# Are there any theorems which mathematically show the importance of random sampling?

This is an obvious question - random sampling seems intuitive as it makes each element having an equal chance of being selected, intuitively reducing possible biases. If someone was to calculate the "lift" of a model, they would intuitively compare it against random selection. Are there any theorems which explicitly state the importance of random sampling in being able to generalize the results (more so than if elements were sampled systematically - e.g. if elements are sampled strictly every 30 minutes, this may cause a bias)?

• Sampling something each 30 minutes in many cases could be equivalent to random sampling. Similar example: you ask every 10th passing pedestrian to fill in your questionnaire - in many cases there is no reason to consider such sample to be non-random sample of pedestrians. Pseudo random number generators are also deterministic functions producing "random" outcomes. – Tim Nov 5 '15 at 15:54
• @tim: On the other hand, if you do the sampling 48 times less frequently, it probably wouldn't be representative. Unless you know that there's no periodicity at the sample frequency, or at a multiple of the sample frequency (and you can't know that for sure), then you can't assume it's random. – naught101 Apr 18 '16 at 6:22