Approaching Inference with a Sample Size Close to the Population

Oct 13, 2025 by wbrum.org 104 views

Experiment with Sample Sizes Close to Population: How to Approach Inference? (+FPC)

When conducting experiments or surveys, researchers often face the challenge of determining the optimal sample size. While a larger sample size typically provides more accurate results, it can be impractical or even impossible to collect data from the entire population. In such cases, researchers may consider using a sample size that is close to the population size. However, this approach raises important questions about how to approach inference and whether the results can be generalized to the larger population.

Let's consider a scenario where we are running a true experiment in all schools of a small country. We want to estimate the average score of students in mathematics, and we have access to data from every school in the country. However, instead of using the entire population, we decide to use a sample size that is close to the population size, say 90% of the total number of schools. This approach may seem reasonable, but it requires careful consideration of the implications for inference.

The Finite Population Correction (FPC)

One of the key challenges in using a sample size close to the population size is the finite population correction (FPC). The FPC is a factor that is used to adjust the standard error of the estimate when the sample size is a significant proportion of the population size. The FPC is given by the formula:

n = N / (N - 1)

where n is the sample size and N is the population size.

Why is the FPC important?

The FPC is important because it takes into account the fact that the sample is not a random selection from the population, but rather a subset of the population. When the sample size is close to the population size, the FPC can have a significant impact on the standard error of the estimate. If the FPC is not applied, the standard error may be underestimated, leading to incorrect conclusions.

Approaching Inference with a Sample Size Close to the Population

So, how do we approach inference when using a sample size close to the population size? Here are some key considerations:

Use the FPC: As mentioned earlier, the FPC is an essential factor to consider when using a sample size close to the population size. Make sure to apply the FPC to adjust the standard error of the estimate.
Consider the design effect: The design effect is a measure of the impact of the sampling design on the variance of the estimate. When the sample size is close to the population size, the design effect can be significant. Consider using a design effect adjustment to account for this.
Use a robust standard error: A robust standard error is a standard error that is less sensitive to outliers and non-normality. When using a sample size close to the population size, a robust standard error may be a good option.
Consider the implications for generalizability: When using a sample size close to the population size, it's essential to consider the implications for generalizability. Can the results be generalized to the larger population, or are there specific characteristics of the sample that may affect the results?

Example: Estimating the Average Score of Students in Mathematics

Let's consider an example where we want to estimate the average score of students in mathematics in a small country. We have access to data from every school in the country, and we decide to use a sample size that is close to the population size, say 90% of the total number of schools. We apply the FPC to adjust the standard error of the estimate and use a robust standard error to account for non-normality.

Table 1: Estimated Average Score of Students in Mathematics

Sample Size	Estimated Average Score	Standard Error
90% of total schools	85.2	0.5

In this example, we can see that the estimated average score of students in mathematics is 85.2, with a standard error of 0.5. However, we must consider the implications of using a sample size close to the population size. Can the results be generalized to the larger population, or are there specific characteristics of the sample that may affect the results?

In conclusion, using a sample size close to the population size can be a challenging but effective approach to inference. However, it requires careful consideration of the implications for inference and generalizability. By applying the FPC, considering the design effect, using a robust standard error, and thinking critically about the implications for generalizability, researchers can ensure that their results are accurate and reliable.

Based on our discussion, here are some recommendations for researchers who want to use a sample size close to the population size:

Use the FPC: Apply the FPC to adjust the standard error of the estimate.
Consider the design effect: Use a design effect adjustment to account for the impact of the sampling design on the variance of the estimate.
Use a robust standard error: Consider using a robust standard error to account for non-normality and outliers.
Think critically about generalizability: Consider the implications of using a sample size close to the population size for generalizability.

By following these recommendations, researchers can ensure that their results are accurate and reliable, even when using a sample size close to the population size.
Experiment with Sample Sizes Close to Population: How to Approach Inference? (+FPC) - Q&A

In our previous article, we discussed the challenges of using a sample size close to the population size and how to approach inference. However, we know that there are many questions and concerns that researchers may have when dealing with this issue. In this article, we will address some of the most frequently asked questions and provide additional insights to help researchers navigate this complex topic.

Q: What is the finite population correction (FPC), and why is it important?

A: The FPC is a factor that is used to adjust the standard error of the estimate when the sample size is a significant proportion of the population size. It is important because it takes into account the fact that the sample is not a random selection from the population, but rather a subset of the population. When the sample size is close to the population size, the FPC can have a significant impact on the standard error of the estimate.

Q: How do I apply the FPC to my data?

A: To apply the FPC, you need to calculate the factor using the formula:

n = N / (N - 1)

where n is the sample size and N is the population size. Then, you need to multiply the standard error of the estimate by the FPC factor.

Q: What is the design effect, and how does it affect my results?

A: The design effect is a measure of the impact of the sampling design on the variance of the estimate. When the sample size is close to the population size, the design effect can be significant. This means that the standard error of the estimate may be underestimated, leading to incorrect conclusions.

Q: How can I account for the design effect in my analysis?

A: There are several ways to account for the design effect, including:

Using a design effect adjustment to adjust the standard error of the estimate
Using a robust standard error to account for non-normality and outliers
Considering the implications of the design effect for generalizability

Q: What is a robust standard error, and how does it differ from a standard error?

A: A robust standard error is a standard error that is less sensitive to outliers and non-normality. It is calculated using a different method than a standard error, which makes it more robust to deviations from normality.

Q: How can I determine if my sample size is close to the population size?

A: To determine if your sample size is close to the population size, you need to calculate the ratio of the sample size to the population size. If the ratio is close to 1, then your sample size is close to the population size.

Q: What are the implications of using a sample size close to the population size for generalizability?

A: When using a sample size close to the population size, it's essential to consider the implications for generalizability. Can the results be generalized to the larger population, or are there specific characteristics of the sample that may affect the results?

Q: How can I ensure that my results are accurate and reliable when using a sample size close to the population size?

A: To ensure that your results are accurate and reliable, you need to:

Apply the FPC to adjust the standard error of the estimate
Consider the design effect and use a design effect adjustment or a robust standard error
Think critically about the implications for generalizability
Use a robust standard error to account for non-normality and outliers

Based on our discussion, here are some recommendations for researchers who want to use a sample size close to the population size:

Use the FPC: Apply the FPC to adjust the standard error of the estimate.
Consider the design effect: Use a design effect adjustment to account for the impact of the sampling design on the variance of the estimate.
Use a robust standard error: Consider using a robust standard error to account for non-normality and outliers.
Think critically about generalizability: Consider the implications of using a sample size close to the population size for generalizability.

By following these recommendations, researchers can ensure that their results are accurate and reliable, even when using a sample size close to the population size.