D. Harvey - Modern Analytical Chemistry (794078), страница 54
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For example, a material scientist interested in the surface chemistry of a metal is more likely to select a freshlyexposed surface, created by fracturing the sample under vacuum, than a surface thathas been exposed to the atmosphere for an extended time. In a qualitative analysisthe sample’s composition does not need to be identical to that of the substancebeing analyzed, provided that enough sample is taken to ensure that all componentscan be detected. In fact, when the goal of an analysis is to identify componentspresent at trace levels, it may be desirable to discriminate against major componentswhen sampling. In a quantitative analysis, however, the sample’s composition mustaccurately represent the target population. The focus of this section, therefore, is ondesigning a sampling plan for a quantitative analysis.Five questions should be considered when designing a sampling plan:1.2.3.4.5.From where within the target population should samples be collected?What type of samples should be collected?What is the minimum amount of sample needed for each analysis?How many samples should be analyzed?How can the overall variance be minimized?Each of these questions is considered below in more detail.7B.1 Where to Sample the Target PopulationSampling errors occur when a sample’s composition is not identical to that of thepopulation from which it is drawn.
When the material being sampled is homogeneous, individual samples can be taken without regard to possible sampling errors.Unfortunately, in most situations the target population is heterogeneous in eithertime or space. As a result of settling, for example, medications available as oral suspensions may have a higher concentration of their active ingredients at the bottomof the container. Before removing a dose (sample), the suspension is shaken to minimize the effect of this spatial heterogeneity. Clinical samples, such as blood orurine, frequently show a temporal heterogeneity. A patient’s blood glucose level, forinstance, will change in response to eating, medication, or exercise.
Other systemsshow both spatial and temporal heterogeneities. The concentration of dissolved O2in a lake shows a temporal heterogeneity due to the change in seasons, whereaspoint sources of pollution may produce a spatial heterogeneity.When the target population’s heterogeneity is of concern, samples must be acquired in a manner that ensures that determinate sampling errors are insignificant.If the target population can be thoroughly homogenized, then samples can be takenwithout introducing sampling errors. In most cases, however, homogenizing the1400-CH07 9/8/99 4:03 PM Page 183Chapter 7 Obtaining and Preparing Samples for Analysis183target population is impracticable. Even more important, homogenization destroysinformation about the analyte’s spatial or temporal distribution within the targetpopulation.Random Sampling The ideal sampling plan provides an unbiased estimate of thetarget population’s properties.
This requirement is satisfied if the sample is collectedat random from the target population.3 Despite its apparent simplicity, a true random sample is difficult to obtain. Haphazard sampling, in which samples are collected without a sampling plan, is not random and may reflect an analyst’s unintentional biases. The best method for ensuring the collection of a random sample is todivide the target population into equal units, assign a unique number to each unit,and use a random number table (Appendix 1E) to select the units from which tosample.
Example 7.3 shows how this is accomplished.EXAMPLE 7.3To analyze the properties of a 100 cm × 100 cm polymer sheet, ten 1 cm × 1 cmsamples are to be selected at random and removed for analysis. Explain how arandom number table can be used to ensure that samples are drawn at random.SOLUTIONAs shown in the following grid, we divide the polymer sheet into 10,000 1 cm ×1 cm squares, each of which can be identified by its row number and itscolumn number.01298 990129899For example, the highlighted square is in row 1 and column 2.
To pick tensquares at random, we enter the random number table at an arbitrary point,and let that number represent the row for the first sample. We then movethrough the table in a predetermined fashion, selecting random numbers forthe column of the first sample, the row of the second sample, and so on until allten samples have been selected. Since our random number table (Appendix 1E)uses five-digit numbers we will use only the last two digits. Let’s begin with thefifth entry and use every other entry after that. The fifth entry is 65423 makingthe first row number 23.
The next entry we use is 41812, giving the first columnnumber as 12. Continuing in this manner, the ten samples are as follows:SampleRowColumn1234523458166461280121701Sample678910RowColumn93914512978317139252random sampleA sample collected at random from thetarget population.1400-CH07 9/8/99 4:03 PM Page 184184Modern Analytical ChemistryA randomly collected sample makes no assumptions about the target population, making it the least biased approach to sampling. On the other hand, randomsampling requires more time and expense than other sampling methods since agreater number of samples are needed to characterize the target population.judgmental samplingSamples collected from the targetpopulation using available informationabout the analyte’s distribution withinthe population.systematic samplingSamples collected from the targetpopulation at regular intervals in time orspace.Figure 7.2Example of a systematic sampling plan forcollecting samples from a lake.
Each soliddot represents a sample collected fromwithin the sampling grid.Nyquist theoremStatement that a periodic signal must besampled at least twice each period toavoid a determinate error in measuringits frequency.systematic–judgmental samplingA sampling plan that combinesjudgmental sampling with systematicsampling.Judgmental Sampling The opposite of random sampling is selective, or judgmental sampling, in which we use available information about the target population to help select samples. Because assumptions about the target population areincluded in the sampling plan, judgmental sampling is more biased than randomsampling; however, fewer samples are required.
Judgmental sampling is commonwhen we wish to limit the number of independent variables influencing the results of an analysis. For example, a researcher studying the bioaccumulation ofpolychlorinated biphenyls (PCBs) in fish may choose to exclude fish that are toosmall or that appear diseased. Judgmental sampling is also encountered in manyprotocols in which the sample to be collected is specifically defined by the regulatory agency.Systematic Sampling Random sampling and judgmental sampling represent extremes in bias and the number of samples needed to accurately characterize the target population. Systematic sampling falls in between these extremes.
In systematicsampling the target population is sampled at regular intervals in space or time. For asystem exhibiting a spatial heterogeneity, such as the distribution of dissolved O2 ina lake, samples can be systematically collected by dividing the system into discreteunits using a two- or three-dimensional grid pattern (Figure 7.2). Samplesare collected from the center of each unit, or at the intersection of gridlines. When a heterogeneity is time-dependent, as is common in clinicalstudies, samples are drawn at regular intervals.When a target population’s spatial or temporal heterogeneity shows aperiodic trend, a systematic sampling leads to a significant bias if samplesare not collected frequently enough.
This is a common problem whensampling electronic signals, in which case the problem is known as aliasing. Consider, for example, a signal consisting of a simple sine wave. Figure 7.3a shows how an insufficient sampling frequency underestimates thesignal’s true frequency.According to the Nyquist theorem, to determine a periodic signal’s true frequency, we must sample the signal at a rate that is at least twice its frequency (Figure 7.3b); that is, the signal must be sampled at least twice during a single cycle orperiod. When samples are collected at an interval of ∆t, the highest frequency thatcan be accurately monitored has a frequency of (2 ∆t)–1. For example, if samples arecollected every hour, the highest frequency that we can monitor is 0.5 h–1, or a periodic cycle lasting 2 h.
A signal with a cycling period of less than 2 h (a frequency ofmore than 0.5 h–1) cannot be monitored. Ideally, the sampling frequency should beat least three to four times that of the highest frequency signal of interest. Thus, ifan hourly periodic cycle is of interest, samples should be collected at least every15–20 min.Systematic–Judgmental Sampling Combinations of the three primary approachesto sampling are also possible.4 One such combination is systematic–judgmentalsampling, which is encountered in environmental studies when a spatial or tempo-1400-CH07 9/8/99 4:03 PM Page 185Chapter 7 Obtaining and Preparing Samples for Analysis185ral distribution of pollutants is anticipated.
For example, a plume of wasteleaching from a landfill can reasonably be expected to move in the same direction as the flow of groundwater. The systematic–judgmental samplingplan shown in Figure 7.4 includes a rectangular grid for systematic samplingand linear transects extending the sampling along the plume’s suspectedmajor and minor axes.5Stratified Sampling Another combination of the three primaryapproaches to sampling is judgmental–random, or stratified sampling.Many target populations are conveniently subdivided into distinct units,or strata.
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