Randomization is an oft-neglected method of statistical inference, and undeservedly so. (It has, after all, received the enviable blessing of Ronald A. Fisher –– the god of statistics). It is simple to learn and so versatile that it arguably provides a rigorous alternative to other methods in any situation. To put it briefly, it is a non-parametric method that creates its own probability distribution under the assumptions of the null hypothesis. Consider the following example. We have two groups of data, one control (C) and another to which a treatment (T) has been applied:

C = {4, 5, 3, 4, 6}
T = {7, 6, 9, 4, 5}

Now, I know that there's an abundance of perfectly valid ways to analyse this data that have nothing to do with randomization. Just bear with me. You want to determine whether or not the treatment has had a significant effect. Formally, this requires a test statistic; let's use one that everybody's familiar with, the mean or average. Therefore, your null hypothesis is that the mean of T is not significantly different from the mean of C. How can you demonstrate that this is or isn't the case, if there is only one measurement (i.e. sample from the "real world") for the mean of C? You have to pretend that you have more.

The data that you do have for C tells you something about what the "real world" might be like, if you had actually gone and taken many, many samples just like the one you have. Let's collapse the "real world" to your sample, and then get lots of new samples (with replacement) out of your sample:

C1 = {5, 4, 6, 6, 4}
C2 = {6, 3, 4, 3, 5}
...

Doing this gives you a bunch of values for the mean of C. Most of them will be close to 4.4. Some of them will have weird values like 3 or 6. This, in fact, gives you a probability distribution of values for the mean of C, against which you can compare the mean you got for T! If I was investing all my attention to writing this instead of (unwisely) trying to do mathematical biology at the same time, I would tell you what that probability would be. Nyah, nyah!


Disclaimer to use against indignant statisticians:
This was written on the fly. Message me and I'll improve on it, or please feel free to post better material!

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