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•Sampling errors are those due to the random selection of sample units for measurement. Nonsampling errors are those from all other sources. The control of both sampling and nonsampling errors is important for all studies, using appropriate protocols and procedures for establishing data quality objectives (DQO) and quality control and assurance (QC/QA).

•Stratified random sampling is sometimes useful for ensuring that the units that are measured are well representative of the population. However, there are potential problems due to the use of incorrect criteria for stratification or the wish to use a different form of stratification to analyze the data after they have been collected.

•Equations for estimating the population mean and total are provided for stratified sampling.

•Post-stratification can be used to analyze the results from simple random sampling as if they had been obtained from stratified random sampling.

•With systematic sampling, the units measured consist of every kth item in a list, or are regularly spaced over the study area. Units may be easier to select than they are with random sampling, and estimates may be more precise than they would otherwise be because they represent the whole population well. Treating a systematic sample as a simple random sample may overestimate the true level of sampling error. Two alternatives to this approach (imposing a stratification and joining sample points with a serpentine line) are described.

•Cluster sampling (selecting groups of close sample units for measurement) may be a cost-effective alternative to simple random sampling.

•Multistage sampling involves randomly sampling large primary units, randomly sampling some smaller units from each selected primary unit, and possibly randomly sampling even smaller units from each secondary unit, etc. This type of sampling plan may be useful when a hierarchical structure already exists in a population.

•Composite sampling is potentially valuable when the cost of measuring sample units is much greater than the cost of selecting and collecting them. It involves mixing several sample units from the same location and then measuring the mixture. If there is a need to identify sample units with extreme values, then special methods are available for doing this without measuring every sample unit.

•Ranked-set sampling involves taking a sample of m units and putting them in order from the smallest to largest. The largest unit only is then measured. A second sample of m units is then selected, and the second largest unit is measured. This is continued until the mth sample of m units is taken and the smallest unit is measured. The mean of the m measured units will then usually be a more accurate

Exercise 2.2

Given the data from question 2.1, suppose that you have to propose a sampling design to estimate the difference between the SO4 levels in two different regions of the same country, both containing approximately 200 small lakes. What sample sizes do you propose to obtain confidence limits of about x1 − x2 ± 0.75?

Exercise 2.3

A study involved taking 40 partly systematic samples from a 1 × 1-m grid in a 1-ha field and measuring the sand content (%) of the soil. The measurements on sand content are as shown in Figure 2.9.

1.Estimate the mean sand content of the field with a standard error and 95% confidence limits, treating the sample as being effectively a random sample of points.

2.Estimate the mean sand content of the field with a standard error and 95% confidence limits using the first of the US EPA methods described in Section 2.9 (combining adjacent points into strata and using the stratified sampling formulae).

3.Estimate the mean sand content in the field, with a standard error and 95% confidence limits, using the second method recommended by the US EPA (joining points with a serpentine line).

4.Compare the results found in items 1 to 3 above and discuss reasons for the differences, if any.

3

Models for Data

3.1  Statistical Models

Many statistical analyses are based on a specific model for a set of data, where this consists of one or more equations that describe the observations in terms of parameters of distributions and random variables. For example, a simple model for the measurement X made by an instrument might be

X = θ + ε

where θ is the true value of what is being measured, and ε is a measurement error that is equally likely to be anywhere in the range from −0.05 to +0.05.

In situations where a model is used, an important task for the data analyst is to select a plausible model and to check, as far as possible, that the data are in agreement with this model. This includes examining both the form of the equation assumed and the distribution or distributions that are assumed for the random variables.

To aid in this type of modeling process there are many standard distributions available, the most important of which are considered in the following two sections of this chapter. In addition, there are some standard types of models that are useful for many sets of data. These are considered in the later sections of this chapter.

3.2  Discrete Statistical Distributions

A discrete distribution is one for which the random variable being considered can only take on certain specific values, rather than any value within some range (Appendix 1, Section A1.2). By far the most common situation in this respect is where the random variable is a count, and the possible values are 0, 1, 2, 3, and so on.

It is conventional to denote a random variable by a capital X and a particular observed value by a lowercase x. A discrete distribution is then defined

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64 Statistics for Environmental Science and Management, Second Edition

Figure 3.1

Examples of hypergeometric, binomial, and Poisson discrete probability distributions.

3.2.3  The Poisson Distribution

One derivation of the Poisson distribution is as the limiting form of the binomial distribution as n tends to infinity and p tends to zero, with the mean μ = np remaining constant. More generally, however, it is possible to derive it as the distribution of the number of events in a given interval of time or a given area of space when the events occur at random, independently of each other at a constant mean rate. The probability function is

The mean and variance are both equal to μ.

In terms of events occurring in time, the type of situation where a Poisson distribution might occur is for counts of the number of occurrences of minor oil leakages in a region per month, or the number of cases per year of

a rare disease in the same region. For events occurring in space, a Poisson distribution might occur for the number of rare plants found in randomly selected meter-square quadrats taken from a large area. In reality, though, counts of these types often display more variation than is expected for the Poisson distribution because of some clustering of the events. Indeed, the ratio of the variance of sample counts to the mean of the same counts, which should be close to 1 for a Poisson distribution, is sometimes used as an index of the extent to which events do not occur independently of each other.

Figure 3.1(c) shows some examples of probabilities calculated for some particular Poisson distributions.

3.3  Continuous Statistical Distributions

Continuous distributions are often defined in terms of a probability density function, f(x), which is a function such that the area under the plotted curve between two limits a and b gives the probability of an observation within this range, as shown in Figure 3.2. This area is also the integral between a and b, so that, in the usual notation of calculus,

The total area under the curve must be exactly 1, and f(x) must be greater than or equal to 0 over the range of possible values of x for the distribution to make sense.

Figure 3.2

The probability density function f(x) for a continuous distribution. The probability of a value between a and b is the area under the curve between these values, i.e., the area between the two vertical lines at x = a and x = b.