D. Harvey - Modern Analytical Chemistry (794078), страница 55
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For example, in determining the concentration of particulatePb in urban air, the target population can be subdivided by particle size.In this case samples can be collected in two ways. In a random sampling,differences in the strata are ignored, and individual samples are collectedat random from the entire target population. In a stratified samplingthe target population is divided into strata, and random samples arecollected from within each stratum.
Strata are analyzed separately, andtheir respective means are pooled to give an overall mean for the targetpopulation.The advantage of stratified sampling is that the composition of each stratum is often more homogeneous than that of the entire target population.When true, the sampling variance for each stratum is less than that when thetarget population is treated as a single unit. As a result, the overall samplingvariance for stratified sampling is always at least as good as, and often betterthan, that obtained by simple random sampling.Convenience Sampling One additional method of sampling deservesbrief mention. In convenience sampling, sample sites are selected usingcriteria other than minimizing sampling error and sampling variance.
Ina survey of groundwater quality, for example, samples can be collectedby drilling wells at randomly selected sites, or by making use of existingwells. The latter method is usually the preferred choice. In this case, cost,expedience, and accessibility are the primary factors used in selecting sampling sites.(a)(b)Figure 7.3Effect of sampling frequency when monitoringperiodic signals. In (a) the sampling frequency is1.2 samples per period.
The dashed line shows theapparent signal, while the solid line shows thetrue signal. In (b) a sampling frequency of fivesamples per period is sufficient to give an accurateestimation of the true signal.7B.2 What Type of Sample to CollectAfter determining where to collect samples, the next step in designing a samplingplan is to decide what type of sample to collect. Three methods are commonly usedto obtain samples: grab sampling, composite sampling, and in situ sampling. Themost common type of sample is a grab sample, in which a portion of the targetpopulation is removed at a given time and location in space. A grab sample, therefore, provides a “snapshot” of the target population.
Grab sampling is easilyadapted to any of the sampling schemes discussed in the previous section. If the target population is fairly uniform in time and space, a set of grab samples collected atrandom can be used to establish its properties. A systematic sampling using grabsamples can be used to characterize a target population whose composition variesover time or space.stratified samplingA sampling plan that divides thepopulation into distinct strata fromwhich random samples are collected.convenience samplingA sampling plan in which samples arecollected because they are easily obtained.grab sampleA single sample removed from the targetpopulation.1400-CH07 9/8/99 4:03 PM Page 186186Modern Analytical ChemistryDirection ofgroundwater flow= Sample siteMajor axisof plumeLandfillFigure 7.4Systematic–judgmental sampling scheme formonitoring the leaching of pollutants from alandfill.
Sites where samples are collectedare represented by the solid dots.composite sampleSeveral grab samples combined to form asingle sample.in situ samplingSampling done within the populationwithout physically removing the sample.A composite sample consists of a set of grab samples that are combined toform a single sample. After thoroughly mixing, the composite sample is analyzed.Because information is lost when individual samples are combined, it is normallydesirable to analyze each grab sample separately. In some situations, however, thereare advantages to working with composite samples. One such situation is in determining a target population’s average composition over time or space. For example,wastewater treatment plants are required to monitor and report the average composition of treated water released to the environment.
One approach is to analyze a series of individual grab samples, collected using a systematic sampling plan, and average the results. Alternatively, the individual grab samples can be combined toform a single composite sample. Analyzing a single composite sample instead ofmany individual grab samples, provides an appreciable savings in time and cost.Composite sampling is also useful when a single sample cannot supply sufficientmaterial for an analysis. For example, methods for determining PCBs in fish oftenrequire as much as 50 g of tissue, an amount that may be difficult to obtain from asingle fish.
Tissue samples from several fish can be combined and homogenized,and a 50-g portion of the composite sample taken for analysis.A significant disadvantage of grab samples and composite samples is the need toremove a portion of the target population for analysis. As a result, neither type of sample can be used to continuously monitor a time-dependent change in the target population. In situ sampling, in which an analytical sensor is placed directly in the targetpopulation, allows continuous monitoring without removing individual grab samples.For example, the pH of a solution moving through an industrial production line can becontinually monitored by immersing a pH electrode within the solution’s flow.1400-CH07 9/8/99 4:03 PM Page 187Chapter 7 Obtaining and Preparing Samples for Analysis7B.3 How Much Sample to CollectTo minimize sampling errors, a randomly collected grab sample must be of an appropriate size. If the sample is too small its composition may differ substantiallyfrom that of the target population, resulting in a significant sampling error.
Samplesthat are too large, however, may require more time and money to collect and analyze, without providing a significant improvement in sampling error.As a starting point, let’s assume that our target population consists of two typesof particles. Particles of type A contain analyte at a fixed concentration, and type Bparticles contain no analyte. If the two types of particles are randomly distributed,then a sample drawn from the population will follow the binomial distribution.* Ifwe collect a sample containing n particles, the expected number of particles containing analyte, nA, isnA = npwhere p is the probability of selecting a particle of type A. The sampling standarddeviation isss =np(1 − p)7.3The relative standard deviation for sampling, ss,r, is obtained by dividing equation7.3 by nA.np(1 − p)s s, r =npSolving for n allows us to calculate the number of particles that must be sampled toobtain a desired sampling variance.n =1−p1× 2ps s, r7.4Note that the relative sampling variance is inversely proportional to the number ofparticles sampled.
Increasing the number of particles in a sample, therefore, improves the sampling variance.EXAMPLE 7.4Suppose you are to analyze a solid where the particles containing analyterepresent only 1 × 10–7% of the population. How many particles must becollected to give a relative sampling variance of 1%?SOLUTIONSince the particles of interest account for 1 × 10–7% of all particles in thepopulation, the probability of selecting one of these particles is only 1 × 10–9.Substituting into equation 7.4 givesn =1 − (1 × 10 −9 )1×= 1 × 10131 × 10 −9(0.01)2Thus, to obtain the desired sampling variance we need to collect 1 × 1013particles.*See Chapter 4 to review the properties of a binomial distribution.1871400-CH07 9/8/99 4:03 PM Page 188188Modern Analytical ChemistryA sample containing 1013 particles can be fairly large.
Suppose this is equivalentto a mass of 80 g. Working with a sample this large is not practical; but does thismean we must work with a smaller sample and accept a larger relative samplingvariance? Fortunately the answer is no. An important feature of equation 7.4 is thatthe relative sampling variance is a function of the number of particles but not theircombined mass. We can reduce the needed mass by crushing and grinding the particles to make them smaller. Our sample must still contain 1013 particles, but sinceeach particle is smaller their combined mass also is smaller. If we assume that a particle is spherical, then its mass is proportional to the cube of its radius.Mass ∝ r3Decreasing a particle’s radius by a factor of 2, for example, decreases its mass by a factor of 23, or 8. Instead of an 80-g sample, a 10-g sample will now contain 1013 particles.EXAMPLE 7.5Assume that the sample of 1013 particles from Example 7.4 weighs 80 g.
By howmuch must you reduce the radius of the particles if you wish to work with asample of 0.6 g?SOLUTIONTo reduce the sample from 80 g to 0.6 g you must change its mass by a factor of80= 133 times0.6This can be accomplished by decreasing the radius of the particles by a factor ofx3 = 133x = 5.1Decreasing the radius by a factor of approximately 5 allows you to decrease thesample’s mass from 80 g to 0.6 g.Treating a population as though it contains only two types of particles is a useful exercise because it shows us that the relative sampling variance can be improvedby collecting more particles of sample. Furthermore, we learned that the mass ofsample needed can be reduced by decreasing particle size without affecting the relative sampling variance.
Both are important conclusions.Few populations, however, meet the conditions for a true binomial distribution. Real populations normally contain more than two types of particles, with theanalyte present at several levels of concentration. Nevertheless, many well-mixedpopulations, in which the population’s composition is homogeneous on the scale atwhich we sample, approximate binomial sampling statistics. Under these conditionsthe following relationship between the mass of a randomly collected grab sample,m, and the percent relative standard deviation for sampling, R, is often valid.6mR2 = Ks7.5where Ks is a sampling constant equal to the mass of sample producing a percentrelative standard deviation for sampling of ±1%.* The sampling constant is evalu*Problem 8 in the end-of-chapter problem set asks you to consider the relationship between equations 7.4 and 7.5.1400-CH07 9/8/99 4:03 PM Page 189Chapter 7 Obtaining and Preparing Samples for Analysisated by determining R using several samples of similar mass.
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