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
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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.
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$\begingroup$ 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. $\endgroup$– dimitriyCommented Jun 17, 2015 at 19:06
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$\begingroup$ @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. $\endgroup$ Commented Jun 17, 2015 at 23:04