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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.)