Is there a precise definition of what a biased sample is? Or is it just a somewhat loose notion used in everyday parlance, but which does not have a precise mathematical definition?


1 Answer 1


Bias is usually defined in the negative, in that we ask whether a sample is unbiased, and an unbiased sample has the following characteristic:

  • Each member of the population has an equal chance of being selected by the sample

If you need that as a math equation:

$$ P_i(S) = p \forall i \in Pop, $$ where $P(S)$ is the probability of being selected, $p$ is some probability s.t. $\sum_{i}^N p_i = 1$

So a biased sample is any sample for which that isn't the case.

  • $\begingroup$ I am pretty sure you definition would designate complex survey data (with sampling weights, one or more stages of clustered sampling, and stratification) as biased, which is surely not always the case. $\endgroup$
    – dimitriy
    Jun 17, 2015 at 19:06
  • $\begingroup$ @Dimitry well, sure, I was working from the simplest case, but you're totally right. For cases of clustered/stratified sampling, you could extend it to say within each cluster, the probability of being sampled should be the same. $\endgroup$
    – potterzot
    Jun 17, 2015 at 23:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.