Uncertainty in Measurements

Two kinds of numbers are encountered in scientific work exact numbers (those whose values are known exactly) and inexact numbers (those whose values have some imcer-tainty). Most of the exact numbers we will encounter in this book have defined values. For example, there are exactly 12 eggs in a dozen, exactly 1000 g in a kilogram, and exactly 2.54 cm in an inch. The number 1 in any conversion factor, such as 1 m = 100 cm or 1 kg = 2.2046 lb, is an exact number. Exact numbers can also result from coimting objects. For example, we can count the exact number of marbles in a jar or the exact number of people in a classroom. 

Numbers obtained by measurement are always inexact. The equipment used to measure quantities always has inherent limitations (equipment errors), and there are differences in how different people make the same measurement (human errors). Suppose ten students with ten balances are to determine the mass of the same dime. The ten measurements will probably vary slightly for various reasons. The balances might be calibrated slightly differently, and there might be differences in how each student reads the mass from the balance. Remember Uncertainties always exist in measured quantities. 

Suppose you determine the mass of a dime on a balance capable of measuring to the nearest 0.0001 g. You could report the mass as 2.2405 0.0001 g. The notation (read plus or minus ) expresses the magnitude of the uncertainty of your measurement. In much scientific work we drop the notation with the understanding that there is always some uncertainty in the last digit reported for any measured quantity. 

What difference exists between the measured values 4.0 and 4.00 g  

The value 4.0 has two significant figures, whereas 4.00 has three. This difference implies that 4.0 has more uncertainty. A mass reported as 4.0 g indicates that the uncertainty is in the first decimal place. Thus, the mass is closer to 4.0 than to 3.9 or 4.1 g. We can represent this uncertainty by writing the mass as 4.0 0.1 g. A mass reported as 4.00 g indicates that the uncertainty is in the second decimal place. In this case the mass is closer to 4.00 than 3.99 or 4.01 g, andwe can represent it as 4.00 0.01 g. (Without further information, we cannot be sure whether the difference in uncertainties of the two measurements reflects the precision or the accuracy of the measurement.) 

Chemistry makes use of two types of numbers exact and inexact. Exact numbers include numbers with defined values, such as 2.54 in the definition 1 inch (in) = 2.54 cm, 1000 in the definition 1 kg = 1000 g, and 12 in the definition 1 dozen = 12 objects. (The number 1 in each of these definitions is also an exact number.) Exact numbers also include those that are obtained by counting. Numbers measured by any method other than counting are inexact 

Measured numbers are inexact because of the measuring devices that are used, the individuals who use them, or both. For example, a ruler that is poorly calibrated will resirlt in measurements that are in error—no matter how carefully it is used. Another ruler may be calibrated properly but have insufficient resolution for the necessary measurement. Finally, whether or not an instrument is properly calibrated or has sufficient resolution, there are unavoidable differences in how different people see and interpret measirrements. 

The nttmber of significant figitres in any number can be determined using the following guidelines  

Any digit that is not zero is significant (112.1 has four significant figures). 

Whenever a measurement is made with a device such as a ruler or a graduated cylinder, an estimate is required. We can illustrate this by measuring the pin shown. 

We can see from the ruler that the pin is a little longer than 2.8 cm and a little shorter than 2.9 cm. Because there are no graduations on the ruler between 2.8 and 2.9, we must estimate the pin s length between 2.8 and 2.9 cm. We do this by imagining that the distance between 2.8 and 2.9 is broken into 10 equal divisions shown in blue and estimating to which division the end of the pin reaches. 

Note that the first two digits in each measurement are the same regardless of who made the measurement these are called the certain numbers of the measurement. However, the third digit is estimated and can vary it is called an uncertain number. When one is making a measurement, the custom is to record all of the certain numbers plus the first uncertain number. It would not make any sense to try to measure the pin to the third decimal place (thousandths of a centimeter), because this mler requires an estimate of even the second decimal place (hundredths of a centimeter). 

For example, if our ruler had marks indicating hundredths of a centimeter, the uncertainty in the measurement of the pin would occur in the thousandths place rather than the hundredths place—but some uncertainty would still exist. 

What do we mean when we say that a measurement always has a degree of uncertainty  

AIMS To understand how uncertainty in a measurement arises. To learn to indicate a measurement s uncertainty by using significant figures. 

Because the last number is based on a visual estimate, it may be different when another person makes the same measurement. For example, if five different people measured the pin, the results might be 

Important Antibiotic Modified to Combat Bacterial Resistance 

If you want to save the world from global warming, you can start by replacing incandescent hghtbulbs. In 2001 approximately 22% of all electricity generated in the United States was used for lighting. This 

The terms precision and accuracy are often used in discussing the uncertainties of measured values. Precision is a measure of how closely individual measurements agree with one another. Accuracy refers to how closely individual measurements agree with the 

Common types of laboratory equipment used to measure liquid volume. 

Measurement of volume using a buret. The volume is read at the bottom of the iiquid cun/e (caiied the meniscus). 

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Uncertainty in measurement is discussed in more detail in Appendix 1.5. 

Do the two grapefruits have the same mass The answer depends on which set of results you consider. Thus a conclusion based on a series of measurements depends on the certainty of those measurements. For this reason, it is important to indicate the uncertainty in any measurement. This is done by always recording the certain digits and the first uncertain digit (the estimated number). These numbers are called the significant figures of a measurement. 

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It was found that that in the case of soft beta and X-ray radiation the IPs behave as an ideal gas counter with the 100% absorption efficiency if they are exposed in the middle of exposure range ( 10 to 10 photons/ pixel area) and that the relative uncertainty in measured intensity is determined primarily by the quantum fluctuations of the incident radiation (1). The thermal neutron absorption efficiency of the present available Gd doped IP-Neutron Detectors (IP-NDs) was found to be 53% and 69%, depending on the thicknes of the doped phosphor layer ( 85pm and 135 pm respectively). No substantial deviation in the IP response with the spatial variation over the surface of the IP was found, when irradiated by the homogeneous field of X-rays or neutrons and deviations were dominated by the incident radiation statistics (1). 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

Precision Precision is generally limited by the uncertainty in measuring the limiting or peak current. Under most experimental conditions, precisions of+1-3% can be reasonably expected. One exception is the analysis of ultratrace analytes in complex matrices by stripping voltammetry, for which precisions as poor as +25% are possible. 

In this expression, p(H) is referred to as the prior probability of the hypothesis H. It is used to express any information we may have about the probability that the hypothesis H is true before we consider the new data D. p(D H) is the likelihood of the data given that the hypothesis H is true. It describes our view of how the data arise from whatever H says about the state of nature, including uncertainties in measurement and any physical theory we might have that relates the data to the hypothesis. p(D) is the marginal distribution of the data D, and because it is a constant with respect to the parameters it is frequently considered only as a normalization factor in Eq. (2), so that p(H D) x p(D H)p(H) up to a proportionality constant. If we have a set of hypotheses that are exclusive and exliaus-tive, i.e., one and only one must be true, then... 

ISO (1993) Guide to the expression of uncertainty in measurement. ISBN 92-67-10188-9, ist Edn., Geneva, Switzerland. 

The proper application of the ISO Guide to the expression of uncertainty in measurement requires that aU sources of rmcertainty are included. In practical terms this means an uncertainty budget has to be developed also by the user. The development of an rmcertainty budget, and the consequences for both analysts and producers is described later in Section 7.2. 

Pauwels (1999) argues that the certified values of CRMs should be presented in the form of an expanded combined uncertainty according to the ISO Guide on the expression of uncertainty in measurement, so that coverage factor should always be clearly mentioned in order to allow an easy recalculation of the combined standard uncertainty. This is needed for uncertainty propagation when the CRM is used for calibration and the ISO Guide should be revised accordingly. The use of the expanded uncertainty has been pohcy in certification by NIST since 1993 (Taylor and Kuyatt 1994). 

As probabilistic exposure and risk assessment methods are developed and become more frequently used for environmental fate and effects assessment, OPP increasingly needs distributions of environmental fate values rather than single point estimates, and quantitation of error and uncertainty in measurements. Probabilistic models currently being developed by the OPP require distributions of environmental fate and effects parameters either by measurement, extrapolation or a combination of the two. The models predictions will allow regulators to base decisions on the likelihood and magnitude of exposure and effects for a range of conditions which vary both spatially and temporally, rather than in a specific environment under static conditions. This increased need for basic data on environmental fate may increase data collection and drive development of less costly and more precise analytical methods. 

At room temperature the relative uncertainty in measuring E over 10 °C temperature intervals is generally about 5% for gas phase reactions and about 3% for liquid phase reactions. 

The more sophisticated treatment of Ingle and Crouch [7] comes very close but also misses the mark for an unexplained reason they insert the condition ... it is assumed there is no uncertainty in measuring Ert and Eot... . Now in fact this could happen (or at least there could be no variation in AEr) for example, if one reference spectrum was used in conjunction with multiple sample spectra using an FTIR spectrometer. However, that would not be a true indication of the total error of the measurement, since the effect of the noise in the reference reading would have been removed from the calculated SD, whereas the true total error of the reading would in... 

Descriptions of the experimental scattering and microscopy conditions have been published elsewhere and are referenced in each section. Throughout this report certain conventions will be used when describing uncertainties in measurements. Plots of small angle scattering data have been calculated from circular averaging of two-dimensional files. The uncertainties are calculated as the estimated standard deviation of the mean. The total combined uncertainty is not specified in each case since comparisons are made with data obtained under... 

Eisinger(55) also noted that it is difficult to obtain accurate data with phenylalanine as the donor and either tryptophan or tyrosine as the acceptor. The source of this problem is the weak S, - S0 absorption of phenylalanine compared to that of tyrosine or tryptophan, which leads to considerable experimental uncertainty in measuring the sensitized acceptor emission. This error may account for the finding of Kupryszewska et al.<56> that the sensitization of the acceptor fluorescence was less than the quenching of the donor fluorescence in their study of phenylalanine-to-tyrosine energy transfer... 

Uncertainty in measurement is a universal feature of all experimental work(2, g.,g.) and must be dealt with if data are to be converted to... 

ANSI and NCSL, U.S. Guide to the Expression of Uncertainty in Measurement, 1800 30 Str., Suite 305B, Boulder, CO 80301, USA, ANSI/NCSLZ540-2-1997, (1997). 

Fig. 16.18 Effect of surface nature and loading on the photodegradation rate of adsorbed BeP. The photolysis experiments were carried out in a rotary cell, and the BeP remaining in the sample was determined by HPLC. Uncertainties in measurements are in the range of 4-10% of the absolute values. Reprinted with permission from Fioressi S, Arce R (2005) Photochemical transformation of benzo(e)pyrene in solution and adsorbed on silica gel and alumina surfaces. Environ Sci Technol 39 3646-3655. Copyright 2005 American Chemical Society...

Figure 24.9 illustrates measured CO and CO2 mole fractions as a function of equivalence ratio. The solid lines represent chemical equilibrium calculations of CO and CO2 mole fractions at measured temperatures. The vertical bars represent the uncertainty in measured CO and CO2 mole fractions due to line... 

The Guide to the Expression of Uncertainty in Measurement (GUM) to some extent is the Bible of uncertainty estimation. Since it was originally made for physical measurements in metrology laboratories, it is quite difficult to translate it into analytical chemistry problems, especially for routine measurements. 

The uncertainty in measured values expressed in units of the measurement. For example, a reaction velocity of 10.2 M/min is presumed to be valid to a tenth, and the absolute uncertainty is 0.1 M/min. See Relative Uncertainty... 

In the 20th century, physicists discovered to their surprise that small particles such as atoms and the components of atoms do not obey Newton s law of motion. Instead of being deterministic—following trajectories determined by the laws of physics—tiny bits of matter behave probabilistically, meaning that their state or trajectory is not precisely determined but can follow one of a number of different options. The German physicist Werner Heisenberg proposed his uncertainty principle in 1927, which states that there is generally some amount of uncertainty in measurements of a particle s state. 

The uncertainty in measurement y is given by a sum of squares with the following form ... 

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