Chapter 2 Introduction to probability sampling

To estimate population parameters like the mean or the total, probability sampling is most appropriate. Probability sampling is random sampling using a random number generator such that all population units have a probability larger than zero of being selected, and these probabilities are known for at least the selected units.

The probability that a unit is included in the sample, in short the inclusion probability of that unit, can be calculated as the sum of the selection probabilities over all samples that can be selected with a given sampling design and that contain this unit. In formula:

\[\begin{equation} \pi_k = \sum_{\mathcal{S} \ni k} p(\mathcal{S}) \;, \tag{2.1} \end{equation}\]

where \(\mathcal{S} \ni k\) indicates that the sum is over all samples that contain unit \(k\), and \(p(\mathcal{S})\) is the selection probability of sample \(\mathcal{S}\). \(p(\cdot)\) is called the sampling design. It is a function that assigns a probability to every possible sample (subset of population units) that can be selected with a given sample selection scheme (sampling algorithm). For instance, consider the following sample selection scheme from a finite population of \(N\) units:

  1. Select with equal probability \(1/N\) a first unit.
  2. Select with equal probability \(1/(N-1)\) a second unit from the remaining \(N-1\) units.
  3. Repeat this until an \(n\)th unit is selected with equal probability from the \(N-(n-1)\) units.

This is a selection scheme for simple random sampling without replacement. With this scheme the selection probability of any sample of \(n\) units is \(1/\binom{N}{n}\) (there are \(\binom{N}{n}\) samples of size \(n\), and each sample has an equal selection probability), and zero for all other samples. There are \(\binom{N-1}{n-1}\) samples of size \(n\) in which unit \(k\) is included. The inclusion probability of each unit \(k\) therefore is \(\binom{N-1}{n-1}/\binom{N}{n}=\frac{n}{N}\). The sampling design plays a key role in the design-based approach as it determines the sampling distribution of random quantities computed from a sample such as the estimator of the population mean, see Section 2.1. The number of selected population units is referred to as the sample size.

In sampling with replacement, each unit can be selected more than once. In this case, the sample size refers to the number of draws, not to the number of unique population units in the sample.

A common misunderstanding is that with probability sampling the inclusion probabilities must be equal. Sampling with unequal inclusion probabilities can be more efficient than with equal probabilities. Unequal probability sampling is no problem as long as the inclusion probabilities are known and proper formulas are used for estimation, see Section 2.1.

There are many schemes for selecting a probability sample. The following sampling designs are described and illustrated in this book:

  1. simple random sampling;
  2. stratified random sampling;
  3. systematic random sampling;
  4. cluster random sampling;
  5. two-stage cluster random sampling;
  6. sampling with probabilities proportional to size;
  7. balanced and well-spread sampling; and
  8. two-phase random sampling.

The first five sampling designs are basic sampling designs. Implementation of these designs is rather straightforward, as well as the associated estimation of the population mean, total, or proportion, and their sampling variance. The final three sampling designs are more advanced. Appropriate use of these designs requires more knowledge of sampling theory and statistics, such as linear regression.

Arbitrary (haphazard) sampling vs. probability sampling

In publications it is commonly stated that the sampling units were selected at random (within strata), without further specification of how the sampling units were precisely selected. In statistical inference, the sampling units are subsequently treated as if they were selected by (stratified) simple random sampling. With probability sampling, all units in the population have a positive probability of being selected, and the inclusion probabilities are known for all units. It is highly questionable whether this also holds for arbitrary and haphazard sampling. In arbitrary and haphazard sampling, the sampling units are not selected by a probability mechanism. So, the selection probabilities of the sampling units and of combinations of sampling units are unknown. Design-based estimation is therefore impossible, because it requires the inclusion probabilities of the population units as determined by the sampling design. The only option for statistical analysis using arbitrarily or haphazardly selected samples is model-based inference, i.e., a model of the spatial variation must be assumed.


  1. Suppose a researcher selects a sample of points from a study area by throwing darts on a map depicting the study area. Is the resulting sample a probability sample? If not, why?

2.1 Horvitz-Thompson estimator

For any probability sampling design, the population total can be estimated as a weighted sum of the observations (measurements) of the study variable on the selected population units:

\[\begin{equation} \hat{t}_{\pi}(z)=\sum_{k \in \mathcal{S} } w_k z_k \;, \tag{2.2} \end{equation}\]

with \(\mathcal{S}\) the sample, \(z_k\) the observed study variable for unit \(k\), and \(w_k\) the design weight attached to unit \(k\):

\[\begin{equation} w_k = \frac{1}{\pi_k}\;, \tag{2.3} \end{equation}\]

with \(\pi_k\) the inclusion probability of unit \(k\). The estimator of Equation (2.2) is referred to as the Horvitz-Thompson estimator or \(\pi\) estimator. The \(z_k/\pi_k\)-values are referred to as the \(\pi\)-expanded values. The \(z\)-value of unit \(k\) in the sample is multiplied by the reciprocal of the inclusion probability of that unit, and the sample sum of these \(\pi\)-expanded values is used as an estimator of the population total. The inclusion probabilities are determined by the type of sampling design and the sample size.

An estimator is not the same as an estimate. Whereas an estimate is a particular value calculated from the sample data, an estimator is a formula for estimating a parameter. An estimator is a random variable and therefore has a probability distribution. For this reason, it is not correct, although very common, to say ‘the variance (standard error) of the estimated population mean equals …’. It is correct to say ‘the variance (standard error) of the estimator of the population mean equals …’.

Also for infinite populations, think of points in a continuous population, the above estimator for the population total can be used, but special attention must then be paid to the inclusion probabilities. Suppose the infinite population is discretised by \(N\) cells of a fine grid, and a simple random sample of \(n\) cells is selected. The inclusion probabilities of the grid cells is then \(n/N\). However, constraining the sampling points to the centres of the cells of the discretisation grid is not needed and even undesirable. To account for the infinite number of points in the population we may adopt a two-step approach, see Figure 1.2. In the first step, \(n\) cells of the discretisation grid are selected by simple random sampling with replacement. In the second step, one point is selected fully randomly from the selected grid cells. If a grid cell is selected more than once, more points are selected in that grid cell. With this selection procedure the inclusion probability density is \(n/A\), with \(A\) the area of the study area. This inclusion probability density equals the expected number of sampling points per unit area, e.g., the expected number of points per ha or per m2. The inclusion probability density can be interpreted as the global sampling intensity. Note that the local sampling intensity may strongly vary; think for instance of cluster random sampling.

The \(\pi\) estimator for the mean of a finite population, \(\bar{z}\), is simply the \(\pi\) estimator for the total, divided by the total number of units in the population, \(N\):

\[\begin{equation} \hat{\bar{z}}_{\pi}=\frac{1}{N} \sum_{k \in \mathcal{S}} \frac{1}{\pi_k}z_k \;. \tag{2.4} \end{equation}\]

For infinite populations discretised by a finite set of points, the same estimator can be used.

For infinite populations, the population total can be estimated by multiplying the estimated population mean by the area of the population \(A\):

\[\begin{equation} \hat{t}_{\pi}(z)=A \hat{\bar{z}}_{\pi} \;. \tag{2.5} \end{equation}\]

The \(\pi\) estimator can be worked out for the different types of sampling design listed above by inserting the inclusion probabilities as determined by the sampling design. For simple random sampling, this leads to the unweighted sample mean (see Chapter 3), and for stratified simple random sampling the \(\pi\) estimator is equal to the weighted sum of the sample means per stratum, with weights equal to the relative size of the strata (see Chapter 4).

2.2 Hansen-Hurwitz estimator

In sampling finite populations, units can be selected with or without replacement. In sampling with replacement after each draw the selected unit is replaced. As a consequence, a unit can be selected more than once. Sampling with replacement is less efficient than sampling without replacement. If a population unit is selected in a given draw, there is no additional information in this unit if it is selected again. One reason that sampling with replacement is still used is that it is easier to implement.

The most common estimator used for sampling with replacement is the Hansen-Hurwitz estimator, referred to as the \(p\)-expanded with replacement (pwr) estimator by Särndal, Swensson, and Wretman (1992). With direct unit sampling, i.e., sampling of individual population units, the pwr estimator is

\[\begin{equation} \hat{t}_{\text{pwr}}(z)=\frac{1}{n}\sum_{k \in \mathcal{S} } \frac{z_k}{p_k} \;, \tag{2.6} \end{equation}\]

with \(p_k\) the draw-by-draw selection probability of population unit \(k\). For instance, in simple random sampling with replacement the draw-by-draw selection probability \(p\) of each unit is \(1/N\). If we select only one unit \(k\), the population total can be estimated by the observation of that unit divided by \(p\), \(\hat{t}(z) = z_k/p_k = N z_k\). If we repeat this \(n\) times, this results in \(n\) estimated population totals. The pwr estimator is the average of these \(n\) elementary estimates. If a unit occurs multiple times in the sample \(\mathcal{S}\), this unit provides multiple elementary estimates of the population total.

A sample obtained by sampling with replacement is referred to as an ordered sample (Särndal, Swensson, and Wretman 1992). Selecting the distinct units from this ordered sample results in the set-sample. Instead of using the ordered sample in the pwr estimator, we may use the set-sample in the \(\pi\) estimator. This requires computation of the inclusion probabilities for sampling with replacement. For instance, for simple random sampling with replacement, the inclusion probability of each unit equals \(1-\left(1-\frac{1}{N}\right)^n\), with \(n\) the number of draws. This probability is smaller than \(n/N\), the inclusion probability for simple random sampling without replacement. There is no general rule on which estimator is most accurate (Särndal, Swensson, and Wretman 1992). In this book I only use the pwr estimator for sampling with replacement.

Sampling with replacement can also be applied at the level of clusters of population units as in cluster random sampling and two-stage cluster random sampling. If the clusters are selected with probabilities proportional to their size and with replacement, estimation of a population parameter is rather simple. This is a second reason why sampling with replacement can be attractive. With cluster sampling, the Hansen-Hurwitz estimator is

\[\begin{equation} \hat{t}_{\text{pwr}}(z)=\frac{1}{n}\sum_{j \in \mathcal{S} } \frac{t_j(z)}{p_j} \;, \tag{2.7} \end{equation}\]

with \(t_j(z)\) the total of the cluster selected in the \(j\)th draw. If not all population units of a selected cluster are observed, but only a sample of population units from a cluster, as in two-stage cluster random sampling, the cluster totals \(t_j(z)\) are replaced by the estimated cluster totals \(\hat{t}_j(z)\).


  1. Consider a population of four units (\(N=4\)). What is the inclusion probability of each population unit for simple random sampling without replacement and simple random sampling with replacement of two units (\(n=2\))?

2.3 Using models in design-based approach

Design-based estimates of population parameters such as the mean, total, or proportion (areal fraction) are model-free: no use is made of a model for the spatial variation of the study variable. However, such a model can be used to optimise the probability sampling design. In Chapter 13 I describe how a model can be used to compare alternative sampling designs at equal costs or equal precision to evaluate which sampling design performs best, to optimise the sample size given a requirement on the precision of the estimated population parameter, or to optimise the spatial strata for stratified random sampling.

A model of the spatial variation can also be used at a later stage, after the data have been collected, in estimating the population parameter of interest. If one or more ancillary variables that are related to the study variable are available, these variables can be used in estimation to increase the accuracy. This leads to alternative estimators, such as the regression estimator, the ratio estimator, and the poststratified estimator (Chapter 10). These estimators together are referred to as model-assisted estimators. In model-assisted estimation the inclusion probabilities, as determined by the random sampling design, play a key role, but besides, modelling assumptions about how the population might have been generated are used to work out an efficient estimator. The role of a model in the model-assisted approach is fundamentally different from its role in the model-based approach. This is explained in Chapter 26.

For novices in geostatistics, Chapters 10 and 13 can be quite challenging, and I recommend skipping these chapters and only return to them after having read the introductory chapter on geostatistics (Chapter 21).


Särndal, C. E., B. Swensson, and J. Wretman. 1992. Model Assisted Survey Sampling. New York: Springer.