RANDOM SAMPLING: METHODOLOGY, ADVANTAGES, DISADVANTAGES, EXAMPLES - MATHS - 2025

The random sampling is how to select a statistically representative sample from a given population. Part of the principle that every element in the sample must have the same probability of being selected.

A draw is an example of random sampling, in which each member of the participant population is assigned a number. To choose the numbers corresponding to the raffle prizes (the sample) some random technique is used, for example extracting from a mailbox the numbers that were recorded on identical cards.

Figure 1. In random sampling, the sample is drawn from the population at random using some technique that ensures that all elements have the same probability of being chosen. Source: netquest.com.

In random sampling, it is essential to choose the sample size appropriately, because a non-representative sample of the population can lead to erroneous conclusions, due to statistical fluctuations.

The size of the sample

There are formulas for determining the proper size of a sample. The most important factor to consider is whether or not the population size is known. Let's look at the formulas to determine the sample size:

Case 1: the size of the population is unknown

When the population size N is unknown, it is possible to select a sample of adequate size n to determine whether a certain hypothesis is true or false.

For this, the following formula is used:

Where:

-p is the probability that the hypothesis is true.

-q is the probability that it is not, therefore q = 1 - p.

-E is the relative margin of error, for example an error of 5% has a margin of E = 0.05.

-Z has to do with the level of confidence required by the study.

In a standardized (or normalized) normal distribution, a confidence level of 90% has Z = 1.645, because the probability that the result is between -1.645σ and + 1.645σ is 90%, where σ is the standard deviation.

Confidence levels and their corresponding Z values

1.- 50% confidence level corresponds to Z = 0.675.

2.- 68.3% confidence level corresponds to Z = 1.

3.- 90% confidence level is equivalent to Z = 1,645.

4.- 95% confidence level corresponds to Z = 1.96

5.- 95.5% confidence level corresponds to Z = 2.

6.- 99.7% confidence level is equivalent to Z = 3.

An example where this formula can be applied would be in a study to determine the average weight of pebbles on a beach.

Clearly it is not possible to study and weigh all the pebbles on the beach, so it is advisable to extract a sample as random as possible and with the appropriate number of elements.

Figure 2. To study the characteristics of the pebbles on a beach, it is necessary to choose a random sample with a representative number of them. (Source: pixabay)

Case 2: the size of the population is known

When the number N of elements that make up a certain population (or universe) is known, if you want to select by simple random sampling a statistically significant sample of size n, this is the formula:

Where:

-Z is the coefficient associated with the confidence level.

-p is the probability of success of the hypothesis.

-q is the probability of failure in the hypothesis, p + q = 1.

-N is the size of the total population.

-E is the relative error of the study result.

Examples

The methodology to extract the samples depends a lot on the type of study that needs to be done. Therefore, random sampling has an infinite number of applications:

Surveys and questionnaires

For example, in telephone surveys, the people to be consulted are chosen using a random number generator, applicable to the region under study.

If you want to apply a questionnaire to the employees of a large company, then you can resort to the selection of the respondents through their employee number, or identity card number.

Said number must also be chosen randomly, using for example a random number generator.

Figure 3. A questionnaire can be applied by randomly selecting the participants. Source: Pixabay.

QA

In the event that the study is on parts manufactured by a machine, parts must be chosen randomly, but from batches manufactured at different times of the day, or on different days or weeks.

Advantage

Simple random sampling:

- It allows reducing the costs of a statistical study, since it is not necessary to study the total population to obtain statistically reliable results, with the desired confidence levels and the level of error required in the study.

- Avoid bias: as the choice of the elements to be studied is completely random, the study faithfully reflects the characteristics of the population, although only part of it was studied.

Disadvantages

- The method is not adequate in cases where you want to know the preferences in different groups or population strata.

In this case, it is preferable to previously determine the groups or segments on which the study is to be carried out. Once the strata or groups have been defined, then if it is convenient for each one of them to apply random sampling.

- It is highly unlikely that information will be obtained on minority sectors, of which it is sometimes necessary to know their characteristics.

For example, if it is a question of campaigning on an expensive product, it is necessary to know the preferences of the wealthiest minority sectors.

Exercise resolved

We want to study the population's preference for a certain cola drink, but there is no previous study in this population, of which its size is unknown.

On the other hand, the sample must be representative with a minimum confidence level of 90% and the conclusions must have a percentage error of 2%.

-How to determine the size n of the sample?

-What would be the sample size if the margin of error is made more flexible to 5%?

Solution

Since the size of the population is unknown, the formula given above is used to determine the sample size:

n = (Z ² p q) / (E ²)

We assume that there is an equal probability of preference (p) for our brand of soft drink as of no preference (q), so p = q = 0.5.

On the other hand, as the result of the study must have a percentage error less than 2%, then the relative error E will be 0.02.

Finally, a Z value = 1,645 produces a confidence level of 90%.

Summarizing, we have the following values:

Z = 1,645

p = 0.5

q = 0.5

E = 0.02

With these data, the minimum sample size is calculated:

n = (1.645 ² 0.5 0.5) / (0.02 ²) = 1691.3

This means that the study with the required margin of error and with the chosen level of confidence, must have a sample of respondents of at least 1692 individuals, chosen by simple random sampling.

If you go from a margin of error of 2% to 5%, then the new sample size is:

n = (1.645 ² 0.5 0.5) / (0.05 ²) = 271

Which is a significantly lower number of individuals. In conclusion, the sample size is very sensitive to the desired margin of error in the study.

References

1. Berenson, M. 1985. Statistics for Management and Economics, Concepts and Applications. Editorial Interamericana.
2. Statistics. Random sampling. Taken from: encyclopediaeconomica.com.
3. Statistics. Sampling. Recovered from: Estadistica.mat.uson.mx.
4. Explorable. Random sampling. Recovered from: explorable.com.
5. Moore, D. 2005. Applied Basic Statistics. 2nd. Edition.
6. Netquest. Random sampling. Recovered from: netquest.com.
7. Wikipedia. Statistical sampling. Recovered from: en.wikipedia.org