Uncertainty, instrument reading

PRM concentration PRM expanded uncertainty Instrument reading WRM concentration WRM expanded uncertainty Instrument reading... 

Analytical measurements are fundamentally subject of uncertainty where various types of deviations (errors) can appear and these may be influenced to varying degree. Even when instrument readings are sufficiently accurate, repeated measurements of a sample lead, in general, to measured results which deviate by varying amounts from each other and from the true value of the sample. 

The purpose of the calibration is to determine a polynomial correction, to be applied to instrument readings, and its uncertainty. The measurands are the polynomial coefficients a, arranged in a column vector a, for the uncertainty estimation, its variance covariance matrix Fa is needed. For any C02 concentration, x, n instrumental readings ytJ are recorded (j= 1 n, n= 15). 

The treatment above assumes that the uncertainty of a measurement has an equal chance of being positive or negative. If, however, an instrument has a zero error, then a constant correction has to be applied to each measurement before we can consider the effect of these random uncertainties. For example, if we know that an instrument reads 0.2 when it should read 0.0, we first need to subtract 0.2 from each reading to give the true value. 

The transmittance for minimum relative error can be derived from Beer s law by calculus, assuming that the error results essentially from the uncertainty in reading the instrument scale and also that the absolute error in reading the transmittance is constant, independent of the value of the transmittance. The result is the prediction that the niinimum relative error in the concentration theoretically occurs when T = 0.368 or A = 0.434. 

The other necessary instrumental component for controlled-current coulometry is an accurate clock for measuring the electrolysis time, fe, and a switch for starting and stopping the electrolysis. Analog clocks can read time to the nearest +0.01 s, but the need to frequently stop and start the electrolysis near the end point leads to a net uncertainty of +0.1 s. Digital clocks provide a more accurate measurement of time, with errors of+1 ms being possible. The switch must control the flow of current and the clock, so that an accurate determination of the electrolysis time is possible. 

The uncertainties refer to 95% confidence limits. These small error limits are indicative of the high precision of the readings obtainable from both instruments. 

We can usually estimate or measure the random error associated with a measurement, such as the length of an object or the temperature of a solution. The uncertainty might be based on how well we can read an instrument or on our experience with a particular method. If possible, uncertainty is expressed as the standard deviation or as a confidence interval, which are discussed in Chapter 4. This section applies only to random error. We assume that systematic error has been detected and corrected. 

To minimize extreme ambient temperature fluctuations. If the laboratory and dew-point instrument temperatures fluctuate by as much as 5°C daily, water activity readings may vary by 0.01 aw. Often, this much uncertainty in sample aw is unacceptable, so there is a need for a temperature-controlled model. 

An error associated with a measurement, called the "uncertainty," is usually the smallest reading that can be read or estimated (by interpolation) from an instrument, or it is the "resolution" of that instrument (the smallest interval of the value measured available on that instrument). Assume that you make N measurements of a quantity x and get results (data) xu x2/..., xN. Of course, you cannot know which datum of these N data is the true value. But you can evaluate the mean, or average, trivially ... 

See absolute uncertainty and relative uncertainty. Uncertainties are always present the experimenter s job is to keep them as small as required for a useful result. We recognize two kinds of uncertainties indeterminate and determinate. Indeterminate uncertainties are those whose size and sign are unknown, and are sometimes (misleadingly) called random. Determinate uncertainties are those of definite sign, often referring to uncertainties due to instrument miscalibration, bias in reading scales, or some unknown influence on the measurement. 

In a test procedure there may be uncertainty due to several volume measurements by pipette or volumetric flask that could affect the final result. In addition there may be an uncertainty due to reading the result on an instrument, such as a colorimeter or an atomic absorption spectrometer. 

Results of the analysis (C) are presented in Table 1. Usually the display of the Perkin-Elmer 5000 instrument in flame atomic absorption analysis is autozeroed for the blank solution. For this reason the values read visually from the display and those stored by GIRAF differ by the value of the blank, a fact that has no influence on the results of the analysis. The uncertainties of the results (U) are shown in Table 1 also. The uncertainties of the calculated analyte concentrations in the test solutions UtT described in the Reference materials section are approximately one third of the corresponding uncertainties of the results of the analysis (for sodium it is not obvious because of rounding), and so the use of the term true values in the context of the validation is permissible. All these data were used for calculation of the acceptance criteria A and B formulated in the Validation plan (see Table 2). The criteria are satisfied for both graphite furnace and flame analysis. 

Concentration Errors When trj = For many photometers and spectrophotometers, the standard deviation in the measurement of T is constant and independent of the magnitude of T. We often see this type of random error in direct-reading instruments with analog meter readouts, which have somewhat limited resolution. The size of a typical scale is such that a reading cannot be reproduced to better than a few tenths of a percent of the full-scale reading, and the magnitude of this uncertainty is the same from one end of the scale to the other. For typical inexpensive instruments, we find standard deviations in transmittance of about 0.003 aj = 0.003). 

In Section 1.5, you read about the issue of uncertainty in measurement and learned to report measured values to reflect this uncertainty. For example, an inexpensive letter scale might show you that the mass of a nickel is 5 grams, but this is not an exact measurement. It is reasonable to assume that the letter scale measures mass with a precision of 1 g and that the nickel therefore has a mass between 4 grams and 6 grams. You could use a more sophisticated instrument with a precision of 0.01 g and report the mass of the nickel as 5.00 g. The purpose of the zeros in this value is to show that this measurement of the nickel s mass has an uncertainty of plus or minus 0.01 g. With this instrument, we can assume that the mass of the nickel is between 4.99 g and 5.01 g. Unless we are told otherwise, we assume that values from measurements have an uncertainty of plus or minus one in the last decimal place reported. Using a far more precise balance found in a chemistry laboratory, you could determine the mass to be 4.9800 g, but this measurement still has an uncertainty of 0.0001 g. Measurements never give exact values. 

Error Analysis and Measurement Assurance. Sources of error in a calibration include (1) difficulty in maintaining the fixed points, (2) accuracy of the standard thermometer, (3) uniformity of the constant temperature medium, (4) accuracy in the signal-reading instrument used, (5) stability of each of the components, (6) hysteresis effects, (7) interpolation uncertainty, and (8) operator error. Techniques for error analysis are described in a number of papers on experimental measurement [104,105]. 

The height of a meniscus in a graduated flask may be read with a greater precision than 2 ml and most probably better than 1 ml (the midpoint between two graduation marks). However, to calculate the interval requires as many as 10 measurements. In the absence of these values, the uncertainty may be assumed to be equal to the half point of the resolution of the instrument. Therefore, as a first approximation, the uncertainty of the 250 ml graduated cylinder with 2 ml graduation marks equals 1 ml. The uncertainty of a 250 ml graduated... 

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