Simple, Stratified Random, and Systematic Sampling With Examples

TL;DR

Simple random sampling requires a complete population frame so every element has an equal and known chance of selection.

Briefing Cornell Notes

Briefing

Sampling methods hinge on one practical requirement: who can be selected, and how selection stays fair. Simple random sampling gives every element an equal, known chance of selection—but only if researchers start with a complete population frame (a full listing of all elements). Without that complete list, “random” selection collapses into convenience or bias, such as handing out questionnaires to people who look more approachable or who happen to be nearby.

In the simple random sampling example, researchers want 5 students out of 20. They first compile a full list of the 20 students (including contact details so selected students can be reached). Then they generate random numbers using a random number generator (the transcript mentions random.org), with the range set to 1–20. The generated numbers determine which students enter the sample—e.g., numbers like 19, 3, 11, 17, and 19 map to the chosen students. If selected students are unavailable, the process repeats: researchers randomly sample again from the remaining students until the required sample size is met.

The limitations of simple random sampling show up when populations contain meaningful subgroups. Because larger groups naturally contribute more elements, simple random sampling can under-represent smaller strata. The transcript illustrates this with a population of 1,000 students split into Bachelor (500), Master (300), MPhil/MS (150), and PhD (50), while the target sample size is 100. Proportionate stratified random sampling fixes the imbalance by drawing a set percentage from each group. Since 100 is 10% of 1,000, the method selects 10% from every stratum: 50 Bachelor students, 30 Master students, 15 MPhil/MS students, and 5 PhD students. The same logic scales to other totals (e.g., 600 out of 3,000 implies 20% from each group).

Stratified sampling can also be adjusted when some groups are effectively over- or under-represented for the study’s goals. The transcript describes changing the sampling proportions (for example, shifting from 50/30/15/5 to other splits) while keeping the overall sample size constant. It also flags a real-world constraint: response rates. If questionnaires are sent to only the “minimum” number implied by the sampling fraction, nonresponse can shrink the usable sample below the target. To counter this, researchers increase the number contacted—doubling or otherwise raising the sent invitations (e.g., sending to 100 instead of 50) so that enough respondents remain.

Systematic sampling offers a different workaround when a complete population frame is unavailable. Instead of listing every element, researchers rely on the total population size (N) and the desired sample size (n). They compute the sampling interval k = N/n and then select every k-th element, starting from a randomly chosen point between 1 and k. The transcript uses bank customers as an example: if 300 customers arrive daily and the target sample is 50, the interval is 300/50 = 6, so every sixth customer is approached until 50 customers are sampled. Similar gate-based approaches apply in malls or universities, such as selecting every 10th student who enters a study area until the required sample size is reached.

Cornell Notes

Simple random sampling requires a complete population frame so each element has an equal, known chance of selection. Researchers list all elements (e.g., 20 students), generate random numbers within the correct range, and select the corresponding students; if someone is unavailable, they resample from the remaining list. Stratified random sampling addresses subgroup imbalance by sampling within each stratum (e.g., Bachelor, Master, MPhil/MS, PhD) using proportions based on the overall sampling fraction, such as 10% from each group when the sample is 100 out of 1,000. When response rates are low, the number contacted should be increased beyond the minimum implied by the sampling fraction. Systematic sampling works without a full list by using the total population size and sample size to choose every k-th element (e.g., every 6th customer when 300 customers and a sample of 50).

Why does simple random sampling fail if researchers don’t have a complete population frame?

Equal and known selection probabilities depend on having a full listing of all elements. Without a complete list, there’s no defensible way to assign each element an equal chance. The transcript contrasts correct random selection (using a full list of 20 students and random numbers) with a biased “random” practice like approaching whoever looks more friendly or more accessible in an office.

How does the transcript’s example operationalize simple random sampling for selecting 5 students from 20?

Researchers first create a list of all 20 students (with contact details so selected students can be reached). They then generate five random numbers using a random number generator, with the range set to 1–20. The students whose numbers match the generated values become the sample. If selected students are unavailable, the method repeats random sampling from the remaining students until five are obtained.

What problem does stratified random sampling solve that simple random sampling can’t?

Simple random sampling can under-represent smaller groups because larger groups contain more elements and therefore dominate the random draws. In the example, PhD students are only 50 out of 1,000, so a simple random sample of 100 would likely include too few PhD students relative to their presence in the population.

How are sampling proportions determined in proportionate stratified random sampling?

Sampling proportions follow the overall sampling fraction. With a population of 1,000 and a target sample of 100, the fraction is 100/1,000 = 10%, so the method selects 10% from each stratum: 50 Bachelor (500×10%), 30 Master (300×10%), 15 MPhil/MS (150×10%), and 5 PhD (50×10%). For a different setup (e.g., 600 out of 3,000), the fraction becomes 20% and the stratum counts scale accordingly.

How does systematic sampling work when there’s no complete list of elements?

Systematic sampling uses only the total population size N and the desired sample size n. It computes an interval k = N/n and then selects every k-th element, starting from a randomly chosen starting point between 1 and k. The bank example uses N=300 customers and n=50, giving k=6, so every sixth arriving customer is approached until 50 customers are sampled.

Review Questions

What specific prerequisite makes equal-probability simple random sampling possible, and what goes wrong without it?
In proportionate stratified random sampling, how do you compute how many units to sample from each stratum?
When using systematic sampling without a population frame, what values are required to determine the sampling interval, and how is the starting point chosen?

Key Points

1
Simple random sampling requires a complete population frame so every element has an equal and known chance of selection.
2
Random selection should be driven by a random number generator over the correct range, not by convenience or who happens to be approachable.
3
If selected units are unavailable, researchers should resample from the remaining population to restore the intended sample size.
4
Stratified random sampling corrects subgroup under-representation by drawing samples within each stratum (group).
5
Proportionate stratified sampling sets stratum sample sizes using the overall sampling fraction (e.g., 10% from each group when sampling 100 out of 1,000).
6
Sampling proportions can be adjusted when study goals require different representation across groups, but the total contacted should still support the minimum usable sample.
7
Systematic sampling can be used without a full list by selecting every k-th element, where k = total population size divided by desired sample size, starting from a random point in 1..k.

Highlights

Simple random sampling only works as “random” when the full population frame exists; otherwise selection probabilities can’t be justified.

Proportionate stratified sampling turns the overall sampling fraction into stratum-specific counts—like taking 10% from every degree level when sampling 100 of 1,000.

Systematic sampling replaces a population list with arithmetic: compute k = N/n and approach every k-th customer or entrant until the sample size is reached.