Quantifying Random Error

If the systematic errors have been eliminated, the measured value will still be distributed about the true value owing to random error. For a given set of measurements, how close is the average value of our measurements to the true value This is where the Gaussian distribution and statistics are used. 

The Gaussian distribution curve assumes that an infinite number of measurements of X, have been made. The maximum of the Gaussian curve occurs atx = /r, the true value of the parameter we are measuring. So, for an infinite number of measurements, the population mean is the true value x,. We assume that any measurements we make are a subset of the Gaussian distribution. As the number of measurements, N, increases, the difference between x and /x tends toward zero. For N greater than 20 to 30 or so, the sample mean rapidly approaches the population mean. For 25 or more replicate measurements, the true value is approximated very well by the experimental mean value. Unfortunately, even 20 measurements of a real sample are not usually possible. Statistics allows us to express the random error associated with the difference between the population mean fi and the mean of a small subset of the population, x. The random error for the mean of a small subset is equal tox — fi. 

The area under any portion of the Gaussian distribution, for example, between a value Xi and a value X2, corresponds to the fraction of the measurements which will yield a measured value of x between and including these two values. The spread of the Gaussian distribution, that is, the width of the bell-shaped curve, is expressed in terms of the population standard deviation cr. The standard deviation cr coincides with 

The precision of analytical results is usually stated in terms of the standard deviation cr. As just explained, in the absence of determinate error, 68.3% of all results can be expected to fall within +cr of the true value, 95.5% of the results will fall within +2cr of the true answer, and 99.7% of our results will fall within +3cr of the true value if we perform enough measurements (more than 20 or so replicates). It is common practice to report analytical results with the mean value and the standard deviation expressed, thereby giving an indication of the precision of the results. 

1 % Si and Analyst B reported 19.6 + 0.2% Si, without telling the customer what the 0.1 and 0.2% mean, the customer might think incorrectly that Analyst A is more precise than Analyst B. 

However, we are usnaUy dealing with a small, finite subset of measuranents, not 20 or more in this case, the standard deviation that should be reported is the sample standard deviation s. For a small finite data set, the sample standard deviation s differs from a in two respects. Look at the equations given in the definitions. The equation for sample standard deviation s contains the sample mean, not the population mean, and nses N - I measurements instead of N, the total number of measurements. The term IV - 1 is called the degrees of freedom. 

You have measured mercury in eight representative samples of biological tissue from herons (which eat fish that may be contaminated with mercury) using cold vapor AAS (which is discussed in Chapter 6). The values in colnmn 2 of the following table were obtained. 

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Recall that the precision quantifies random errors. Let s consider the titration curve especially about the equivalence point (Fig. 10.1). It appears that the lower the slope is at the equivalence point, the higher the propagation of the error due to the pH measurement is. This is sufficient to justify the introduction of the parameter Ti to quantify the phenomenon, q is named the sharpness index. It is defined as being the magnitude of the slope of the titration curve ... 

The standard uncertainty arising from random effects is typically measured from precision studies and is quantified in terms of the standard deviation of a set of measured values. For example, consider a set of replicate weighings performed in order to determine the random error associated with a weighing. If the true mass of the object being weighed is 10 g exactly, then the values obtained might be as follows ... 

Method validation seeks to quantify the likely accuracy of results by assessing both systematic and random effects on results. The properly related to systematic errors is the trueness, i.e. the closeness of agreement between the average value obtained from a large set of test results and an accepted reference value. The properly related to random errors is precision, i.e. the closeness of agreement between independent test results obtained under stipulated conditions. Accnracy is therefore, normally studied as tmeness and precision. 

The estimated slope and intercept provide an estimate of the systematic difference or error between two methods over the analytical measurement range. Additionally an estimate of the random error is important. As mentioned above, it is commonplace to consider the dispersion around the line in the vertical direction, which is quantified as SD, ( (here denoted SD21). SD21 has originally been introduced in the context of OLR, but it may as well be considered in relation to Deming regression analysis. 

Precision The precision of an analytical procedure expresses the closeness of agreement (usually expressed as coefficient of variation) between a series of measurements obtained from multiple sampling of the same homogeneous sample (independent assays) under prescribed conditions. It quantifies the random errors produced by the procedure and is evaluated with respect to three levels repeatability, intermediate precision (within laboratory), and reproducibility (between laboratories). 

The CMB model combines the chemical and physical characteristics of particles or gases measured at the sources and the receptors to quantify the source contributions to the receptor (Winchester and Nifong 1971 Miller et al. 1972). CMB is a method for the solution of the set of equations (26.9) to determine the unknown sj. The source profiles atj, that is, the fractional amount of the species in the emissions from each source type, and the receptor concentrations, with appropriate uncertainty estimates, serve as input to the CMB model. We start by analyzing the case where one particulate sample is available. The first assumption of CMB is that all sources contributing to the measured concentrations c, in the receptor have been identified. Each measured concentration c,- can then be expressed as the sum of the true value c and a random error e, ... 

There is some uncertainty in all data, and model building must take this error into account. The first step in error management is error detection, error reduction, and error quantification. There are three types of error systematic error, random error, and blunders. Improved experimental protocol can reduce all these, but designing progressively better experiments eventually leads to diminishing returns so that at some point it is necessary to use some kind of error analysis to manage the uncertainty in the variable being quantified. 

A second reason to fit data to a function is to test whether the data are consistent with a model or to use a theoretical function to extrapolate experimental results to conditions otherwise not attainable. The goodness of fit of the data to a theoretical function can be gauged by the coefficient of determination EP), which is the correlation coefficient squared. B is often interpreted as the fraction of the variability of the response variable that is explained by its functional relationship to the independent variable. For example, if R = 0.8 then 80% of the variation in the response variable is explained by the model and 20% of the variation is the result of factors, including random error, that are not part of the model. Some geochemical processes occur under conditions or over time intervals that are difficult to simulate by experiments, making it necessary to use a theoretical model to predict their behavior. This approach uses data from experiments performed under easily attainable conditions to quantify parameters that are used to calibrate a theoretical model. These parameters are then used to predict the value of the variable at... 

Accuracy refers to the correctness of data, i.e., it is related to the deviation of the determined value from the true value. Inaccuracy results from imprecision (random error) and bias (systematic error) in the measurement process. Whereas precision is easier to determine, accuracy is more informative and important but the most difficult quaUty criterion to quantify. Certified reference materials are most useful in evaluating the laboratory analysis (see subsequent chapters). In addition, the recovery of spikes of the analyte from a sample can also give... 

In an attempt to quantify the random error ( noise ) and the systematic error ( reproduction error ) with comparable metrics the signal observed overall is... 

Quantification of analytes can be performed by the internal standard method, using an isotopically labeled internal standard added to the sample in a precise amount at the early stage of the analytical treatment. By selectively monitoring specific ions of the molecule to be quantified and the corresponding shifted ions of the labeled homologue as reference signal, the precision of the analytical results of GC-MS can be improved. Moreover, random errors during different steps of the sample preparation are minimized. This so-called isotopic dilution technique has been widely used in numerous fields. Several examples of this are reported below. 

The NESR is a measure of the random errors of the instrument, expressed in radio-metric units. It represents the one-sigma uncertainty in an individual spectrum. This fully quantifies the random errors of the spectrometer. Examples of the responsivity... 

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