sy, or analytical results, sx, for a given sample or the population, oy and ox. The SD is the square root of the variance. 

Fig.2. Cumulative frequency distribution of measurements in 47 houses during sunnier and winter.

Figure 3. Frequency distributions of measured indoor radon concentrations for Nagasaki, Hiroshima, Misasa and Mihama.

Table I. Distribution of Measured Radon Concentration, Bq/m, in 21 Detached Houses Designed According to Figure 3...

Table III. Distribution of Measured Radon Concentration in 24 Detached Houses Designed with Mechanical Exhaust Ventilation through the Crawl Space...

Figure 4. Frequency distribution of measured radon concentrations in U.S. houses derived from multiple surveys (taken from Nero et al., 1984, and used with permission).

Fig. 5.3 Statistical distribution of measured poplar yields in Germany divided into subspecies. Result of a survey of a total of n = 357 yield data of various poplar stands of 2-18 years on 25 different sites in Germany. The high frequency of the lower yields is caused by the high share of young stands, as the yield of poplar grows up to an age of 5-10 years...

Another problem encountered when registering precipitation in mountainous regions is the spatial distribution of measurement stations (e.g. [26]). The volume and intensity of precipitation in mountainous regions is highly variable even over small areas. Yet these very regions often suffer from a lack of dense precipitation measurement networks, since measurement stations are mainly to be found in valleys. Consequently, up to now the data collected in order to measure the spatial variability and volume of precipitation in higher-altitude regions has been insufficient. 

UV/VIS/NIR spectroscopy and ESR spectroscopy. The UV/VIS/NIR spectrum shows a sharp peak at 983 nm and a broad peak at 846 nm. These two absorbances are attributed to allowed NIR-transitions and these values are consistent with spectra of the cation obtained with other methods [2]. EPR spectroscopy of Cgg-cations, produced by different methods, leads to a broad distribution of measured g-values. These differences are caused by the short lifetime of the cation, the usually low signal-noise ratio and the uncertainty of the purity. The most reliable value imtil now is probably the one obtained by Reed and co-workers for the salt Cgg"(CBiiHgClg)-(g= 2.0022) [2,9] (see also Section 8.5). Ex situ ESR spectroscopy of above-mentioned bulk electrolysis solutions led to a g-value of2.0027 [8], which is very close to that of the salt, whereas the ESR spectra of this electro lyticaUy formed cation shows features not observed earlier. The observed splitting of the ESR signal at lower modulation amplitudes was assigned to a rhombic symmetry of the cation radical at lower temperatures (5-200 K). 

The experiments summarized in Table 3.1 are done under conditions in which the applied force increases with time. The most-probable mpture force (F ), which is taken from a distribution of measured values, reflects the competition between force loading and bond mpture and, with the exception of the CD inclusion compounds, mpture force increases with loading rate (Evans and Ritchie 1997). The data emphasize that the mechanical strength of the interactions do not necessarily reflect the thermodynamic strength, or association constant, of the interaction. This point is made by comparison of the DNA duplexes (i eq... 

The observed concentration ranges of TBT in sediment or SPM in Dutch open waters and 19 Dutch harbours are shown in Table 2. The highest observed TBT-Sn concentration in Dutch harbours was a factor 30 higher than the highest TBT-Sn concentrations in open waters. The lowest TBT concentrations were more or less in the same range. In order to normalise the sediment and suspended matter concentrations to water concentrations a value for the (organic carbon partition coefficient) needs to be estimated. From the distribution of measured sediment and... 

Figure 8 3. Distributions of measurements of analyte of concentration, Lcrit, showing that half the results at this concentration will be rejected using Lcrit as the decision point.

Figure 5-2 Detection limit. Curves show distribution of measurements expected for a blank and a sample whose concentration is at the detection limit. The area of any region is proportional to the number of measurements in that region. Only 1% of measurements for a blank are expected to exceed the detection limit. However, 60% of measurements for a sample containing analyte at the detection limit will be below the detection limit. There is a 1% chance of concluding that a blank has analyte above the detection limit. If a sample contains analyte at the detection limit, there is a 50% chance of concluding that analyte is absent because its signal is below the detection limit. Curves in this figure are Student s t distributions, which are broader than the Gaussian distribution.

Gaussian distribution Theoretical bell-shaped distribution of measurements when all error is random. The center of the curve is the mean, p, and the width is characterized by the standard deviation, a. A nortnalized Gaussian distribution, also called the normal error curve, has an area of unity and is given by... 

Figure 15 illustrates the distribution of measurements obtained using this scheme. By assembling the signals recorded for a... 

Hence the hypothesis on normal distribution of measured temperature values in the chemical reactor is accepted. 

Distribution of measured concentrations into the various bound and free forms can in principle be examined by inserting sulfide into equilibrium models as a... 

Figure A2.3 Probability plot showing the distribution of measured BCF values and the lognormal distribution used to represent the variation in these measured values...

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