Hi, I'm Richard Nisbett, and this course is Critical Thinking for the Information Age. And this is lesson one, it's on statistics. We'll be talking about some basic statistical concepts that are fundamental to the course and essential for everyday life. If you've had no statistics or just one course, I think you'll get a lot from this video. If you've had several courses you may want to watch for a couple of minutes and see if it seems worthwhile, otherwise go straight to lesson two. In a standard introductory statistics course, in college, you'd spend 95% of the time on definitions, and formulas, and calculations, and maybe 5% of the time on examples. This whirlwind tour you're about to take is the reverse. It's light on definitions and heavy on examples. First concept is concept of a variable. A variable is something that varies, as opposed to a constant. The current temperature is a value for the variable temperature, 27, 30, 19, 29. As opposed to freezing temperature, which is constant, it's always the same. Some variables would be height, some people are taller than others. IQ, some people have higher IQs than others. The number of eggs laid by farmer Jones' hen house each week. And the number of errors per week in the manufacture of Ford transmissions. Anything that varies about a thing, event or person can be a variable. Variables are distributed in some way. One of the ways they're most frequently distributed is the normal distribution pattern. It's called the normal distribution because it's so common. It's also sometimes called the bell curve because it looks like a bell. In this kind of distribution, the mean is the most common value, and that's in the middle. As you get further and further away, from the mean, cases become rarer and rarer. So let's create a distribution. How about the distribution of cooking ability. Think of somebody who's a really good cook, that might be your grandma. Think of somebody who's not such a good cook, maybe your roommate or your cousin. Somebody who has trouble with grilled cheese sandwiches. And then, somebody who's at the mean is the average Joe. And there's lots of average Joe's compared to good old grandma's cooking and your poor roommate's cooking. Cases become fewer and fewer as you get further from the mean. An important point about the normal distribution, you can describe it in terms of standard deviations from the mean. That's almost like the average deviation but not quite. The mean for the number of eggs laid by farmer Jones' hens, each week, is six, let's say. Hen # 28 laid 8 eggs last week, so the deviation there, from the mean, is 2. And hen # 17 laid 5 eggs last week, so the deviation there is -1, that's 5 minus 6. And the average deviation is |2|+|-1| divided by 2, which is 3, divided by 2, which is 1.5. And so let's look at the rendering of the normal distribution, which has standard deviations indicated. And let's think about a particular kind of distribution. Let's take the distribution of heights of American males. The average American male is a little less that 5'10". And the average deviation is a little less than 3", and so is the standard deviation. So here's the mean, here is -1 standard deviation, here is +1 standard deviation. Now, standard deviation fact number one is that 68% of all cases, eggs laid per week, transmission errors per day, male heights, are within -1 standard deviation and +1 standard deviation. So slightly more than two-thirds of all American males are between 5'7" and 6'1". Standard deviation fact number two is that 84% of all cases are between, The bottom of the distribution -4 or 5 standard deviations, where there are almost no cases, up to 1 standard deviation, there you find 84% of cases. So 84% of all males are less than 6'1", and about 16% of American males are taller than 6'1". Standard deviation fact number three is that 96% of all cases lie between -2 standard deviations and +2 standard deviations. So 96% of all American males are taller than 5'4" and shorter than 6'4". Standard deviation fact number four is that you can convert standard deviations to percentiles. The mean is always at the 50th percentile. 1 standard deviation is always at the 84th percentile. And 83% of cases are below the 84th percentile. And 16% of cases are above, The 84th percentile. Well, IQ scores are represented in terms of standard deviation from the mean. The mean is arbitrarily defined as 100. So if the average score on that IQ test was 223 questions gotten right, we say the average is 100. And that's also what we say if the number of questions gotten right on average was 43, or if it was 767, it's just the convention. We know 100 corresponds to the average score. The standard deviation is arbitrarily defined as 15. That means we know that about two-thirds of people have IQs between 85 and 115. We know that a person with an IQ of 115 is about at the 84th percentile. And we know that a person with an IQ of 130 is about at the 98th percentile. So a person with an IQ of 130 is higher than 97% of the population. The same as 1% and lower than 2%. Standard deviations are useful for thinking about all kinds of things, for example, finance. Imagine you're considering a stock with a 4% average per year return over the last few years, and an average standard deviation of 3%. If the future is like the past and if I were your financial adviser, I would tell you right now that past performance is no guarantee of future performance, but if it were to be, you could assume the following. So let's look at a normal distribution of these gains. The mean is 4% and the standard deviation is 3. So one standard deviation above the mean is going to be 7, and one standard deviation below the mean is going to be 1. And we know that 68% of all cases are going to be between 1 and 7% gains. We also know that about 16% of the time, gains are going to be greater than 7%, and about 16% of the time, gains are going to be less than 1%. That's pretty stable, but if the standard deviation is 12, we get this kind of pattern. We find that 68% of the time, the gains are going to be between -8% and +16%. And we know that 16% of the time gains are going to be more than 16% and 16% of the time they're going to be worse than -8%. That's pretty volatile, so what are the implications for buying stock? If you know the standard deviation you have some pretty precise numbers to think about. If you're not counting on the stock for current income, why not go for volatility and buy the highest standard deviation stock. In the longer run you'll probably make more money. On the other hand, if you're counting on income from the stock for the next few years or if there's a possibility you may have to sell that stock for ready cash you may want the low standard deviation stock, because that's more stable. You're not going to lose your shirt, although, you're not going to get rich either. Standard deviations are also very useful for thinking about effect size. The effect size is the magnitude of difference in standard deviation terms. There other ways of calculating it too, but the most common way to calculate effect size is in standard deviation terms. So the effect size is what is the difference between one condition and another. For example, imagine you've designed a new way of teaching algebra. Kids taught by the old method get 72 on the exam and kids taught by the new method get 78 on the exam. Is that a big deal or not? It completely depends on the standard deviation. So the mean is 72, if the standard deviation is 6, that's 78. That's a big gain, because that takes the average kid from the 50th percentile to about the 84th percentile, which is no joke. On the other hand, assume that the standard deviation is 18. If so, it's not such a big deal. Because the gain is only one-third of a standard deviation, which is the equivalent of going from the 50th percentile to just the 64th percentile, which is not such a big deal. And you might want to take into consideration whether there are added costs if that's all the gain you're getting. Notice that many journalists don't understand that raw means are not very informative. It would be pretty common for a journalist to say, well, the old technique allowed students to get, on average, about 72 problems right. The new technique allowed about 78. The difference between 78 and 72 is 6, so 6 divided by 72 is 8.3%. So students got 8% better. That's literally meaningless, because you don't know how big an effect is unless you have a sensible metric. And the standard deviation gives you the metric. It tells you that 78 is worth moving 34 percentiles up from average. To give percentages in a case like this is almost as meaningless as to say that the new Star Wars movie was 8% better than the new Batman movie. So that's what we've done today for a single variable. In the next session, we'll be talking about two variables at the same time, namely, correlation.
