# Bias of functions defined on samples for population

• MHB
• mathinator
In summary, the estimate of the population parameter using the simple random sample may not always be unbiased for the population parameter, as not all sample parameters are unbiased for the population parameter. The sample variance correction is a specific example of this, and it is possible to find a counter-example for any arbitrary f. Additional details may be needed for a more thorough analysis.
mathinator
Let X1, · · · , Xn be a simple random sample from some finite population of values {x1, · · · xN }.
Is the estimate $$\displaystyle \frac{1}{n} \sum_{i}^{n} f(Xi)$$ always unbiased for $$\displaystyle \frac{1}{N} \sum_{i}^{N} f(xi)$$ no matter what f is?My thinking: I don't think all f's are unbiased, because not all sample parameters (ex: variance, or s^2) are unbiased for the population parameter (unless they are corrected for finite population sampling). I am confused if I am interpreting the question correctly, i.e f refers to parameters we can kind about the population :(

Thank you for your help in advance!

Hi mathinator,

Welcome to MHB! :)

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.

Jameson said:
Hi mathinator,

Welcome to MHB! :)

Yep I fully agree with your thought process. The sample variance correction is a great example of how this won't work for any arbitrary $f$. I think one counter-example is sufficient to wrap this problem, unless more detail is explicitly specified.

Thank you for your response!

## 1. What is the definition of bias in relation to functions defined on samples for population?

Bias refers to the systematic error or deviation of a function's estimated values from the true values of the population. It can occur due to the sampling process or the choice of the function itself.

## 2. How does bias affect the accuracy of a function's estimates?

Bias can lead to an overestimation or underestimation of the true values of the population. This means that the estimates provided by the function will not be accurate and may not reflect the true characteristics of the population.

## 3. What are some common sources of bias in functions defined on samples for population?

Some common sources of bias include sampling bias, measurement bias, and selection bias. These can arise due to flaws in the sampling process, measurement methods, or the selection of the function itself.

## 4. How can bias be reduced in functions defined on samples for population?

Bias can be reduced by using a representative and unbiased sample, ensuring accurate and precise measurements, and carefully selecting the appropriate function for the data. Additionally, using multiple methods or approaches can help to identify and reduce bias.

## 5. Can bias ever be completely eliminated in functions defined on samples for population?

No, it is not possible to completely eliminate bias in functions defined on samples for population. However, by minimizing bias through careful sampling and selection of the function, we can improve the accuracy of the estimates and make them more representative of the population.

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