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Random Samples from a Property Can Miss Problems

 People sometimes analyze soil samples for pesticides, chemicals, or other pollution. This is unreliable if the pollution varies inside the property. For example if you analyze 18 samples from 18 acres and find no arsenic, you still have a 10% chance of missing scattered arsenic that covers fraction .1201 of the site (21/6 acres out of 18). You'd have a 50-50 chance of missing arsenic on 2/3 of an acre (.0378 of 18 acres). Your 18 samples have a 90% chance of missing pollution on fraction .0058 of the site (which is 1/9 of an acre, or over 4,000 square feet). With larger numbers of samples, the area you might miss gets smaller. For example with 500 samples, scattered arsenic totaling 3,600 square feet (.0046 of the 18 acres) would still have a 10% chance of being missed, and 165 square feet would have a 90% chance of being missed. This shows both the importance of large numbers of samples, and the need to know where sources of pollution are, since random samples are not very good at finding small concentrations of pollution. If the 500 samples are spread over 1,000 acres instead of 18 acres, then 200,000 square feet of arsenic scattered on the property would have a 10% chance of being missed, and 9,000 square feet would have a 90% chance of being missed. So when you hear, "We took samples and didn't find any ...", that doesn't mean there's no problem. All this text applies to DDT, lead, or any other pollutant that stays put in soils. We don't mean to pick on arsenic. If you take this many samples and find no problem in them Then you have a 10% 50% 90% chance of missing a problem that's present in the following fraction of the land: 1 .9000 .5000 .1000 10 .2057 .0670 .0105 18 .1201 .0378 .0058 20 .1087 .0341 .0053 30 .0739 .0228 .0035 40 .0559 .0172 .0026 50 .0450 .0138 .0021 60 .0376 .0115 .0018 70 .0324 .0099 .0015 80 .0284 .0086 .0013 90 .0253 .0077 .0012 100 .0228 .0069 .0011 200 .0114 .0035 .0005 300 .0076 .0023 .0004 400 .0057 .0017 .0003 500 .0046 .0014 .0002 600 .0038 .0012 .0002 700 .0033 .0010 .0002 800 .0029 .0009 .0001 900 .0026 .0008 .0001 1000 .0023 .0007 .0001 Methodology: Let P represent the proportion of the site that is contaminated (by some item, beyond some acceptable level; choose your poison). Then each sample has P probability of hitting the contamination, and 1-P probability of missing it. n independent samples have (1-P)n chance that they all miss the contamination. Let C represent this chance that all samples miss the contamination, so C = (1-P)n Solving for P gives the formula P = 1-C1/n , which was used to calculate the numbers above. (This multiplication of independent probabilities assumes scattered contamination, such as loading stations. A systematic grid of 1 sample per acre would have a better chance of finding a continuous area of contamination, such as flooding by a contaminant, or air emissions from a neighboring smelter.)