Samples, sampling methods and the sampling distribution

The population is every unit you want to know about: all sixty households in Ashanagar, all voters in a constituency, all firms in a sector. Each unit is an element of it (CUET 2021). A number that describes the whole population, such as its mean income μ or its standard deviation σ, is a parameter. It is fixed, and normally unknown.

A sample is the part you actually observe. A number computed from it, such as the sample mean x̄ or the sample standard deviation s, is a statistic. It is known, and it changes from sample to sample. Statistical inference is the step from one to the other: a university newspaper that surveys 300 students, finds that 60 are economics majors, and reports that 20% of all students are economics majors has made an inference (CUET 2018).

One consequence is easy to miss. A statistic is a random variable, because which units land in the sample is a matter of chance. That is why a statistic has a distribution, and a parameter does not. The rest of this lesson, and much of the module, comes from taking that sentence seriously.

A sampling method is probability sampling when every unit has a known, non-zero chance of selection, decided by a random device. Only then can the theory of this module say how far the statistic is likely to stray. Everything else is non-probability sampling, where the researcher’s judgement or the respondent’s willingness decides.

Past papers test the classification again and again. Systematic sampling is a probability method (CUET 2021). Cluster sampling is therefore not a non-random method (CUET 2025), and multistage, double and area sampling are all probability designs, while spot sampling is not (UGC NET 2026). When the study population is heterogeneous, stratified random sampling is the appropriate technique (Hyderabad 2015). A matching question (UGC NET 2024) pairs simple random sampling with equal probability of selection, systematic sampling with a random first unit and a fixed gap after it, quota sampling with non-probability sampling, and stratified sampling with random choice inside every stratum.

Reading about methods only goes so far. The lab draws twelve of Ashanagar’s sixty households in five ways, a thousand times each, and compares where the sample means land with the truth, ₹28,783. The sixty households sit in six lanes of ten that differ in how well off they are, as real neighbourhoods do. The data and the lanes are invented.

Read the readouts, not just the pictures. Three things should stand out. Simple random, systematic, stratified and cluster sampling are all unbiased: their sample means average out at the truth. They differ in precision. Stratifying on lanes cuts the scatter to well under half, because it guarantees that rich and poor lanes are both heard; the two-stage cluster design is far more scattered than simple random sampling for the opposite reason. And the volunteer design is biased as well as noisy: it overstates the mean by several thousand rupees, and no amount of repetition changes that, because the error has a direction.

A random method protects against one kind of error: the luck of the draw, called sampling error. It does not protect against everything else. The common failures are all non-sampling errors, and none of them shrinks with a bigger sample.

• A bad frame. The list you draw from is not the population. A phone survey misses households without phones.
• Non-response. The households that decline differ from the ones that answer, usually in exactly the way the survey cares about. The volunteer design in the lab is non-response in its purest form: reply chance rises with income.
• Self-selection. People who choose to reply to an online poll are not a sample of anyone.
• Measurement error. People misreport their income, or the question is worded badly.

The classic example is the 1936 United States election, when a magazine polled millions of its readers, car owners and telephone subscribers and predicted that the challenger would win. The incumbent won easily. The sample was enormous and the frame was the problem: in 1936 the households with cars and telephones were not the households that voted. A far smaller poll drawn with more care called it correctly. A big sample cures sampling error and nothing else.

Take every possible sample of size n from a population and compute the statistic for each one. The list of values, together with how often each occurs, is the sampling distribution of that statistic: the distribution of the different possible values of a statistic with their respective probabilities (JNU 2013). It is a distribution of statistics, not of parameters and not of individual population or sample values (CUET 2019). The standard deviation of that distribution is the standard error, the next lesson’s subject, and it is the standard deviation of a statistic, not of a sample (Hyderabad 2015). CUET 2022 goes further and asks for the “square root of the standard deviation”, which is not the standard error, and marks the statement wrong.

It sounds abstract until you build one. The lab holds a population of six households, A to F, with incomes of ₹12k to ₹58.5k. Choose n and list every sample. The bars show the sampling distribution of the mean.

Two facts come out of the lab, and both hold for any population.

The first says that the sample mean is unbiased: it is right on average, the property the lab shows when the mean of all sample means equals μ to the last decimal (UGC NET 2023). The second says the spread of the sample mean shrinks as the sample grows. When units are drawn without replacement from a finite population, the variance is multiplied by a correction.

Since (N − n)/(N − 1) is less than 1, sampling with replacement gives the larger variance: σ²/n is greater than σ²/n × (N − n)/(N − 1) (IIT JAM 2022). Intuitively, with replacement the same household can turn up twice, which wastes information; without it every draw tells you something new. When N is huge compared with n the correction is close to 1 and is ignored, which is why most formulas leave it out. Switch the lab between the two settings and check the readouts against these formulas.

The lab listed fifteen samples of two from six households. That count is a combination, the number of ways to choose n units from N when order does not matter.

Six households, samples of two: C(6, 2) = 15 without replacement, and 6² = 36 ordered samples with replacement, the two counts the lab shows. The count rises very fast with n, which is why nobody lists the samples in practice and the theory has to do the work instead. Even twelve households out of Ashanagar’s sixty could be chosen in about 1.4 × 10¹² ways.