Hack #2. Describe the World Using Just Two Numbers
Most of the statistical solutions and tools presented in this book work only because you can look at a sample and make accurate inferences about a larger population. The Central Limit Theorem is the meta-tool, the prime directive, the king of all secrets that allows us to pull off these inferential tricks.
Statistics provide solutions to problems whenever your goal is to describe a group of scores. Sometimes the whole group of scores you want to describe is in front of you. The tools for this task are called descriptivestatistics. More often, you can see only part of the group of the scores you want to describe, but you still want to describe the whole group. This summary approach is called inferentialstatistics. In inferential statistics, the part of the group of scores you can see is called a sample, and the whole group of scores you wish to make inferences about is the population.
It is quite a trick, though, when you think about it, to be able to describe with any confidence a population of values when, by definition, you are not directly observing those values. By using three pieces of information—two sample values and an assumption about the shape of the distribution of scores in the population—you can confidently and accurately describe those invisible populations. The set of procedures for deriving that eerily accurate description is collectively known as the Central Limit Theorem.
Inferential statistics tend to use two values to describe populations, the mean and the standard deviation.
Rather than describe a sample of values by showing them all, it is simply more efficient to report some fair summary of a group of scores instead of listing every single score. This single number is meant to fairly represent all the scores and what they have in common. Consequently, this single number is referred to as the central tendency of a group of scores.
Typically, the best measure of central tendency, for a variety of reasons, is the mean [Hack #21]. The mean is the arithmetic average of all the scores and is calculated by adding together all the values in a group, and then dividing that total by the number of values. The mean provides more information about all the scores in a group than other central tendency options (such as reporting the middle score, the most common score, and so on).
In fact, mathematically, the mean has an interesting property. A side effect of how it is created (adding up all scores and dividing by the number of scores) produces a number that is as close as possible to all the other scores. The mean will be close to some scores and far away from some others, but if you add up those distances, you get a total that is as small as possible. No other number, real or imagined, will produce a smaller total distance from all the scores in a group than the mean.
Just knowing the mean of a distribution doesn't quite tell us enough. We also need to know something about the variability of the scores. Are they mostly close to the mean or mostly far from the mean? Two wildly different distributions could have the same mean but differ in their variability. The most commonly reported measure of variability summarizes the distances between each score and the mean.
As with the mean, the more informative measure of variability would be one that uses all the values in a distribution. A measure of variability that does this is the standard deviation. The standard deviation is the average distance of each score from the mean. A standard deviation calculates all the distances in a distribution and averages them. The "distances" referred to are the distance between each score and the mean.
TIP
Another commonly reported value that summarizes the variability in a distribution is the variance. The variance is simply the standard deviation squared and is not particularly useful in picturing a distribution, but it is helpful when comparing different distributions and is frequently used as a value in statistical calculations, such as with the independent t test [Hack #17].
The formula for the standard deviation appears to be more complicated than it needs to be, but there are some mathematical complications with summing distances (negative distances always cancel out the positive distances when the mean is used as the dividing point). Consequently, here is the equation:
S means to sum up. The x means each score, and the n means the number of scores.
The Central Limit Theorem is fairly brief, but very powerful. Behold the truth:
If you randomly select multiple samples from a population, the means of each of those samples will be normally distributed.
Attached to the theorem are a couple of mathematical rules for accurately estimating the descriptive values for this imaginary distribution of sample means:
These mathematical rules produce more accurate results, and the distribution is closer to the normal curve as the sample size within any sample gets bigger.
TIP
30 or more in a sample seems to be enough to produce accurate applications of the Central Limit Theorem.
Okay, so the Central Limit Theorem appears somewhat intellectually interesting and no doubt makes statisticians all giggly and wriggly, but what does it all mean? How can anyone use it to do anything cool?
As discussed in "Know the Big Secret" [Hack #1], the secret trick that all statisticians know is how to solve problems statistically by taking known information about the distribution of some values and expressing that information as a statement of probability. The key, of course, is how one knows the distribution of all these exotic types of values that might interest a statistician. How can one know the distribution of average differences or the distribution of the size of a relationship between two sets of variables? The Central Limit Theorem, that's how.
For example, to estimate the probability that any two groups would differ on some variable by a certain amount, we need to know the distribution of means in the population from which those samples were drawn. How could we possibly know what that distribution is when the population of means is invisible and might even be only theoretical? The Central Limit Theorem, Bub, that's how! How can we know the distributions of correlations (an index of the strength of a relationship between two variables) which could be drawn from a population of infinite possible correlations? Ever hear of the Central Limit Theorem, dude?
Because we know the proportion of values that reside all along the normal curve [Hack #23], and the Central Limit Theorem tells me that these summary values are normally distributed, I can place probabilities on each statistical outcome. I can use these probabilities to indicate the level of statistical significance (the level of certainty) I have in my conclusions and decisions. Without the Central Limit Theorem, I could hardly ever make statements about statistical significance. And what a drab, sad life that would be.
To apply the Central Limit Theorem, I need start with only a sample of values that I have randomly drawn from a population. Imagine, for example, that I have a group of eight new Cub Scouts. It's my job to teach them knot tying. I suspect, let's say, that this isn't the brightest bunch of Scouts who have ever come to me for knot-tying guidance.
Before I demand extra pay, I want to determine whether they are, in fact, a few badges short of a bushel. I want to know their IQ. I know that the population's average IQ is 100, but I notice that no one in my group has an intelligence test score above 100. I would expect at least some above that score. Could this group have been selected from that average population? Maybe my sample is just unusual and doesn't represent all Cubbies. A statistical approach, using the Central Limit Theorem, would be to ask:
Is it possible that the mean IQ of the population represented by this sample is 100?
If I want to know something about the population from which my Scouts were drawn, I can use the Central Limit Theorem to pretty accurately estimate the population's mean IQ and its standard deviation. I can also figure out how much difference there is likely to be between the population's mean IQ and the mean IQ in my sample.
I need some data from my scouts to figure all this out. Table 1-1 should provide some good information.
The descriptive statistics for this sample of eight IQ scores are:
- Mean IQ = 91.75
- Standard deviation = 4.53
So, I know in my sample that most scores are within about 41/2 IQ points of 91.75. It is the invisible population they came from, though, that I am most interested in. The Central Limit Theorem allows me to estimate the population's mean, standard deviation, and, most importantly, how far sample means will likely stray from the population mean:
- Mean IQ
- Our sample mean is our best estimate, so the population mean is likely close to 91.75.
- Standard deviation of IQ scores in the population
- The formula we used to calculate our sample standard deviation is designed especially to estimate the population standard deviation, so we'll guess 4.53.
- Standard deviation of the mean
- This is the real value of interest. We know our sample mean is less than 100, but could that be by chance? How far would a mean from a sample of eight tend to stray from the population mean when chosen randomly from that population? Here's where we use the equation from earlier in this hack. We enter our sample values to produce our standard deviation of the mean, which is usually called the standard error of the mean:
We now know, thanks to the Central Limit Theorem, that most samples of eight Scouts will produce means that are within 1.6 IQ points of the population mean. It is unlikely, then, that our sample mean of 91.75 could have been drawn from a population with a mean of 100. A mean of 93, maybe, or 94, but not 100.
Because we know these means are normally distributed, we can use our knowledge of the shape of the normal distribution [Hack #23] to produce an exact probability that our mean of 91.75 could have come from a population with a mean of 100. It will happen way less than 1 out of 100,000 times. It seems very likely that my knot-tying students are tougher to teach than normal. I might ask for extra money.
A fuzzy version of the Central Limit Theorem points out that:
Data that are affected by lots of random forces and unrelated events end up normally distributed.
As this is true of almost everything we measure, we can apply the normal distribution characteristics to make probability statements about most visible and invisible concepts.
We haven't even discussed the most powerful implication of the Central Limit Theorem. Means drawn randomly from a population will be normally distributed, regardless of the shape of the population. Think about that for a second. Even if the population from which you draw your sample of values is not normal—even if it is the opposite of normal (like my Uncle Frank, for example)—the means you draw out will still be normally distributed.
This is a pretty remarkable and handy characteristic of the universe. Whether I am trying to describe a population that is normal or non-normal, on Earth or on Mars, the trick still works.
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