Multistage Sampling Method

In two-stage cluster sampling, a simple random sample of clusters is selected and then a simple random sample is selected from the units in each sampled cluster.

Multi stage sampling is a generalisation of two stage sampling. As the name suggests, multi stage sampling is carried out in different stages. In each stage progressively smaller (population) geographic areas will be randomly selected. 

Example:

A political pollster interested in assembly elections in Andhra Pradesh may first divide the state into different assembly units and a sample of assembly constituencies may be selected in the first stage.

In the second stage, each of the sampled assembly constituents are divided into a number of segments and a second stage sampled assembly segments may be selected.

In the third stage within each sampled assembly segment either all the house-holds or a sample random of households would be interviewed.

In this sampling method, it is possible to take as many stages as are necessary to achieve a representative
sample. Each stage results in a reduction of sample size.

In a multi stage sampling at each stage of sampling a suitable method of sampling is used. More number of stages are used to arrive at a sample of desired sampling units.

Advantages

a) Multistage sampling provides cost gains by reducing the data collection on costs.

b) Multistage sampling is more flexible and allows us to use different sampling procedures in different stages of sampling.

c) If the population is spread over a very wide geographical area, multistage sampling is the only sampling method available in a number of practical situations.

Limitations

a) If the sampling units selected at different stages are not representative multistage sampling becomes less precise and efficient.

How systematic sampling is differ from Stratified Random Sampling

Systematic sampling and stratified random sampling are both probability sampling techniques used to select a representative sample from a population, but they go about it in different ways:

Stratified Random Sampling:

• Divides the population: Here, you first divide the entire population (let's say, all students in a school) into subgroups (strata) based on shared characteristics (like grade level). These subgroups should be mutually exclusive and collectively exhaustive (every member of the population belongs to exactly one subgroup).
• Random selection within subgroups: Then, you randomly select a sample from each subgroup. This ensures all relevant subgroups are represented in the final sample.

Systematic Sampling:

• Ordering the population: This method treats the population as a single list in some order (like a list of students in alphabetical order).
• Fixed interval selection: You decide on a sampling interval by dividing the total population size by your desired sample size. Then, you pick a random starting point from the list and select every nth element thereafter based on the interval.

Here's a table summarizing the key differences:

Feature	Stratified Random Sampling	Systematic Sampling
Divides population	Yes, into subgroups (strata)	No
Basis for selection	Random selection within subgroups	Fixed interval selection from ordered list
Risk of bias	Lower, ensures all subgroups are represented	Higher if the ordering coincides with a pattern in the population

Choosing the right method:

• Use stratified random sampling if you have a diverse population with subgroups you want to be sure are represented in the sample.
• Use systematic sampling if ordering the population is easy and there's no underlying pattern or cyclical trend within the population that might bias your selection. It can also be slightly more efficient to implement than stratified sampling.

Example:

Imagine you want to survey students about their preferred lunch options.

• Stratified Random Sampling: You could divide the students into subgroups by grade level (strata). Then, randomly select a sample from each grade level to ensure all grade levels have a voice.
• Systematic Sampling: If the student list is in alphabetical order (not ideal, but possible), you could choose a random starting student and then survey every 10th student on the list (assuming you want a 10% sample). However, this could be biased if, for example, taller students tend to be placed alphabetically later (and perhaps have different lunch preferences).

Stratified sampling method - proportional stratified sample and disproportional stratified sample

The stratified sampling method is used when the population is heterogeneous rather than homogeneous.

A heterogeneous population is composed of unlike elements such as male/female, rural/urban, literate/illiterate, high income/low income groups, etc.

In such cases, use of simple random sampling may not always provide a representative sample of the
population.

In stratified sampling, we divide the population into relatively homogeneous groups called strata. Then we select a sample using simple random sampling from each stratum.

There are two approaches to decide the sample size from each stratum, namely, proportional stratified sample and disproportional stratified sample. With either approach, the stratified sampling guarantees that every unit in the population has a chance of being selected. 

We will now discuss these two approaches of selecting samples.

Imagine you're a researcher studying political opinions, and you want to ensure your findings represent the whole population. But the population itself is diverse - there might be young voters, retirees, and working professionals. A simple random sample might miss a group entirely!

This is where stratified sampling comes in. You divide the population (voters) into subgroups (strata) based on relevant characteristics (age groups). Then, you draw a random sample from each subgroup.

Choosing sample sizes from these subgroups is where proportional and disproportional stratified sampling differ:

• Proportional Stratified Sample: Here, the sample size within each subgroup reflects its proportion in the entire population. Imagine voters are 60% young, 30% retirees, and 10% working professionals. In a sample of 100, you'd take 60 from young voters, 30 from retirees, and 10 from professionals. This ensures all groups are represented proportionally.

  • Example: A school administrator wants to survey students about a new lunch program. The school has 60% freshmen, 20% sophomores, 15% juniors, and 5% seniors. If the administrator wants a total sample of 200 students, they would take a proportional stratified sample: 120 freshmen, 40 sophomores, 30 juniors, and 10 seniors.

• Disproportional Stratified Sample: This approach focuses on specific subgroups you want to study more intensely. Maybe you're particularly interested in the opinions of young voters (60% of the population). You could take a larger sample, say 70, from them, while taking a smaller sample, like 20 each, from retirees and professionals. This way, you get more data on the group of interest (young voters) for better analysis.

  • Example: Researchers want to understand the movie preferences of different age groups. They know teenagers are a smaller population segment (15%) but have a strong influence on movie trends. So, they take a disproportional stratified sample: 40 teenagers, 30 young adults (25% of population), 20 middle-aged adults (40% of population), and 10 seniors (20% of population). This allows a more focused study on teenagers' preferences.