# Math vs Probability vs Statistics

For a certain job interview, I gave myself a 6 in SQL and 8 in Statistics. I love math and probability but I always found significance testing and confidence intervals rather dry.

What is the difference between math, probability and statistics again? This is a soft question, but it should have a clear answer.

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 Hard answer: en.m.wikipedia.org/wiki/Probability. Two more hard answers somewhere near it. – gnometorule Jan 25 at 15:59 don't laugh! I think it's just that Statistics is evidence-based, while Probability is about models. – john mangual Jan 25 at 16:06 Probability is physics, see Hilbert's sixth problem ;-) – dtldarek Jan 25 at 16:19 @johnmangual : Statistics is much more heavily about models than probability theory is. – Michael Hardy Jan 25 at 16:35

A probabilist can tell you that getting 100 heads in a row is no less likely than any other outcome when tossing a fair coin 100 times. A statistician will suspect the coin is biased.

Statistics is a discipline that relies heavily on mathematics, but is not within mathematics.

For example, consider the Behrens–Fisher problem: What should one infer about the difference between the means of two normally distributed populations, which may have different variances, when one observes a random sample from each?

Bartlett criticized Fisher's "fiducial" solution to this problem on the grounds that Fisher's fiducial intervals are not confidence intervals, i.e. they don't have constant coverage rates. That is certainly a mathematical fact. But Fisher disputed the idea that they ought to have constant coverage rates. That's essentially a philosophical position. Suppose you had prior probability distributions on the means and variances of the two populations, and then asked what's the conditional distribution of the difference between the two means, given the observed samples? That's just a math problem, and the posterior probability intervals that you get don't have constant coverage rates, so under some circumstances it clearly makes sense not to have constant coverage rates. Just which of several math problems should be used to model the Behrens–Fisher problem? That's more akin to a philosophical question than to a math problem. It's certainly not itself a mathematical question. But in a sense, it is the Behrens–Fisher problem.

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Only some very poor statistician would not know that getting 100 heads in a row is no less likely than any other outcome when tossing a fair coin 100 times. Only some very poor probabilist would not suspect the coin is biased. – Did Jan 25 at 17:40
@did : Certainly that's true. But I was addressing the question of what the difference is. – Michael Hardy Jan 26 at 6:19

Do not look at math as some 'equations' or 'formulas'. Math is an infinite-sized construction set, sort of like LEGO, but much, much more exciting. It is a way of combining beautiful logical structures to derive new beautiful logical structures.

Statistics can be seen as an inverse of probability theory, to some extent at least. If, say, you are given some database, you can estimate parameters of the distribution that this data set follows. Probability theory concerns itself with random variables and their distributions. So you assume that some phenomena (e.g. the number of infected species in the population) follows some distribution, and from this you can derive some property of this phenomena (e.g. mean time until extinction).

If time is also a factor, you should look at random/stochastic processes too.

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A probabilist, a statistician and a mathematician sit in a cafe across from a parked sedan. They see five people get into the car. After a minute passes, six people exit the car. The probabilist says that the event is non measurable since the sigma algebra of the car accounts only for 5 seats. The statistician says that six people walking out qualifies as a significant statistic against the null hypothesis that there were no people inside the car to begin with. The mathematician says that if another person were to enter the car now, then there would be exactly zero people in the car.

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Probability: deals with predicting the likelihood of future events

Statistics - involves the analysis of frequency of past events

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 Please do not plagiarize. This is essentially a verbatim copy of a quote from this page, which is an excerpt from this passage. – robjohn♦ Mar 16 at 8:43