# Precise definition of a 'biased sample'?

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

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.

• 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. Jun 17, 2015 at 19:06
• @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. Jun 17, 2015 at 23:04