Sampling methods for surveys
Survey sampling: frames and errors, simple random, systematic, stratified, cluster and multistage sampling, sample size, and crop-cutting surveys.
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India estimates the yield of rice in every district each season without harvesting every field. It estimates livestock numbers without counting every animal on one day, and farm incomes without interviewing every household. All of this depends on sampling: studying a carefully chosen part of a population and using it to draw conclusions about the whole. Well-designed samples give estimates of known precision at a fraction of the cost and time of a complete count, or census, and they are often more accurate, because a smaller survey can be done more carefully.
Indian statisticians shaped modern survey sampling. P. C. Mahalanobis at the Indian Statistical Institute pioneered large-scale sample surveys and the National Sample Survey. P. V. Sukhatme at the Indian Council of Agricultural Research developed the crop-cutting experiment method for estimating crop yields.
Basic terms
- Population: the whole set of units of interest, such as all farm households in a district.
- Sampling unit: the unit selected, such as a village, a household or a field.
- Sampling frame: the list or map from which units are drawn, such as village lists or land records. A poor frame means some units can never be selected.
- Parameter: a population value, such as the true mean yield. A statistic or estimate is the value calculated from the sample.
- Sampling error: the difference between the estimate and the true value that arises because only a sample was observed. It can be measured and reduced by increasing the sample size.
- Non-sampling errors: errors from non-response, measurement mistakes, poor frames, interviewer bias and data processing. They occur in censuses too, and are often larger than sampling error.
Probability and non-probability sampling
In probability sampling, every unit has a known, non-zero chance of selection, so precision can be calculated. Non-probability methods, such as purposive, convenience, quota and snowball sampling, are cheap and useful for pilot studies or qualitative work, but their estimates cannot be given valid standard errors.
Simple random sampling (SRS)
Each possible sample of size n is equally likely. Units are drawn using random number tables or software. Sampling can be with replacement or, more commonly, without replacement (SRSWOR).
The term (1 − f) is the finite population correction. It can be ignored when f is below about 5%.
Sample size
To estimate a mean within a margin d at 95% confidence:
How many fields to sample?
Paddy yields in a block have a CV of about 30% across fields. To estimate the mean yield within ±5% of its true value at 95% confidence: n0 = (1.96 × 30 ⁄ 5)2 = 11.762 ≈ 139 fields. Accepting ±10% reduces this to about 35 fields. Halving the permissible error quadruples the sample size, which is why survey designers think hard about the precision they really need.
Systematic sampling
Systematic sampling selects every k-th unit after a random start between 1 and k, where k = N ⁄ n. It is simple to apply in the field, for example along rows of trees or in lists of households. It spreads the sample evenly, which often gives better precision than SRS. It can be badly biased, though, if the list has a periodic pattern that matches k.
Stratified sampling
Stratified sampling divides the population into strata: groups that are internally similar but different from each other, such as farm-size classes, irrigated and rainfed land, or agro-climatic zones. An independent random sample is drawn from each stratum.
- Precision gains come because variation between strata is removed from the error.
- Separate estimates for each stratum become available.
- Different methods can be used in different strata.
The stratified mean is ȳst = Σ Wh ȳh, where Wh = Nh ⁄ N is the stratum weight. Sample sizes can be allocated in three ways:
- Equal: the same n in each stratum.
- Proportional: nh ∝ Nh.
- Optimum (Neyman): nh ∝ Nh Sh. More sample goes to large and variable strata, and, if costs differ, to cheaper ones.
Cluster and multistage sampling
Cluster sampling selects whole groups of units, such as villages, and surveys every unit within them. It is cheap when travel is costly and no list of individual units exists. Units within a cluster tend to be similar, however, so it is usually less precise than SRS of the same number of units.
Multistage sampling selects at several levels. For example:
- first-stage units are villages;
- second-stage units are households or fields within the selected villages;
- third-stage units are plots within fields.
Most national agricultural and socio-economic surveys are multistage and stratified. Units are often selected with probability proportional to size (PPS), so that larger villages have a higher chance of selection.
Crop-cutting experiments
The General Crop Estimation Surveys (GCES) estimate yields of major crops in India using crop-cutting experiments (CCEs):
- Within each district or block (the stratum), villages are selected at random.
- Within selected villages, fields growing the crop are selected at random.
- Within each field, a plot of fixed size and shape (for example, a 5 m × 5 m square, located using random coordinates) is harvested, threshed and weighed.
- The yield is converted to a dry-grain basis and to kg per hectare, then combined with crop area to give production.
Millions of CCEs are conducted each year. They are used for production statistics and for settling crop insurance claims under the Pradhan Mantri Fasal Bima Yojana. Smartphone apps, remote sensing and technology-based yield estimation are now used to improve their timeliness and accuracy.
Key points
- Sampling estimates population values from part of the population. A census counts everything.
- Sampling error falls with sample size; non-sampling errors (frame, non-response, measurement) need careful field work.
- SRS: SE(ȳ) = √[(1 − f)s2 ⁄ n]; sample size n0 = (z s ⁄ d)2 or (z CV ⁄ e)2.
- Systematic sampling takes every k-th unit (k = N ⁄ n); watch for periodicity.
- Stratified sampling improves precision. Allocation can be equal, proportional or Neyman (NhSh).
- Cluster sampling is cheap but less precise; multistage and PPS designs underlie national surveys, including crop-cutting experiments.
What the exams ask
- Define population, sampling frame, sampling unit, and sampling and non-sampling errors.
- Compare SRS, systematic, stratified and cluster sampling.
- Calculate the sample size needed for a given precision.
- Explain proportional and Neyman allocation.
- Describe the crop-cutting experiment method.
Described in general terms: the kind of examination, not a past paper. Laisha never reproduces exam questions.
Questions people ask
Is a bigger sample always better?
It gives smaller sampling error, but costs more and can increase non-sampling error if field staff are stretched. A well-designed, carefully conducted moderate sample usually beats a large, sloppy one. Precision also improves roughly with √n, so doubling n reduces the standard error by only about 30%.
When should I stratify?
Stratify when the population contains distinct groups that differ in the variable being measured, and when you know which group each unit belongs to before sampling. Examples include irrigated and rainfed fields, or small and large farms. Stratification also guarantees that every group is represented.
Why are crop-cutting plots placed at random?
People tend to choose plots that look typical, which often means better than average or away from the edges. Random coordinates remove this bias. The rule "harvest wherever the random point falls" is essential for an unbiased estimate.
Sources
- Cochran, W. G. Sampling Techniques, 3rd edition. Wiley, 1977.
- Sukhatme, P. V., Sukhatme, B. V., Sukhatme, S. and Asok, C. Sampling Theory of Surveys with Applications, 3rd edition. Iowa State University Press and Indian Society of Agricultural Statistics, 1984.
- Mahalanobis, P. C. "Recent experiments in statistical sampling in the Indian Statistical Institute." Journal of the Royal Statistical Society 109, 325–378 (1946).
- Singh, D. and Chaudhary, F. S. Theory and Analysis of Sample Survey Designs. New Age International, 1986.
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