Errors in Measurements PDF

The causes of measurement errors are numerous and their magnitudes are variable. This leads to uncertainties in reported results. However, measurement errors can be minimized and some types can be eliminated altogether by careful experimental design and control. Their effects can be assessed by th...

The causes of measurement errors are numerous and their magnitudes are variable. This leads to uncertainties in reported results. However, measurement errors can be minimized and some types can be eliminated altogether by careful experimental design and control. Their effects can be assessed by the application of statistical methods of data analysis and chemometrics. Gross errors may arise from faulty equipment or bad laboratory practice; proper equipment maintenance and appropriate training and supervision of personnel should eliminate these. Importance of studying errors Whether it is reading a burette or thermometer, weighing a sample or timing events, or monitoring an electrical signal or liquid flow, there will always be inherent variations in the measured parameter if readings are repeated a number of times under the same conditions. In addition, errors may go undetected if the true or accepted value is not known for comparison purposes. Errors must be controlled and assessed so that valid analytical measurements can be made and reported. The reliability of such data must be demonstrated so that an end-user can have an acceptable degree of confidence in the results of an analysis. Types of Errors The absolute error, EA, in a measurement or result xM is given by the equation EA = xM - xT where xT is the true or accepted value. Examples are shown in Figure 1 where a 200 mg aspirin standard has been analyzed a number of times. The absolute errors range from -5 mg to +10 mg. The relative error, ER, in a measurement or result, xM, is given by the equation ER = (xM - xT)/xT Often, ER is expressed as a percentage relative error, 100ER. Thus, for the aspirin results shown in Figure 1, the relative error ranges from -2.5% to +5%. Relative errors are particularly useful for comparing results of differing magnitude Types of Errors There are three basic sources of determinate or systematic errors that lead to a bias in measured values or results: the analyst or operator; the equipment (apparatus and instrumentation) & the laboratory environment; the method or procedure Operator errors can arise through carelessness, insufficient training, illness or disability. Equipment errors include substandard volumetric glassware, faulty or worn mechanical components, incorrect electrical signals and a poor or insufficiently controlled laboratory environment. Method or procedural errors are caused by inadequate method validation, the application of a method to samples or concentration levels for which it is not suitable or unexpected variations in sample characteristics that affect measurements It should be possible to eliminate errors of this type by careful observation and record keeping, equipment maintenance and training of laboratory personnel Determinate errors that lead to a higher value or result than a true or accepted one are said to show a positive bias; those leading to a lower value or result are said to show a negative bias. Particularly large errors are described as gross errors; these should be easily apparent and readily eliminated. Determinate errors can be proportional to the size of sample taken for analysis. If so, they will have the same effect on the magnitude of a result regardless of the size of the sample, and their presence can thus be difficult to detect. For example, copper(II) can be determined by titration after reaction with potassium iodide to release iodine according to the equation 2Cu2+ + 4I- 2CuI + I2 However, the reaction is not specific to copper(II), and any iron(III) present in the sample will react in the same way. Results for the determination of copper in an alloy containing 20%, but which also contained 0.2% of iron are shown in Figure 2 for a range of sample sizes The same absolute error of +0.2% or relative error of 1% (i.e. a positive bias) occurs regardless of sample size, due to the presence of the iron. This type of error may go undetected unless the constituents of the sample and the chemistry of the method are known. Constant determinate errors are independent of sample size, and therefore become less significant as the sample size is increased. For example, where a visual indicator is employed in a volumetric procedure, a small amount of titrant is required to change the color at the end-point, even in a blank solution (i.e. when the solution contains none of the species to be determined). This indicator blank is the same regardless of the size of the titer when the species being determined is present The relative error, therefore, decreases with the magnitude of the titer, as shown graphically in Figure 3. Thus, for an indicator blank of 0.02 cm3, the relative error for a 1 cm3 titer is 2%, but this falls to only 0.08% for a 25 cm3 titer. -Known also as random errors, these arise from random fluctuations in measured quantities, which always occur even under closely controlled conditions. -It is impossible to eliminate them entirely, but they can be minimized by careful experimental design and control. -Environmental factors such as temperature, pressure and humidity, and electrical properties such as current, voltage and resistance are all susceptible to small continuous and random variations described as noise. -These contribute to the overall indeterminate error in any physical or physico-chemical measurement, but no one specific source can be identified. A series of measurements made under the same prescribed conditions and represented graphically is known as a frequency distribution. The frequency of occurrence of each experimental value is plotted as a function of the magnitude of the error or deviation from the average or mean value. For analytical data, the values are often distributed symmetrically about the mean value, the most common being the normal error or Gaussian distribution curve The curve shows that small errors are more probable than large ones, positive and negative errors are equally probable, and the maximum of the curve corresponds to the mean value. The normal error curve is the basis of a number of statistical tests that can be applied to analytical data to assess the effects of indeterminate errors, to compare values and to establish levels of confidence in results. Errors are associated with every measurement made in an analytical procedure, and these will be aggregated in the final calculated result. The accumulation or propagation of errors is treated similarly for both determinate (systematic) and indeterminate (random) errors Determinate (systematic) errors can be either positive or negative, hence some cancellation of errors is likely in computing an overall determinate error, and in some instances this may be zero The overall error is calculated using one of two alternative expressions, that is where only a linear combination of individual measurements is required to compute the result, the overall absolute determinate error, ET, is given by ET = E1 + E2 + E3 + ……. E1 and E2 etc., being the absolute determinate errors in the individual measurements taking sign into account where a multiplicative expression is required to compute the result, the overall relative determinate error, ETR, is given by ETR = E1R + E2R + E3R + ……. E1R and E2R etc., being the relative determinate errors in the individual measurements taking sign into account The accumulated effect of indeterminate (random) errors is computed by combining statistical parameters for each measurement