Precision of the mean

There is a 95% probability that any weight value picked at random has a value that lies within 0.0509 g of the average. The precision of the mean is given by... 

The least-squares procedure just described is an example of a univariate calibration procedure because only one response is used per sample. The process of relating multiple instrument responses to an analyte or a mixture of analytes is known as multivariate calibration. Multivariate calibration methods have become quite popular in recent years as new instruments become available that produce multidimensional responses (absorbance of several samples at multiple wavelengths, mass spectrum of chromatographically separated components, and so forth). Multivariate calibration methods are very powerful. They can be used to determine multiple components in mixtures simultaneously and can provide redundancy in measurements to improve precision. Recall that repeating a measurement N times provides a Vn improvement in the precision of the mean value. These methods can also be used to detect the presence of interferences that would not be identified in a univariate calibration. 

The apparent duration of the remagnetization event and the high precision of the mean direction from the Bowers Supergroup are most consistent with locking in of the remanence during the Cretaceous normal polarity superchron. 

The standard deviation of the mean and of the noise tell us about the precision of the mean and the noise. The values for data set B (based on 20 measurements) are about twice as good (half as large) as those for data set A (from 5 measurements). 

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. 

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. 

Statistically, a similar Indication of precision could be achieved by utilising the 95% probability level if the results fell on a "Gaussian" curve, viz., the confidence would lie within two standard deviations of the mean. R 2 x SD = 56.3 24.8... 

Effect of Uncertainties in Thermal Design Parameters. The parameters that are used ia the basic siting calculations of a heat exchanger iaclude heat-transfer coefficients tube dimensions, eg, tube diameter and wall thickness and physical properties, eg, thermal conductivity, density, viscosity, and specific heat. Nominal or mean values of these parameters are used ia the basic siting calculations. In reaUty, there are uncertainties ia these nominal values. For example, heat-transfer correlations from which one computes convective heat-transfer coefficients have data spreads around the mean values. Because heat-transfer tubes caimot be produced ia precise dimensions, tube wall thickness varies over a range of the mean value. In addition, the thermal conductivity of tube wall material cannot be measured exactiy, a dding to the uncertainty ia the design and performance calculations. 

Automated methods are more rehable and much more precise than the average manual method dependence on the technique of the individual technologist is eliminated. The relative precision, or repeatabiUty, measured by the consistency of the results of repeated analyses performed on the same sample, ranges between 1% and 5% on automated analy2ers. The accuracy of an assay, defined as the closeness of the result or of the mean of repHcate measurements to the tme or expected value (4), is also of importance in clinical medicine. 

In its most common mode of operation, STM employs a piezoelectric transducer to scan the tip across the sample (Figure 2a). A feedback loop operates on the scanner to maintain a constant separation between the tip and the sample. Monitoring the position of the scanner provides a precise measurement of the tip s position in three dimensions. The precision of the piezoelectric scanning elements, together with the exponential dependence of A upon c/means that STM is able to provide images of individual atoms. 

It is, therefore, necessary to establish the precision of the results, by which we mean the extent to which they are reproducible. This is commonly expressed in terms of the numerical difference between a given experimental value and the mean value of all the experimental results. The spread or range in a set of results is the numerical difference between the highest and lowest results this... 

When a new analytical method is being developed it is usual practice to compare the values of the mean and precision of the new (test) method with those of an established (reference) procedure. 

Table 11 shows the precision obtained with the Eagle-Picher Turbidimeter. Column 4 is the standard deviation of the specific surface values, and column 5 gives these as percentage of the mean specific surface values... 

This means that the precision of the prediction decreases with the square root of time. This describes the random walk model. A drift can be easily built into such a model by the addition of some constant drift function at each successive time period. 

The result of this analysis provides a measure of the precision of the estimate of the mean plus confidence limits for the estimate. 

The quantities AUMC and AUSC can be regarded as the first and second statistical moments of the plasma concentration curve. These two moments have an equivalent in descriptive statistics, where they define the mean and variance, respectively, in the case of a stochastic distribution of frequencies (Section 3.2). From the above considerations it appears that the statistical moment method strongly depends on numerical integration of the plasma concentration curve Cp(r) and its product with t and (r-MRT). Multiplication by t and (r-MRT) tends to amplify the errors in the plasma concentration Cp(r) at larger values of t. As a consequence, the estimation of the statistical moments critically depends on the precision of the measurement process that is used in the determination of the plasma concentration values. This contrasts with compartmental analysis, where the parameters of the model are estimated by means of least squares regression. 

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... 

As an example, we can set alpha- and beta- levels to the same value, which makes for a simple computation of the number of samples needed, at least for the simple case we have been considering the comparison of means. If we use the 95% value for both (a very stringent test), which corresponds to a Z-value of 1.96 (as we know), then if we let D represent the difference in means between the two values (sample data and population mean), and S is the precision of the data, we find that... 

In words, we would need 15 samples for 95% confidence on both alpha and beta, to distinguish a difference of the means equal to the precision of the measurement, and the number increases as the square of any decrease in difference we want to detect. 

To compute the power for a hypothesis test based on standard deviation, we would have to read off the corresponding probability points from a chi-square table for 95% confidences on both alpha and beta, the square root of the ratio of 2(0.95, v) and 2(0.05, v) (v = the degrees of freedom, close enough to n for now) is the ratio of standard deviations that can be distinguished at that level of power. Similarly to the case of the means, v would also be related to the square of that ratio, but x2 would still have to be read from tables (or computed numerically). As an example, for 35 samples, the precision of the instrument could not be tested to be better than... 

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