We have an experiment where we observe a random number of objects (samples) and we measure how many of those are defective. Imagine that this is done in different factories that they produce these objects in different speeds, so 1 factory might produce 1 object a day, and others 1000 or more. Is there any formal way (or any ideas) to produce an ordering of the factories that reflects how defective products they have (from the worse to the best)? The problem is that if we take into account only the ratio
$\frac{defectives}{observed}$ it does not produce a "fair" ordering since
$\frac{1~defective}{2~observed}=\frac{20~defective}{40~observed}$ while the second ratio should be considered worse since it has a bigger sample.
