Replicate measurements

The following data were recorded during the preparation of a calibration curve, where S eas and s are the mean and standard deviation, respectively, for three replicate measurements of the signal. 

Four replicate measurements were made at the center of the factorial design, giving responses of 0.334, 0.336, 0.346, and 0.323. Determine if a first-order empirical model is appropriate for this system. Use a 90% confidence interval when accounting for the effect of random error. 

Control charts were originally developed in the 1920s as a quality assurance tool for the control of manufactured products.Two types of control charts are commonly used in quality assurance a property control chart in which results for single measurements, or the means for several replicate measurements, are plotted sequentially and a precision control chart in which ranges or standard deviations are plotted sequentially. In either case, the control chart consists of a line representing the mean value for the measured property or the precision, and two or more boundary lines whose positions are determined by the precision of the measurement process. The position of the data points about the boundary lines determines whether the system is in statistical control. 

In order to apply weighting, estimates of the [Pg.45]

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

Precision The closeness of replicate measurements on the same sample. 

In UV-spectroscopy the weighing and dilution steps usually introduce more error than does the measurement itself and thus the wish to obtain a replicate measurement involves a second weighing and dilution sequence. 

Data. Several groups of n replicate measurements of a given property on each of m different samples (for a total of n = l (ni). The group sizes n,- need not be identical. 

Three parallel collection lines were located 50 ft apart and perpendicular to the application line. This pattern allowed three replicate measurements to be made at... 

Hepatitis D infection requires the presence of HBV for HDV viral replication. Measuring HDV RNA levels in the serum by polymerase chain reaction (PCR) confirms the presence of... 

To perform this analysis, we first prepare a dilute solution of polymer with an accurately known concentration. We then inject an aliquot of this solution into a viscometer that is maintained at a precisely controlled temperature, typically well above room temperature. We calculate the solution s viscosity from the time that it takes a given volume of the solution to flow through a capillary. Replicate measurements are made for several different concentrations, from which the viscosity at infinite dilution is obtained by extrapolation. We calculate the viscosity average molecular weight from the Mark-Houwink-Sakurada equation (Eq. 5.5). 

The number of calibration points, p, their distance and measure at the concentration scale, the number of replicate measurements,... 

To compute the results shown in Tables 34-3 and 34-4, the precision of each set of replicates for each sample, method, and location are individually calculated using the root mean square deviation equation as shown (Equations 34-1 and 34-2) in standard symbolic and MathCad notation, respectively. Thus the standard deviation of each set of sample replicates yields an estimate of the precision for each sample, for each method, and for each location. The precision is calculated where each ytj is an individual replicate (/ ) measurement for the ith sample yt is the average of the replicate measurements for the ith sample, for each method, at each location and N is the number of replicates for each sample, method, and location. The results of these computations for these data... 

The section following shows a statistical test (text for the Comp Meth MathCad Worksheet) for the efficient comparison of two analytical methods. This test requires that replicate measurements be made on two different samples using two different analytical methods. The test will determine whether there is a significant difference in the precision and accuracy for the two methods. It will also determine whether there is significant systematic error between the methods, and calculate the magnitude of that error (as bias). 

This efficient statistical test requires the minimum data collection and analysis for the comparison of two methods. The experimental design for data collection has been shown graphically in Chapter 35 (Figure 35-2), with the numerical data for this test given in Table 38-1. Two methods are used to analyze two different samples, with approximately five replicate measurements per sample as shown graphically in the previously mentioned figure. 

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,... 

Our precision as the standard deviation (s) of these five replicate measurements is calculated as 0.114 with n- 1=4 degrees of freedom. The /-value from the t table, a = 0.95, degrees of freedom as 4, is 2.776. 

A method has been developed for differentiating hexavalent from trivalent chromium [33]. The metal is electrodeposited with mercury on pyrolytic graphite-coated tubular furnaces in the temperature range 1000-3000 °C, using a flow-through assembly. Both the hexa- and trivalent forms are deposited as the metal at pH 4.7 and a potential at -1.8 V against the standard calomel electrode, while at pH 4.7, but at -0.3 V, the hexavalent form is selectively reduced to the trivalent form and accumulated by adsorption. This method was applied to the analysis of chromium species in samples of different salinity, in conjunction with atomic absorption spectrophotometry. The limit of detection was 0.05 xg/l chromium and relative standard deviation from replicate measurements of 0.4 xg chromium (VI) was 13%. Matrix interference was largely overcome in this procedure. 

Precision FIA measurements typically show low relative standard deviations (RSD) on replicate measurements, mainly due to the definite and reproducible way of sample introduction. This is a very important feature especially for CL, which is very sensitive to several environmental factors and sensitivity relies greatly on the rate of the reaction. 

Precision studies should mirror the operating conditions used during routine use of the method. For example, the range of operating conditions, such as the variation in the laboratory temperature, needs to reflect that which will occur in practice. In addition the same number of replicate measurements per test portion should be used. Where a range of analytes is measured by a single method (e.g. pesticides by GC or trace elements by ICP-MS), or where different matrix types are encountered, it is necessary to determine the precision parameters... 

For tests designed to detect the presence or absence of an analyte, the threshold concentration that can be detected can be determined from replicate measurements over a range of concentrations. These data can be used to establish at what concentration a cut-off point can be drawn between reliable detection and non-detection. At each concentration level, it may be necessary to measure approximately ten replicates. The cut-off point depends on the number of false negative results that can be tolerated. It can be seen from Table 4.7 that for the given example the positive identification of the analyte is not reliable below 100 xg g-1. 

A detailed treatment of linearity evaluation is beyond the scope of this present book but a few general points are made below. It is important to establish the homogeneity of the variance ( homoscedasticity ) of the method across the working range. This can be done by carrying out ten replicate measurements at the extreme ends of the range. The variance of each set is calculated and a statistical test (F test) carried out to check if these two variances are statistically significantly different [9]. 

These ten results represent a sample from a much larger population of data as, in theory, the analyst could have made measurements on many more samples taken from the tub of low-fat spread. Owing to the presence of random errors (see Section 6.3.3), there will always be differences between the results from replicate measurements. To get a clearer picture of how the results from replicate measurements are distributed, it is useful to plot the data. Figure 6.1 shows a frequency plot or histogram of the data. The horizontal axis is divided into bins , each representing a range of results, while the vertical axis shows the frequency with which results occur in each of the ranges (bins). 

Bias is a measure of trueness . It tells us how close the mean of a set of measurement results is to an assumed true value. Precision, on the other hand, is a measure of the spread or dispersion of a set of results. Precision applies to a set of replicate measurements and tells us how the individual members of that set are distributed about the calculated mean value, regardless of where this mean value lies with respect to the true value. 

V = (1000 0.4) ml (from supplier s catalogue), iq (V) = 0.10 ml (standard deviation of replicate measurements of volume of liquid in flask when filled to the calibration mark). 

If you answered (b), perhaps you were thinking of the spread of values obtained from replicate measurements. While these do indeed form a range, one such range will relate to only one source of uncertainty and there may be several sources of uncertainty affecting a particular measurement. The precision of a measurement is an indication of the random error associated with it. This takes no account of any systematic errors that may be connected with the measurement. It is important to realize that uncertainty covers the effects of both random error and systematic error and, moreover, takes into account multiple sources of these effects where they are known to exist and are considered significant. 

We have a second standard uncertainty for the volume of the volumetric flask. This estimate was obtained by making replicate measurements of the volume. The two standard uncertainties relating to the volume must be combined to produce a single value. This is achieved by a straightforward application of equation (6.12) ... 

In an investigation of a determinate error a series of replicate measurements were made using a range of sample weights. The results obtained are tabulated below. 

Replicate measurements Cathode = Cu grid Anode = Ti/Ti coated ... 

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