There are three types of lies: Lies, damned lies, and statistics.
I think that one of the problems with math education in this country is that not even math instructors understand why the math is important.
Nowhere is this more true than with the field of statistics, which I teach from time to time at the University of Arizona.
So the question I will address today is:
One reason math gets a bad rep is because most students think that it’s just about numbers… or arithmetic in other words. Fractions here, percentages there… carry the one, move the decimal place to the right, or is it the left?
But mathematics is not just about formulas. Mathematics is about patterns. What do patterns have to do with statistics? What is statistics?
It’s very simple, too simple in fact. In the end, statistics is about one thing only.
Statistics is simply about deciding to go with choice A, or not A. That’s it! Anytime you make any kind of decision, no matter how trivial or grand, you brain is cranking away at the statistics without you knowing it!
Briefly, this is how the Bayesian theory of the brain works. We will use the example of deciding where to go for lunch:
You and a friend decide to go for lunch on University Avenue. You will decide where to go as you walk there. How does your brain frame the decision process?
Basically it amounts to a binary process. It’s either yes or no question that your brain must solve for each restaurant. So your brain may make a list somehow, perhaps by mapping the restaurants as you walk down the mental street:
- Does Chinese sound good? No.
- How about Indian? Nah.
- Pizza? Nope.
- Thai? Yes!
Sounds pretty simple right? That’s because your brain is a pro at running the stats, but make no mistake, it is actually a statistical analysis your brain is running. How so?
Let’s start with 1 above. Why did we decide no on Chinese? Perhaps it’s not Chinese food you don’t like, but just that particular restaurant… but why don’t you like it?
Because you have a previous dataset in your mind, the so-called “priors” that help you decide the “likelihood” of making a yes or no decision.
The priors may be as follows:
- I’ve been to that Chinese restaurant twice, and both times I didn’t care for it much, or maybe it was just “OK”
- I’ve never been to that restaurant, but from what I’ve heard it’s not very good.
Notice that in both cases, neither one can really predict what the food will be like for lunch today. Maybe they don’t specialize in the plates you got last time? Maybe the chef was mad and didn’t care? Maybe your friends are liars? Maybe they only like Pei Wei’s and anything Chinese restaurant not named Pei Wei is horrible to them. Or maybe the last time they went there, their girlfriend broke up with them afterwards, and now they have a bad memory associated with that restaurant.
There are a plethora of reasons, but the truth can very well be that had you decided to eat there for lunch today, and got their specialty of the day, it would have been the best meal you’ve had all year!
But you won’t know, because the past, or prior knowledge, has biased your decision. And it could also be that you are not rejecting that place outright, but you were only 45% in favor of going there while Thai food was a 90%.
The world our brain lives in is a fuzzy logic world, where concepts like “I will go to Chinese restaurant with a 45% Yes” exist. In the real world, this isn’t the case; you either go there for lunch today, or you don’t.
It’s either A, or not A… if you choose A, you are done. If you choose not A, you go on to a different decision, a domain that now excludes A.
When abstract meets reality
Is this fuzzy world of the brain just an abstract idea, or does it make any sense in the real world?
The answer is that it does make sense, and this is where more statistics comes in. If there is a 40% chance of rain today… this doesn’t really make sense in the sense that it will either rain, or it won’t rain. Binary world. Two possibilities.
But in the long run, if you had 100 days exactly like today, 40 of those days it would rain, 60 of those days it won’t. So after many repeated experiments we can turn a binary world (Bernoulli random variable) into a fuzzy truth, a chance or probability of happening.
Now some may think of the truth as this holy and ideal concept. But truth is very subjective, especially if it’s in the future.
Thinking of truth as THE TRUTH is like destiny. Thinking of truth as fuzzy truth is like free will.
And this affects other scientific fields such as physics.
The quantum world is one of the most bizarre realities that exists, but the evidence supporting it is overwhelming. Einstein was once asked about this and he has a famous quote in response, “God does not play dice with the universe.”
Einstein was wrong, and came around in later years.
One idea of the world is that of a deterministic world. This is the world of physics we learn about in high school. You drop a rock and you can predict or calculate when it will hit the ground. Note the word “predict” here. The future is determined, and once we know enough of the parameters, we can predict the future perfectly.
The quantum world is nothing like this, and rather than be deterministic, events are probabilistic. There is no way to say for sure that anything will happen. You can only give the chances of it occuring.
Even more bizarre is that the more you try to measure the parameters to predict what will happen, you actually alter the parameters, thus changing the probabilities or the predictions. It is as though some particles have a mind of their own and you can’t predict what they will do, as though they have free will also…
So it seems that the world is more non-deterministic than we previously thought, and even things that we know “for sure” may only be 99% confidence. There is always something unexpected that can occur. Always.
And to understand the world of uncertainty, you need probability and statistics. You don’t even have to be a quantum physicists to “need” this type of math. Other uncertain events include:
- Chinese for lunch?
- Buy more stock in Google?
- Propose to my love?
- Buy the latest New Kids on the Block cassette tape?
- Read any more Tucson Science articles?
For each of these decisions, your brain is constantly calculating probabilities. And even as we speak, or rather as you read, the probabilities are changing, especially with the last bullet point above.
What are the chances?
That’s where statistics enters in!
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