 So, before we start talking about statistics, it's, I think it's important to ask the question, why do we use statistics at all? These are painful things, nobody likes to study, well some people like to study them. Generally, people aren't interested in these things, right? So why mess with them? The reason we mess with statistics is because the world is divided into two sets of relationships and we tend to think in terms of one set, ignoring the other, and here's what I mean by that. There's virtually all laws of physics or what we call deterministic relationships. That is they always hold, right? So we've got Bernoulli's law, this is the thing that enables us to fly in airplanes. If you have an airplane wing and it's going through the air at a certain speed, by going through the air at a certain speed, the air goes over it and creates pressure in the airplane lifts and if air pressure and temperature is held constant, this wing which goes through at a certain speed gets a certain lift every single time, it works this way all the time. If you increase the speed, you get more lift all the time. So this is a deterministic relationship, it's always true in every single instance. And this is what we're used to dealing with, right? So you drop a glass and it drops to the floor at a certain speed and it smashes and that happens every single time you drop the glass. This is the way we're used to dealing with life deterministic relationships. And that's well and good for things like the laws of physics, but it doesn't apply to the laws of economics. Laws of economics are stochastic relationships and stochastic relationships are very different. They're true on average, they're not true in every instance. So for example, if you raise the minimum wage, it is the case that on average you'll get less employment. Now what happens is it doesn't happen in every single case, you might have a friend who, you know, when the minimum wage didn't rise, he lost his job, when the minimum wage did rise later on, he didn't lose his job. And you point to your friend and say, well look, this minimum wage nonsense must not apply because I know people who did not lose their jobs when the minimum wage went up. The reason they didn't lose their jobs when the minimum wage went up is not an indictment of minimum wage, rather what's going on is you're dealing with a stochastic relationship, that it's true on average even if it isn't true every single time, right? So the good example of the difference between the two, deterministic relationship is four is greater than three. Four is always greater than three, right? It doesn't matter what your circumstances are, what the temperature is outside, whether or not you woke up in the left side of the bed, four is always greater than three. It's a deterministic relationship. If you ever find an instance in which four is not greater than three, just one example, you have disproven the relationship, that's deterministic. A stochastic relationship, dogs are bigger than cats, generally that's true. Now you can find some cats that are particularly large and some dogs that are particularly small, and you can point to this example and say, look at this, this is gargantuan cat and a very small dog. I've disproven your statement that dogs are bigger than cats. No, you haven't because the statement dogs are bigger than cats is a stochastic statement. It talks about a stochastic relationship. On average, dogs are bigger than cats even though you'll find some examples in which it's not true. All right, so why does this matter? Well, it matters because to understand stochastic relationships, we have to do more than simply observe what's happening. We have to process data about what's happening and that's the nastiness of stats. And the reason we have to do this is because our usual method of talking about relationships is anecdotes, right? We use anecdotes. We say things like, well, you know, the tax rate went up and because of that, I can see that I have less money in my checking account. This is an anecdote. The minimum wage went up and my uncle lost his job. This is an anecdote. Anecdotes work very well when you're talking about deterministic relationships because the deterministic relationship is true in every single instance. So every anecdote you find is a perfect encapsulation of this deterministic relationship. Stochastic relationships are only true on average. They might be true in some specific cases, right? Other cases they may not be, but they're true on average, which means that anecdotes really don't work. You can pick an anecdote that fits the stochastic relationship. You can also pick an anecdote that doesn't fit the stochastic relationship. And this is the thing people, often when they talk, debate back and forth about economic issues, one side or the other at some point will accuse the other one of cherry picking the data. And that's what cherry picking means, that I've picked an anecdote that happens to support my position, right? Anecdotes, when it comes to stochastic relationships, are very useful for illustrating the relationship. But they aren't at all good for understanding the relationship. To understand the relationship, we need lots of data and we need to apply the rules of statistics to analyze this. All right. So one of the things that happens when we talk about statistics is that we run into a roadblock that statistics is discussed in the language of mathematics. But when we talk about things to which statistics apply, we use English, right? So we say things like, China's economy is bigger than the US economy, right? And some people don't care about that. Some people get better at shape about it, right? But we say this thing. When we say it in English, China's economy is bigger than the US economy. What we're missing when we say this in English are the particular nuances that statistics in the language of mathematics can convey, but the language of English has problems with. So for example, it is the case that if you add up all of the productivity that goes on in China, the total productivity is somewhat larger than total productivity in the US. But if you measure the productivity on a per-person basis, so take the total economic productivity in China divided by the number of Chinese workers, take the total amount of economic productivity in the United States, divide it by the number of American workers. And what you find now is the numbers are very different, right? American productivity per person is something like $54,000. So the average American worker, the average American worker, generates about $54,000 worth of goods and services. The average Chinese worker generates about $13,000 worth of goods and services. So what's happening here? What's happening here is we have to be very careful when we talk about statistics to remember that we're talking about them in English, but their native language is mathematics. So be careful when you say things like the economy, Chinese economy is larger than the US economy. Be careful to underline precisely what you mean by that, right? And that's going to take more than just a single sentence. You've got to say things like, well, I'm talking about on a per capita basis, and I'm adjusting for differences in costs of living, right? All of that's nasty stuff that cause people's eyes to glaze over, but it's important in understanding what it is you're really trying to convey. People, and when I say people, I point the finger largely at the media here, tend to be comfortable talking about averages and even talking about medians, right? So people tend to understand that the average income in the US is somewhat different than the median income in the US, and that there's a fundamental difference in these things that deserves note, right? So if we have the five of us, six of us in a room, and Jack's income is $2 million a year, our incomes are all $50,000 a year. On average, the average income in this room is quite high. And of course, the problem is, well, but that's not really indicative of what's going on here, right? Because what's happening is Jack's, this one guy with a very high income, this is dragging up the numbers. So rather than talking about averages, we can talk about medians. And with median, we line us all up, poorest person to richest person, and take the guy in the middle. And if we're all earning $50,000 and Jack's over here is earning $2 million, the guy in the middle, well, he's a guy with a $50,000 income. Jack's $2 million income doesn't affect this median, right? So the median in some sense, under some circumstances, is more indicative of what it is we're trying to convey, people's average income. So the media and people in general tend to be comfortable talking about averages and even on occasion talking about medians. What you find people almost never talking about outside of statisticians is standard deviation or variance. And standard deviation or variance tells us the degree to which a random phenomenon wanders away from its average. Give you an example. Imagine a picture that you have a guitar string and you pluck this guitar string. What happens if you look at it, you see it vibrating, right? And it's moving very fast. And what you see is it kind of forms this pattern of this, you know, you can see the edges where it vibrates to left and right. And, you know, you can see kind of the center point where it is. And it's kind of a blur. That center point, that's the average. So you see this string vibrating back and forth. On average, its location is this center point. The blur around it, this blur is the standard deviation. And it turns out this blur, the standard deviation, becomes under some circumstances incredibly important. Sometimes more important than the average. Give you an example. The average temperature in February in Fairbanks, Alaska is about minus 2 Fahrenheit. The average temperature on the moon is 5 Fahrenheit. And if you look at those two averages, you would conclude that the environment on the moon, absent the fact there's no air, is more hospitable than the environment in Fairbanks, Alaska. We're looking just at the averages. Well, here's what the averages don't tell you. In February, in Fairbanks, Alaska, the temperature varies. Of course, the average is 2. But it can vary as high as maybe, you know, 15 degrees and maybe as low as minus 20. So the average is 3. And you've got this plus or minus 10, 15 degrees on it. And the plus or minus represents this kind of like that vibrating guitar string that usually we find the February temperature in Fairbanks, Alaska to be minus 2, sometimes a little bit higher, sometimes a little bit lower, but it's in this range minus 2, plus or minus, you know, 10, 15, 20 degrees. On the moon, the average temperature is 5. But it varies by about 250 degrees. So on the highs, you're getting 250 degrees Fahrenheit. On the lows, you're getting minus 200 some degrees Fahrenheit. So what you see is, although the average temperature on the moon is higher than the average temperature in Fairbanks, Alaska, when you account for the standard deviation, you find that the environment on the moon is way less hospitable, right? In Fairbanks, you're going to be incredibly uncomfortable, but you'll survive. On the moon, no. And the difference is not the averages, it's the standard deviations, the degree to which this thing we're measuring vibrates around its mean. I'm no physicist, but I'm curious if it's a contentious subject to say that laws and physics are deterministic, in a sense. Yeah, that's a good question. I'm not a physicist either, right? But I can tell you a little bit about it. There are relationships in the physical world that are not deterministic, they're stochastic. And in fact, a good example of this is the field of fluid dynamics, a set of physical rules that describe how fluids behave as they're flowing through whatever it is that fluids flow through, right? And here's the interesting thing. The laws of fluid dynamics work very well. It's like Bernoulli's law, right? And the equations predict in a deterministic fashion what's going on, provided that your fluid has lots of molecules in it. If you have a bit of fluid that has very few molecules in it, right, like a thimble full of water or something, you'll find that the laws of fluid dynamics start to break down, not because the laws don't apply, but rather because the random behavior of the individual molecules starts to become more noticeable because you have fewer of them. And this is an interesting thing with economics, when people say, well, the problem with economics is it's really not a science because people behave randomly, to which as an economist I reply, it is, and people do behave partially randomly, but they also partially behave in predictable ways. And so when you get lots of humans, like having lots of molecules of the fluid, the individual random behaviors cancel out. And what you start to see more predominantly is the underlying predictable behavior, similarly with the fluids. So yes, there's this, I described it as this binary thing of relationships that are either deterministic or stochastic, in fact, there's a continuum. Physical laws tend to fall over here, social science laws tend to fall over here, but there's also a point where we start to overlap depending on how many people you're dealing with.