Why?
Mathematics is based on the logic of the deductive argument.
Logical deduction tells us that A implies B.
A
B.
Statisticians have a generalisation for this.
Statisticians may be able to infer B from C.
But statistics always depends on a context, supplementary facts, such as is t-he are a rainy one? (Would a rainy month mean rain nearly every day or rain twice in the month? This has to do with the reference set.)
Statisticians always want more data so they can decrease the risk of making a mistake. (If we had the record of rainfall for all days but 2, we would be able to make a pretty close prediction.)
How can inferences be made?
Statistics uses mathematics.
We use mathematics, in the sense that supposing that something general happens we can look at what possible outcomes would look like. If the data we see is among the possible outcomes, we we have data and the general assumption that are consistent.
We need probability theory because in each situation a component of the passage from the general to a part with be a random selection.
Randomness is often included in studies containing uncertainty.
Chance is only the measure of our ignorance.
Henri Poincaré
Statistics uses psychology. We are going to see how different experiences lead to different 'baseline beliefs'. How people adhere to experiments. How the placebo effect works all have effect on statistics. Experimenter prior beliefs can effect (often unconsciously) the outcome of the studies.
Statistics uses economics.
If we could collect as much data as we want to we could make our approximations much better.But we can't; this is where design of experiments comes in.