5 Week 5: Estimation and Sampling Theory¹⁷

Remember, the role of descriptive statistics is to concisely summarize what we do know. In contrast, the purpose of inferential statistics is to “learn what we do not know from what we do.” What kinds of things would we like to learn about? And how do we learn them? These are the questions that lie at the heart of inferential statistics.

Sampling theory plays a huge role in specifying the assumptions upon which your statistical inferences rely. And in order to talk about “making inferences” the way statisticians think about it, we need to be a bit more explicit about what it is that we’re drawing inferences from (the sample) and what it is that we’re drawing inferences about (the population).

In almost every situation of interest, what we have available to us as researchers is a sample of data. We might have run experiment with some number of participants; a polling company might have phoned some number of people to ask questions about voting intentions; etc. Regardless: the data set available to us is finite, and incomplete. We can’t possibly get every person in the world to do our experiment; a polling company doesn’t have the time or the money to ring up every voter in the country etc. In our earlier discussion of descriptive statistics, the sample was the only thing we were interested in. Our only goal was to find ways of describing, summarizing and graphing that sample. This is about to change.

5.1 Defining a population

A sample is a concrete thing. You can open up a data file, and there’s the data from your sample. A population, on the other hand, is a more abstract idea. It refers to the set of all possible people, or all possible observations, that you want to draw conclusions about, and is generally much bigger than the sample. In an ideal world, the researcher would begin the study with a clear idea of what the population of interest is, since the process of designing a study and testing hypotheses about the data that it produces does depend on the population about which you want to make statements. However, that doesn’t always happen in practice: usually the researcher only has a vague idea of what the population is and designs the study as best they can on that basis.

Sometimes it’s easy to state the population of interest. For instance, in the “polling company” example given last week, the population consisted of all voters enrolled at the a time of the study – millions of people. The sample was a set of 1000 people who all belong to that population. In most situations the situation is much less simple. In a typical a psychological experiment, determining the population of interest is a bit more complicated. Suppose I run an experiment using 100 undergraduate students as my participants. My goal is to try to learn something about how the mind works. So, which of the following would count as “the population”:

• All of the undergraduate psychology students at the Erasmus University?
• Undergraduate psychology students in general, anywhere in the world?
• Dutch people currently living?
• People of similar ages to my sample?
• Anyone currently alive?
• Any human being, past, present or future?
• Any biological organism with a sufficient degree of intelligence operating in a terrestrial environment?
• Any intelligent being?

Each of these defines a real group of mind-possessing entities, all of which might be of interest to me as a cognitive scientist, and it’s not at all clear which one ought to be the true population of interest.

5.1.1 Simple random samples

Irrespective of how we define the population, the critical point is that the sample is a subset of the population, and our goal is to use our knowledge of the sample to draw inferences about the properties of the population. The relationship between the two depends on the procedure by which the sample was selected. This procedure is referred to as a sampling method, and it is important to understand why it matters.

To keep things simple, imagine we have a bag containing 10 chips. Each chip has a unique letter printed on it, so we can distinguish between the 10 chips. The chips come in two colors, black and white.

Figure 5.1: Simple random sampling without replacement from a finite population

This set of chips is the population of interest, and it is depicted graphically on the left of Figure 5.1.

As you can see from looking at the picture, there are 4 black chips and 6 white chips, but of course in real life we wouldn’t know that unless we looked in the bag. Now imagine you run the following “experiment”: you shake up the bag, close your eyes, and pull out 4 chips without putting any chips back into the bag. First out comes the \(a\) chip (black), then the \(c\) chip (white), then \(j\) (white) and then finally \(b\) (black). Each time you repeat this procedure (the right-hand side of Figure 5.1) you get different results, but the procedure is identical in each case. The fact that the same procedure can lead to different results each time, we refer to it as a random process. Because we shook the bag before pulling any chips out, it seems reasonable to think that every chip has the same chance of being selected. A procedure in which every member of the population has the same chance of being selected is called a simple random sample. The fact that we did not put the chips back in the bag while sampling, means you can’t pull out the same chip twice, and in such cases the observations are said to have been sampled without replacement.

To help make sure you understand the importance of the sampling procedure, consider an alternative way in which the experiment could have been run. Suppose that my son had opened the bag, and decided to pull out only black chips without putting any of them back in the bag. This biased sampling scheme is depicted in Figure 5.2.

Figure 5.2: Biased sampling without replacement from a finite population

Now consider the evidentiary value of seeing 4 black chips and 0 white chips. Clearly, it depends on the sampling scheme, does it not? If you know that the sampling scheme is biased to select only black chips, then a sample that consists of only black chips doesn’t tell you very much about the population! For this reason, statisticians really like it when a data set can be considered a simple random sample, because it makes the data analysis much easier.

A third procedure is worth mentioning. This time around we close our eyes, shake the bag, and pull out a chip. This time, however, we record the observation and then put the chip back in the bag. Again we close our eyes, shake the bag, and pull out a chip. We then repeat this procedure until we have 4 chips. Data sets generated in this way are still simple random samples, but because we put the chips back in the bag immediately after drawing them it is referred to as a sample with replacement. The difference between this situation and the first one is that it is possible to observe the same chip multiple times while sampling, as illustrated in Figure 5.3.

Figure 5.3: Simple random sampling with replacement from a finite population

Most experiments tend to be sampling without replacement, because the same person is not allowed to participate in the experiment twice. However, most statistical theory is based on the assumption that the data arise from a simple random sample with replacement. In real life, this very rarely matters. If the population of interest is large (e.g., has more than 10 entities!) the difference between sampling with- and without- replacement is too small to be concerned with. The difference between simple random samples and biased samples, on the other hand, is not such an easy thing to dismiss.

5.1.2 Most samples are not simple random samples

As you can see from looking at the list of possible populations that I showed above, it is almost impossible to obtain a simple random sample from most populations of interest. When I run experiments, I’d consider it a minor miracle if my participants turned out to be a random sampling of the undergraduate psychology students at university, even though this is by far the narrowest population that I might want to generalize to. A thorough discussion of other types of sampling schemes is beyond the scope of this book, but to give you a sense of what’s out there I’ll list a few of the more important ones:

• Stratified sampling. Suppose your population is (or can be) divided into several different sub-populations, or strata. Perhaps you’re running a study at several different sites, for example. Instead of trying to sample randomly from the population as a whole, you try to collect a separate random sample from each of the strata. Stratified sampling is sometimes easier to do than simple random sampling, especially when the population is already divided into the distinct strata. It can also be more efficient that simple random sampling, especially when some of the sub-populations are rare. For instance, when studying schizophrenia it would be much better to divide the population into two strata (schizophrenic and not-schizophrenic), and then sample an equal number of people from each group. If you selected people randomly, you would get so few schizophrenic people in the sample that your study would be useless. This specific kind of of stratified sampling is referred to as oversampling because it makes a deliberate attempt to over-represent rare groups.

• Snowball sampling is a technique that is especially useful when sampling from a “hidden” or hard to access population, and is especially common in social sciences. For instance, suppose the researchers want to conduct an opinion poll among transgender people. The research team might only have contact details for a few trans folks, so the survey starts by asking them to participate (stage 1). At the end of the survey, the participants are asked to provide contact details for other people who might want to participate. In stage 2, those new contacts are surveyed. The process continues until the researchers have sufficient data. The big advantage to snowball sampling is that it gets you data in situations that might otherwise be impossible to get any. On the statistical side, the main disadvantage is that the sample is highly non-random, and non-random in ways that are difficult to address. On the real life side, the disadvantage is that the procedure can be unethical if not handled well, because hidden populations are often hidden for a reason. I chose transgender people as an example here to highlight this: if you weren’t careful you might end up outing people who don’t want to be outed (very, very bad form), and even if you don’t make that mistake it can still be intrusive to use people’s social networks to study them. It’s certainly very hard to get people’s informed consent before contacting them, yet in many cases the simple act of contacting them and saying “hey we want to study you” can be hurtful. Social networks are complex things, and just because you can use them to get data doesn’t always mean you should.

• Convenience sampling is more or less what it sounds like. The samples are chosen in a way that is convenient to the researcher, and not selected at random from the population of interest. Snowball sampling is one type of convenience sampling, but there are many others. A common example in psychology are studies that rely on undergraduate psychology students. These samples are generally non-random in two respects: firstly, reliance on undergraduate psychology students automatically means that your data are restricted to a single sub-population. Secondly, the students usually get to pick which studies they participate in, so the sample is a self selected subset of psychology students not a randomly selected subset. In real life, most studies are convenience samples of one form or another. This is sometimes a severe limitation, but not always.

5.1.3 How much does it matter if you don’t have a simple random sample?

Okay, so real world data collection tends not to involve nice simple random samples. Does that matter? A little thought should make it clear to you that it can matter if your data are not a simple random sample: just think about the difference between Figures 5.1 and 5.2. However, it’s not quite as bad as it sounds. Some types of biased samples are entirely unproblematic. For instance, when using a stratified sampling technique you actually know what the bias is because you created it deliberately, often to increase the effectiveness of your study, and there are statistical techniques that you can use to adjust for the biases you’ve introduced (not covered in this book!). So in those situations it’s not a problem.

More generally though, it’s important to remember that random sampling is a means to an end, not the end in itself. Let’s assume you’ve relied on a convenience sample, and as such you can assume it’s biased. A bias in your sampling method is only a problem if it causes you to draw the wrong conclusions. When viewed from that perspective, I’d argue that we don’t need the sample to be randomly generated in every respect: we only need it to be random with respect to the psychologically-relevant phenomenon of interest. Suppose I’m doing a study looking at working memory capacity. In study 1, I actually have the ability to sample randomly from all human beings currently alive, with one exception: I can only sample people born on a Monday. In study 2, I am able to sample randomly from the Dutch population. I want to generalize my results to the population of all living humans. Which study is better? The answer, obviously, is study 1. Why? Because we have no reason to think that being “born on a Monday” has any interesting relationship to working memory capacity. In contrast, I can think of several reasons why “being Dutch” might matter. The Netherlands is a wealthy, industrialized country with a very well-developed education system. People growing up in that system will have had life experiences much more similar to the experiences of the people who designed the tests for working memory capacity. This shared experience might easily translate into similar beliefs about how to “take a test,” a shared assumption about how psychological experimentation works, and so on. These things might actually matter. For instance, “test taking” style might have taught the Dutch participants how to direct their attention exclusively on fairly abstract test materials relative to people that haven’t grown up in a similar environment; leading to a misleading picture of what working memory capacity is.

There are two points hidden in this discussion. Firstly, when designing your own studies, it’s important to think about what population you care about, and try hard to sample in a way that is appropriate to that population. In practice, you’re usually forced to put up with a “sample of convenience” (e.g., psychology lecturers sample psychology students because that’s the least expensive way to collect data, and our coffers aren’t exactly overflowing with gold), but if so you should at least spend some time thinking about what the dangers of this practice might be.

Secondly, if you’re going to criticize someone else’s study because they’ve used a sample of convenience rather than laboriously sampling randomly from the entire human population, at least have the courtesy to offer a specific theory as to how this might have distorted the results. Remember, everyone in science is aware of this issue, and does what they can to alleviate it. Merely pointing out that “the study only included people from group BLAH” is entirely unhelpful, and borders on being insulting to the researchers, who are aware of the issue. They just don’t happen to be in possession of the infinite supply of time and money required to construct the perfect sample. In short, if you want to offer a responsible critique of the sampling process, then be helpful. Rehashing the blindingly obvious truisms that I’ve been rambling on about in this section isn’t helpful.

5.1.4 Population parameters and sample statistics

Okay. Setting aside the thorny methodological issues associated with obtaining a random sample, let’s consider a slightly different issue. Up to this point we have been talking about populations the way a scientist might. To a psychologist, a population might be a group of people. To an ecologist, a population might be a group of bears. In most cases the populations that scientists care about are concrete things that actually exist in the real world.

Statisticians, however, are a funny lot. On the one hand, they are interested in real world data and real science in the same way that scientists are. On the other hand, they also operate in the realm of pure abstraction in the way that mathematicians do. As a consequence, statistical theory tends to be a bit abstract in how a population is defined. In much the same way that psychological researchers operationalize our abstract theoretical ideas in terms of concrete measurements, statisticians operationalize the concept of a “population” in terms of mathematical objects that they know how to work with. You’ve already come across these objects: they’re called probability distributions (remember, the place where data comes from).

The idea is quite simple. Let’s say we’re talking about IQ scores. To a psychologist, the population of interest is a group of actual humans who have IQ scores. A statistician “simplifies” this by operationally defining the population as the probability distribution depicted in Figure 5.4a.

Figure 5.4: The population distribution of IQ scores (panel a) and two samples drawn randomly from it. In panel b we have a sample of 100 observations, and panel c we have a sample of 10,000 observations.

IQ tests are designed so that the average IQ is 100, the standard deviation of IQ scores is 15, and the distribution of IQ scores is normal. These values are referred to as the population parameters because they are characteristics of the entire population. That is, we say that the population mean \(\mu\) is 100, and the population standard deviation \(\sigma\) is 15.

Now suppose we collect some data. We select 100 people at random and administer an IQ test, getting a simple random sample from the population. The sample would consist of a collection of numbers like this:

106 101 98 80 74 ... 107 72 100

Each of these IQ scores is sampled from a normal distribution with mean 100 and standard deviation 15. So if I plot a histogram of the sample, I get something like the one shown in Figure 5.4b. As you can see, the histogram is roughly the right shape, but it’s a very crude approximation to the true population distribution shown in Figure 5.4a. The mean of the sample is fairly close to the population mean 100 but not identical. In this case, it turns out that the people in the sample have a mean IQ of 98.5, and the standard deviation of their IQ scores is 15.9. These sample statistics are properties of the data set, and although they are fairly similar to the true population values, they are not the same. In general, sample statistics are the things you can calculate from your data set, and the population parameters are the things you want to learn about. Later on in this chapter we’ll talk about how you can estimate population parameters using your sample statistics and how to work out how confident you are in your estimates but before we get to that there’s a few more ideas in sampling theory that you need to know about.

5.2 The law of large numbers

We just looked at the results of one fictitious IQ experiment with a sample size of \(N=100\). The results were somewhat encouraging: the true population mean is 100, and the sample mean of 98.5 is a pretty reasonable approximation to it. In many scientific studies that level of precision is perfectly acceptable, but in other situations you need to be a lot more precise. If we want our sample statistics to be much closer to the population parameters, what can we do about it?

The obvious answer is to collect more data. Suppose that we ran a much larger experiment, this time measuring the IQ’s of 10,000 people. We can simulate the results of this experiment using R, using the rnorm() function, which generates random numbers sampled from a normal distribution. For an experiment with a sample size of n = 10000, and a population with mean = 100 and sd = 15, R produces our fake IQ data using these commands:

IQ <- rnorm(n=10000, mean=100, sd=15) # generate 10000 IQ scores
IQ <- round(IQ) # round to nearest whole number

Cool, we just generated 10,000 fake IQ scores, but 10,000 numbers is too many numbers to look at. We can look at the first 100 like this:

print(IQ[1:100])

##   [1]  95  96 103 110 115  86 114 107 126 116  94  68  78 116 104 105 104 105
##  [19]  67 104  78 101 109  90 104 120  72 117 116  76  65  98  99  71  95  77
##  [37]  86 126 107  94  61 118  93  91  92 132 112 129 131  66 130  98 110 128
##  [55] 102  92 103 104 117  89 117 115 133  88 102  77 105  83  85  92 109  94
##  [73]  84 121  92 107  92 107 128 101  79 107 111 100  86 101 124  87 113 100
##  [91]  85 107 102 116 118  95  99 118 102 101

We can also compute the mean IQ using mean(IQ) and the standard deviation using sd(IQ), or draw a histogram. The histogram of this much larger sample is shown in Figure 5.4c. Even a moment’s inspections makes clear that the larger sample is a much better approximation to the true population distribution than the smaller one. This is reflected in the sample statistics: the mean IQ for the larger sample turns out to be 99.9, and the standard deviation is 15.1. These values are now very close to the true population.

I feel a bit silly saying this, but the thing I want you to take away from this is that large samples generally give you better information. I feel silly saying it because it’s so bloody obvious that it shouldn’t need to be said. In fact, it’s such an obvious point that when Jacob Bernoulli – one of the founders of probability theory – formalized this idea back in 1713, he was kind of a jerk about it. Here’s how he described the fact that we all share this intuition:

For even the most stupid of men, by some instinct of nature, by himself and without any instruction (which is a remarkable thing), is convinced that the more observations have been made, the less danger there is of wandering from one’s goal (see Stigler, 1986, p65).

Okay, so the passage comes across as a bit condescending (not to mention sexist), but his main point is correct: it really does feel obvious that more data will give you better answers. The question is, why is this so? Not surprisingly, this intuition that we all share turns out to be correct, and statisticians refer to it as the law of large numbers. The law of large numbers is a mathematical law that applies to many different sample statistics, but the simplest way to think about it is as a law about averages. The sample mean is the most obvious example of a statistic that relies on averaging (because that’s what the mean is… an average), so let’s look at that. When applied to the sample mean, what the law of large numbers states is that as the sample gets larger, the sample mean tends to get closer to the true population mean. Or, to say it a little bit more precisely, as the sample size “approaches” infinity (written as \(N \rightarrow \infty\)) the sample mean approaches the population mean (\(\bar{X} \rightarrow \mu\)).

I don’t intend to subject you to a proof that the law of large numbers is true, but it’s one of the most important tools for statistical theory. The law of large numbers is the thing we can use to justify our belief that collecting more and more data will eventually lead us to the truth. For any particular data set, the sample statistics that we calculate from it will be wrong, but the law of large numbers tells us that if we keep collecting more data those sample statistics will tend to get closer and closer to the true population parameters.

5.3 Sampling distributions

The law of large numbers is a very powerful tool, but it’s not going to be good enough to answer all our questions. Among other things, all it gives us is a “long run guarantee.” In the long run, if we were somehow able to collect an infinite amount of data, then the law of large numbers guarantees that our sample statistics will be correct. But as John Maynard Keynes famously argued in economics, a long run guarantee is of little use in real life:

[The] long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons they can only tell us, that when the storm is long past, the ocean is flat again. Keynes (1923, 80)

As in economics, so too in psychology and statistics. It is not enough to know that we will eventually arrive at the right answer when calculating the sample mean. Knowing that an infinitely large sample will tell me the exact value of the population mean is cold comfort when my actual sample has a sample size of \(N=100\). In real life, then, we must know something about the behavior of the sample mean when it is calculated from a more modest sample!

5.3.1 Sampling distribution of the sample mean

With this in mind, let’s abandon the idea that our studies will have sample sizes of 10000, and consider a very modest experiment indeed. This time around we’ll sample \(N=5\) people and measure their IQ scores. As before, I can simulate this experiment in R using the rnorm() function:

> IQ.1 <- round( rnorm(n=5, mean=100, sd=15 ))
> IQ.1
[1]  90  82  94  99 110

The mean IQ in this sample turns out to be exactly 95¹⁸. Not surprisingly, this is much farther away from the population mean than the previous experiment (with a sample size of 10000). Now imagine that I decided to replicate the experiment. That is, I repeat the procedure as closely as possible: I randomly sample 5 new people and measure their IQ. Again, R allows me to simulate the results of this procedure:

> IQ.2 <- round( rnorm(n=5, mean=100, sd=15 ))
> IQ.2
[1]  78  88 111 111 117

This time around, the mean IQ in my sample is 101. If I repeat the experiment 10 times I obtain the results shown in Table 5.1, and as you can see the sample mean varies from one replication to the next.

Table 5.1: Ten replications of the IQ experiment, each with a sample size of \(N=5\).

Replication	Person.1	Person.2	Person.3	Person.4	Person.5	Sample.Mean
Replication 1	90	82	94	99	110	95.0
Replication 2	78	88	111	111	117	101.0
Replication 3	111	122	91	98	86	101.6
Replication 4	98	96	119	99	107	103.8
Replication 5	105	113	103	103	98	104.4
Replication 6	81	89	93	85	114	92.4
Replication 7	100	93	108	98	133	106.4
Replication 8	107	100	105	117	85	102.8
Replication 9	86	119	108	73	116	100.4
Replication 10	95	126	112	120	76	105.8

Now suppose that I decided to keep going in this fashion, replicating this “five IQ scores” experiment over and over again. Every time I replicate the experiment I write down the sample mean. Over time, I’d be amassing a new data set, in which every experiment generates a single data point. The first 10 observations from my data set are the sample means listed in Table 5.1, so my data set starts out like this:

                      95.0 101.0 101.6 103.8 104.4 ...

What if I continued like this for 10,000 replications, and then drew a histogram? Using the magical powers of R that’s exactly what I did, and you can see the results in Figure 5.5. As this picture illustrates, the average of 5 IQ scores is usually between 90 and 110. But more importantly, what it highlights is that if we replicate an experiment over and over again, what we end up with is a distribution of sample means! This distribution has a special name in statistics: it’s called the sampling distribution of the sample mean.

Figure 5.5: The sampling distribution of the sample mean for the “five IQ scores experiment.” If you repeatedly sample 5 people at random and calculate their average IQ, you’ll almost certainly get a number between 80 and 120, even though there are quite a lot of individuals who have IQs above 120 or below 80. For comparison, the black line plots the population distribution of IQ scores.

Sampling distributions are an important theoretical idea in inferential statistics, and they’re crucial for understanding the behaviour of small samples. For instance, when I ran the very first “five IQ scores” experiment, the sample mean turned out to be 95. What the sampling distribution in Figure 5.5 tells us, though, is that the “five IQ scores” experiment is not very accurate. If I repeat the experiment, the sampling distribution tells me that I can expect to see a sample mean anywhere between 80 and 120.

5.3.2 Sampling distributions exist for any sample statistic

One thing to keep in mind when thinking about sampling distributions is that any sample statistic you might care to calculate has a sampling distribution. For example, suppose that each time I replicated the “five IQ scores” experiment I wrote down the largest IQ score in the experiment. This would give me a data set that started out like this:

                      110 117 122 119 113 ... 

Doing this over and over again would give me a very different sampling distribution, namely the sampling distribution of the maximum. The sampling distribution of the maximum of 5 IQ scores is shown in Figure 5.6. Not surprisingly, if you pick 5 people at random and then find the person with the highest IQ score, they’re going to have an above average IQ. Most of the time you’ll end up with someone whose IQ is measured in the 100 to 140 range.

Figure 5.6: The sampling distribution of the maximum for the “five IQ scores experiment.” If you sample 5 people at random and select the one with the highest IQ score, you’ll probably see someone with an IQ between 100 and 140.

5.4 The central limit theorem

At this point I hope you have a pretty good sense of what sampling distributions are, and in particular what the sampling distribution of the sample mean is. In this section I want to talk about how the sampling distribution of the sample mean changes as a function of sample size. Intuitively, you already know part of the answer: if you only have a few observations, the sample mean is likely to be quite inaccurate: if you replicate a small experiment and recalculate the mean you probably get a very different answer. In other words, the sampling distribution is quite wide. If you replicate a large experiment and recalculate the sample mean you’ll probably get a similar answer to the one you got last time, so the sampling distribution will be narrow(er). You can see this visually in Figure 5.7: the bigger the sample size, the narrower the sampling distribution gets.

In the upper left corner of the Figure 5.7, our (fictitious) sample contained only a single observation, so the mean of each sample is just one person’s IQ score. As a consequence, the sampling distribution of the sample mean is of course identical to the population distribution of IQ scores. When we raise the sample size to 2, the mean of any one sample tends to be closer to the population mean than a one person’s IQ score, and so the histogram (i.e., the sampling distribution) is a bit narrower than the population distribution. By the time we raise the sample size to 10, we can see that the distribution of sample means tend to be fairly tightly clustered around the true population mean.

Figure 5.7: An illustration of the how sampling distribution of the mean depends on sample size. In each panel, I generated 10,000 samples of IQ data, and calculated the mean IQ observed within each of these data sets. The histograms in these plots show the distribution of these means (i.e., the sampling distribution of the sample mean). Each individual IQ score was drawn from a normal distribution with mean 100 and standard deviation 15, which is shown as the solid black line).

We can quantify this effect by calculating the standard deviation of the sampling distribution, which is referred to as the standard error. The standard error of a statistic is often denoted SE, and since we’re usually interested in the standard error of the sample mean, we often use the acronym SEM. As you can see just by looking at the picture, as the sample size \(N\) increases, the SEM decreases.

Okay, so that’s one part of the story. However, there’s something I’ve been glossing over so far. All my examples up to this point have been based on the “IQ scores” experiments, and because IQ scores are roughly normally distributed, I’ve assumed that the population distribution is normal. What if it isn’t normal? What happens to the sampling distribution of the mean?

The remarkable thing is this: no matter what shape your population distribution is, as \(N\) increases the sampling distribution of the mean starts to look more like a normal distribution. To give you a sense of this, I ran some simulations using R. To do this, I started with the “ramped” distribution shown in the upper left plot of Figure 5.8. As you can see by comparing the triangular shaped histogram to the bell curve plotted by the black line, the population distribution doesn’t look very much like a normal distribution at all.

Next, I used R to simulate the results of a large number of experiments. In each experiment I took \(N=2\) samples from this distribution, and then calculated the sample mean. The upper right corner of Figure 5.8 plots the histogram of these sample means (i.e., the sampling distribution of the sample mean for \(N=2\)). This time, the histogram produces a \(\cap\)-shaped distribution: it’s still not normal, but it’s a lot closer to the black line than the population distribution in the upper left corner. When I increase the sample size to \(N=4\), the sampling distribution of the mean is very close to normal, and by the time we reach a sample size of \(N=8\) it’s almost perfectly normal. In other words, as long as your sample size isn’t tiny, the sampling distribution of the sample mean will be approximately normal no matter what your population distribution looks like!

Figure 5.8: A demonstration of the central limit theorem. In the upper left panel, we have a non-normal population distribution; the other panels show the sampling distribution of the sample mean for samples of size 2,4 and 8, for data drawn from the distribution in the upper left panel. As you can see, even though the original population distribution is non-normal, the sampling distribution of the sample mean becomes pretty close to normal by the time you have a sample of even 4 observations.

On the basis of these figures, it seems like we have evidence for all of the following claims about the sampling distribution of the mean:

• The mean of the sampling distribution is the same as the mean of the population
• The standard deviation of the sampling distribution (i.e., the standard error) gets smaller as the sample size increases
• The shape of the sampling distribution becomes normal as the sample size increases

As it happens, not only are all of these statements true, there is a very famous theorem in statistics that proves all three of them, known as the central limit theorem. Among other things, the central limit theorem tells us that if the population distribution has mean \(\mu\) and standard deviation \(\sigma\), then the sampling distribution of the sample mean also has mean \(\mu\), and the standard error of its mean is: \[ \mbox{SEM} = \frac{\sigma}{ \sqrt{N} } \] Because we divide the population standard deviation \(\sigma\) by the square root of the sample size \(N\), the SEM gets smaller as the sample size increases. It also tells us that the shape of the sampling distribution becomes normal.¹⁹

This result is useful for all sorts of things. It tells us why large experiments are more reliable than small ones, and because it gives us an explicit formula for the standard error it tells us how much more reliable a large experiment is. It tells us why the normal distribution is, well, normal. In real experiments, many of the things that we want to measure are actually averages of lots of different quantities (e.g., arguably, “general” intelligence as measured by IQ is an average of a large number of “specific” skills and abilities), and when that happens, the averaged quantity should follow a normal distribution. Because of this mathematical law, the normal distribution pops up over and over again in real data.

5.5 Estimating population parameters

In all the IQ examples in the previous sections, we actually knew the population parameters ahead of time. As every undergraduate gets taught in their very first lecture on the measurement of intelligence, IQ scores are defined to have mean 100 and standard deviation 15. However, this is a bit of a lie. How do we know that IQ scores have a true population mean of 100? Well, we know this because the people who designed the tests have administered them to very large samples, and have then “rigged” the scoring rules so that their sample has mean 100. That’s not a bad thing of course: it’s an important part of designing a psychological measurement. However, it’s important to keep in mind that this theoretical mean of 100 only attaches to the population that the test designers used to design the tests. Good test designers will actually go to some lengths to provide “test norms” that can apply to lots of different populations (e.g., different age groups, nationalities etc).

This is very handy, but of course almost every research project of interest involves looking at a different population of people to those used in the test norms. For instance, suppose you wanted to measure the effect of low level lead poisoning on cognitive functioning in Port Pirie, a South Australian industrial town with a lead smelter. Perhaps you decide that you want to compare IQ scores among people in Port Pirie to a comparable sample in Whyalla, a South Australian industrial town with a steel refinery.²⁰ Regardless of which town you’re thinking about, it doesn’t make a lot of sense simply to assume that the true population mean IQ is 100. No-one has, to my knowledge, produced sensible norming data that can automatically be applied to South Australian industrial towns. We’re going to have to estimate the population parameters from a sample of data. So how do we do this?

5.5.1 Estimating the population mean

Suppose we go to Port Pirie and 100 of the locals are kind enough to sit through an IQ test. The average IQ score among these people turns out to be \(\bar{X}=98.5\). So what is the true mean IQ for the entire population of Port Pirie? Obviously, we don’t know the answer to that question. It could be \(97.2\), but if could also be \(103.5\). Our sampling isn’t exhaustive so we cannot give a definitive answer. Nevertheless if I was forced at gunpoint to give a “best guess” I’d have to say \(98.5\). That’s the essence of statistical estimation: giving a best guess.

In this example, estimating the unknown poulation parameter is straightforward. I calculate the sample mean, and I use that as my estimate of the population mean. It’s pretty simple, and in the next section I’ll explain the statistical justification for this intuitive answer. However, for the moment what I want to do is make sure you recognise that the sample statistic and the estimate of the population parameter are conceptually different things. A sample statistic is a description of your data, whereas the estimate is a guess about the population. With that in mind, statisticians often different notation to refer to them. For instance, the true population mean is denoted by \(\mu\), and we use \(\hat\mu\) (say: mu-hat) to refer to our estimate of the population mean. In contrast, the sample mean is denoted \(\bar{X}\) (say:x-bar) or sometimes \(m\). However, in simple random samples, the estimate of the population mean is identical to the sample mean: if I observe a sample mean of \(\bar{X} = 98.5\), then my estimate of the population mean is also \(\hat\mu = 98.5\). To help keep the notation clear, here’s a handy table:

Symbol	What is it?	Do we know what it is?
\(\bar{X}\)	Sample mean	Yes - calculated from your sample
\(\mu\)	True population mean	Almost never known for sure
\(\hat{\mu}\)	Estimate of the population mean	Yes - identical to the sample mean

5.5.2 Estimating the population standard deviation

So far, estimation seems pretty simple, and you might be wondering why I forced you to read through all that stuff about sampling theory. In the case of the mean, our estimate of the population parameter (i.e. \(\hat\mu\)) turned out to identical to the corresponding sample statistic (i.e. \(\bar{X}\)). However, that’s not always true. To see this, let’s have a think about how to construct an estimate of the population standard deviation, which we’ll denote \(\hat\sigma\). What shall we use as our estimate in this case? Your first thought might be that we could do the same thing we did when estimating the mean, and just use the sample statistic as our estimate. That’s almost the right thing to do, but not quite.

Here’s why. Suppose I have a sample that contains a single observation. For this example, it helps to consider a sample where you have no intutions at all about what the true population values might be, so let’s use something completely fictitious. Suppose the observation in question measures the cromulence of my shoes. It turns out that my shoes have a cromulence of 20. So here’s my sample:

20

This is a perfectly legitimate sample, even if it does have a sample size of \(N=1\). It has a sample mean of 20, and because every observation in this sample is equal to the sample mean (obviously!) it has a sample standard deviation of 0. As a description of the sample this seems quite right: the sample contains a single observation and therefore there is no variation observed within the sample. A sample standard deviation of \(s = 0\) is the right answer here. But as an estimate of the population standard deviation, it feels completely insane, right? Admittedly, you and I don’t know anything at all about what “cromulence” is, but we know something about data: the only reason that we don’t see any variability in the sample is that the sample is too small to display any variation! So, if you have a sample size of \(N=1\), it feels like the right answer is just to say “no idea at all.”

Notice that you don’t have the same intuition when it comes to the sample mean and the population mean. If forced to make a best guess about the population mean, it doesn’t feel completely insane to guess that the population mean is 20. Sure, you probably wouldn’t feel very confident in that guess, because you have only the one observation to work with, but it’s still the best guess you can make.

Let’s extend this example a little. Suppose I now make a second observation. My data set now has \(N=2\) observations of the cromulence of shoes, and the complete sample now looks like this:

20, 22

This time around, our sample is just large enough for us to be able to observe some variability: two observations is the bare minimum number needed for any variability to be observed! For our new data set, the sample mean is \(\bar{X}=21\), and the sample standard deviation is \(s=1\). What intuitions do we have about the population? Again, as far as the population mean goes, the best guess we can possibly make is the sample mean: if forced to guess, we’d probably guess that the population mean cromulence is 21. What about the standard deviation? This is a little more complicated. The sample standard deviation is only based on two observations, and if you’re at all like me you probably have the intuition that, with only two observations, we haven’t given the population “enough of a chance” to reveal its true variability to us. It’s not just that we suspect that the estimate is wrong: after all, with only two observations we expect it to be wrong to some degree. The worry is that the error is systematic. Specifically, we suspect that the sample standard deviation is likely to be smaller than the population standard deviation.

This intuition feels right, but it would be nice to demonstrate this somehow. There are in fact mathematical proofs that confirm this intuition, but unless you have the right mathematical background they don’t help very much. Instead, what I’ll do is use R to simulate the results of some experiments. With that in mind, let’s return to our IQ studies. Suppose the true population mean IQ is 100 and the standard deviation is 15. I can use the rnorm() function to generate the the results of an experiment in which I measure \(N=2\) IQ scores, and calculate the sample standard deviation. If I do this over and over again, and plot a histogram of these sample standard deviations, what I have is the sampling distribution of the standard deviation. I’ve plotted this distribution in Figure 5.9. Even though the true population standard deviation is 15, the average of the sample standard deviations is only 8.5. Notice that this is a very different result to what we found in Figure 5.7 when we plotted the sampling distribution of the mean. If you look at that sampling distribution, what you see is that the population mean is 100, and the mean of the sampling distribution of sample means is also 100.

Figure 5.9: The sampling distribution of the sample standard deviation for a “two IQ scores” experiment. The true population standard deviation is 15 (dashed line), but as you can see from the histogram, the vast majority of experiments will produce a much smaller sample standard deviation than this. On average, this experiment would produce a sample standard deviation of only 8.5, well below the true value! In other words, the sample standard deviation is a biased estimate of the population standard deviation.

Now let’s extend the simulation. Instead of restricting ourselves to the situation where we have a sample size of \(N=2\), let’s repeat the exercise for sample sizes from 1 to 10. If we plot the average sample mean and average sample standard deviation as a function of sample size, you get the results shown in Figure 5.10. On the left hand side (panel a), I’ve plotted the average sample mean and on the right hand side (panel b), I’ve plotted the average standard deviation. The two plots are quite different: on average, the average sample mean is equal to the population mean. It is an unbiased estimator, which is essentially the reason why your best estimate for the population mean is the sample mean.²¹ The plot on the right is quite different: on average, the sample standard deviation \(s\) is smaller than the population standard deviation \(\sigma\). It is a biased estimator. In other words, if we want to make a “best guess” \(\hat\sigma\) about the value of the population standard deviation \(\sigma\), we should make sure our guess is a little bit larger than the sample standard deviation \(s\).

Figure 5.10: An illustration of the fact that the sample mean is an unbiased estimator of the population mean (panel a), but the sample standard deviation is a biased estimator of the population standard deviation (panel b). To generate the figure, I generated 10,000 simulated data sets with 1 observation each, 10,000 more with 2 observations, and so on up to a sample size of 10. Each data set consisted of fake IQ data: that is, the data were normally distributed with a true population mean of 100 and standard deviation 15. On average, the sample means turn out to be 100, regardless of sample size (panel a). However, the sample standard deviations turn out to be systematically too small (panel b), especially for small sample sizes.

The fix to this systematic bias turns out to be very simple. Here’s how it works. Before tackling the standard deviation, let’s look at the variance. If you recall from Section 2.6.3, the variance is defined to be the average of the squared deviations from the mean. That is: \[ s^2 = \frac{{\sum_{i=1}^N} (X_i - \bar{X})^2}{N} \] However, the sample variance \(s^2\) is a biased estimator of the population variance \(\sigma^2\). But as it turns out, we only need to make a tiny tweak to transform this into an unbiased estimator. All we have to do is divide by \(N-1\) rather than by \(N\). If we do that, we obtain the following formula: \[ \hat\sigma^2 = \frac{{\sum_{i=1}^N} (X_i - \bar{X})^2}{N-1} \] This is an unbiased estimator of the population variance \(\sigma^2\). Moreover, this finally answers the question we raised in Section 2.5.3.2 of the labs: why does R give us slightly different answers when we use the var() function than using the formula in Section 2.6.3? A similar story applies for the standard deviation. If we divide by \(N-1\) rather than \(N\), our estimate of the population standard deviation becomes: \[ \hat\sigma = \sqrt{\frac{{\sum_{i=1}^N} (X_i - \bar{X})^2}{N-1}} \] and when we use R’s built in standard deviation function sd(), what it’s doing is dividing by \(N-1\), not \(N\).²²

One final point: in practice, a lot of people tend to refer to \(\hat{\sigma}\) (i.e., the formula where we divide by \(N-1\)) as the sample standard deviation. Technically, this is incorrect: the sample standard deviation should be equal to \(s\) (i.e., the formula where we divide by \(N\)). These aren’t the same thing, either conceptually or numerically. One is a property of the sample, the other is an estimated characteristic of the population. However, in almost every real life application, what we actually care about is the estimate of the population parameter, and so people always report \(\hat\sigma\) rather than \(s\). This is the right number to report, of course, it’s that people tend to get a little bit imprecise about terminology when they write it up, because “sample standard deviation” is shorter than “estimated population standard deviation.” It’s no big deal, and in practice I do the same thing everyone else does. Nevertheless, I think it’s important to keep the two concepts separate: it’s never a good idea to confuse “known properties of your sample” with “guesses about the population from which it came.” The moment you start thinking that \(s\) and \(\hat\sigma\) are the same thing, you start doing exactly that.

To finish this section off, here’s another couple of tables to help keep things clear:

Symbol	What is it?	Do we know what it is?
\(s\)	Sample standard deviation	Yes - calculated from your sample
\(\sigma\)	Population standard deviation	Almost never known for sure
\(\hat{\sigma}\)	Estimate of the population standard deviation	Yes - but not the same as the sample standard deviation
\(s^2\)	Sample variance	Yes - calculated from your sample
\(\sigma^2\)	Population variance	Almost never known for sure
\(\hat{\sigma}^2\)	Estimate of the population variance	Yes - but not the same as the sample variance

5.6 Estimating a confidence interval

Statistics means never having to say you’re certain – Unknown origin

Up to this point in this chapter, I’ve outlined the basics of sampling theory which statisticians rely on to make guesses about population parameters on the basis of a sample of data. As this discussion illustrates, one of the reasons we need all this sampling theory is that every data set leaves us with a some of uncertainty, so our estimates are never going to be perfectly accurate. The thing that has been missing from this discussion is an attempt to quantify the amount of uncertainty that attaches to our estimate. It’s not enough to be able guess that, say, the mean IQ of undergraduate students is 115 (yes, I just made that number up). We also want to be able to say something that expresses the degree of certainty that we have in our guess. For example, it would be nice to be able to say that there is a 95% chance that the true mean lies between 109 and 121. The name for this is a confidence interval for the mean.

Armed with an understanding of sampling distributions, constructing a confidence interval for the mean is actually pretty easy. Here’s how it works. Suppose the true population mean is \(\mu\) and the standard deviation is \(\sigma\). I’ve just finished running my study that has \(N\) participants, and the mean IQ among those participants is \(\bar{X}\). We know from our discussion of the central limit theorem (Section 5.4) that the sampling distribution of the mean is approximately normal. We also know from our discussion of the normal distribution last week (Section 4.5.3) that there is a 95% chance that a normally-distributed quantity will fall within two standard deviations of the true mean. To be more precise, we can use the qnorm() function to compute the 2.5th and 97.5th percentiles of the normal distribution

qnorm( p = c(.025, .975) )

## [1] -1.959964  1.959964

Okay, so I was slightly imprecise earlier on. The more correct answer is that there is a 95% chance that a normally-distributed quantity will fall within 1.96 standard deviations of the true mean. Next, recall that the standard deviation of the sampling distribution is referred to as the standard error, and the standard error of the mean is written as SEM. When we put all these pieces together, we learn that there is a 95% probability that the sample mean \(\bar{X}\) that we have actually observed lies within 1.96 standard errors of the population mean.

Mathematically, we can write this as: \[ \mu - \left( 1.96 \times \mbox{SEM} \right) \ \leq \ \bar{X}\ \leq \ \mu + \left( 1.96 \times \mbox{SEM} \right) \] where the SEM is equal to \(\sigma / \sqrt{N}\), and we can be 95% confident that this is true.

However, that’s not answering the question that we’re actually interested in. The equation above tells us what we should expect about the sample mean, given that we know what the population parameters are. What we want is to have this work the other way around: we want to know what we should believe about the population parameters, given that we have observed a particular sample. However, it’s not too difficult to do this. Using a little high school algebra, a sneaky way to rewrite our equation is like this:

\[ \bar{X} - \left( 1.96 \times \mbox{SEM} \right) \ \leq \ \mu \ \leq \ \bar{X} + \left( 1.96 \times \mbox{SEM}\right) \]

What this is telling, is that the range of values has a 95% probability of containing the population mean \(\mu\). We refer to this range as a 95% confidence interval, denoted \(\mbox{CI}_{95}\). In short, as long as \(N\) is sufficiently large – large enough for us to believe that the sampling distribution of the mean is normal – then we can write this as our formula for the 95% confidence interval:

\[ \mbox{CI}_{95} = \bar{X} \pm \left( 1.96 \times \frac{\sigma}{\sqrt{N}} \right) \]

Of course, there’s nothing special about the number 1.96: it just happens to be the multiplier you need to use if you want a 95% confidence interval. If I’d wanted a 70% confidence interval, I could have used the qnorm() function to calculate the 15th and 85th quantiles:

qnorm( p = c(.15, .85) )

## [1] -1.036433  1.036433

and so the formula for \(\mbox{CI}_{70}\) would be the same as the formula for \(\mbox{CI}_{95}\) except that we’d use 1.04 as our “magic number” rather than 1.96. The “magic number” is also referred to as the critical value.

5.6.1 Degrees of freedom

The formula that I’ve given above for the 95% confidence interval is approximately correct, but I glossed over an important detail in the discussion. Notice my formula requires you to use the standard error of the mean, SEM, which in turn requires you to use the true population standard deviation \(\sigma\). Yet, in Section 5.5 I stressed the fact that we don’t actually know the true population parameters. Because we don’t know the true value of \(\sigma\), we have to use an estimate of the population standard deviation \(\hat{\sigma}\) instead. This is pretty straightforward to do, but this has the consequence that we need to use the quantiles of the \(t\)-distribution rather than the normal distribution to calculate our magic number; and the answer depends on the sample size. When \(N\) is very large, we get pretty much the same value using qt() that we would if we used qnorm()…

n <- 10000   # suppose our sample size is 10,000
qt( p = .975, df = n-1)   # calculate the 97.5th quantile of the t-dist

## [1] 1.960201

But when \(N\) is small, we get a much bigger number as the critical value when we use the \(t\) distribution:

n <- 10   # suppose our sample size is 10
qt( p = .975, df = n-1)   # calculate the 97.5th quantile of the t-dist

## [1] 2.262157

There’s nothing too mysterious about what’s happening here. Larger critical values mean that the confidence interval is wider, indicating that we’re more uncertain about what the true value of \(\mu\) actually is. When we use the \(t\) distribution instead of the normal distribution, we get bigger numbers, indicating that we have more uncertainty. And why do we have that extra uncertainty? Well, because our estimate of the population standard deviation \(\hat\sigma\) might be wrong! If it’s wrong, it implies that we’re a bit less sure about what our sampling distribution of the mean actually looks like… and this uncertainty ends up getting reflected in a wider confidence interval.

The amount of uncertainty is informed by the degrees of freedom of our estimate. The degrees of freedom (df) of an estimate is the number of independent pieces of information on which the estimate is based. The degrees of freedom is both a concept, and a correction (as seen above). The concept is that if you estimate a property of numbers, and you use this estimate, you will be forcing some constraints on your numbers. To illustrate: let’s pick two random numbers: 51 and -3. I used my personal freedom to pick those two numbers. Now, if our three numbers are 51, -3, and x, and the mean of these three numbers is 2, there is only one solution to x: x has to be -42, otherwise the mean will not be 2. The degrees of freedom for these three numbers is \(N-1 = 3-1= 2\), because 2 of the numbers can be free, but the last number has no freedom, it becomes fixed after the first two are decided. Statisticians often apply degrees of freedom to their calculations. For example, when we calculate the standard deviation of a sample, we first calculate the mean of the sample. By estimating the mean, we are fixing an aspect of our sample, and so, our sample now has \(N-1\) degrees of freedom when we calculate the standard deviation.

While this explains degrees of freedom as a concept, the why of it all will remain unanswered during this course. Maybe pursue a master in statistics if you are really interested in a more detailed explanation of degrees of freedom. I’m sorry. Degrees of freedom are important during this course when working with the \(t\)-distribution, which we will do more extensively in the next chapter. Simply put, in this course, the degrees of freedom are an additional parameter (like the mean and standard deviation) informing the shape of a \(t\) distribution.

5.6.2 Interpreting a confidence interval

The hardest thing about confidence intervals is understanding what they mean. Whenever people first encounter confidence intervals, the first instinct is almost always to say that “there is a 95% probabaility that the true mean lies inside the confidence interval.” It’s simple, and it seems to capture the common sense idea of what it means to say that I am “95% confident.” Unfortunately, it’s not quite right. The intuitive definition relies very heavily on your own personal beliefs about the value of the population mean. I say that I am 95% confident because those are my beliefs. In everyday life that’s perfectly okay, but if you remember back to Section 4.3, you’ll notice that talking about personal belief and confidence is a Bayesian idea. Personally (speaking as a Bayesian) I have no problem with the idea that the phrase “95% probability” is allowed to refer to a personal belief. However, confidence intervals are not Bayesian tools. Like everything else in this chapter, confidence intervals are frequentist tools, and if you are going to use frequentist methods then it’s not appropriate to attach a Bayesian interpretation to them. If you use frequentist methods, you must adopt frequentist interpretations!

Okay, so if that’s not the right answer, what is? Remember what we said about frequentist probability: the only way we are allowed to make “probability statements” is to talk about a sequence of events, and to count up the frequencies of different kinds of events. From that perspective, the interpretation of a 95% confidence interval must have something to do with replication. Specifically: if we replicated the experiment over and over again and computed a 95% confidence interval for each replication, then 95% of those intervals would contain the true mean. More generally, 95% of all confidence intervals constructed using this procedure should contain the true population mean. This idea is illustrated in Figure 5.11, which shows 50 confidence intervals constructed for a “measure 10 IQ scores” experiment (top panel) and another 50 confidence intervals for a “measure 25 IQ scores” experiment (bottom panel). A bit fortuitously, across the 100 replications that I simulated, it turned out that exactly 95 of them contained the true mean.

Figure 5.11: 95% confidence intervals. The top (panel a) shows 50 simulated replications of an experiment in which we measure the IQs of 10 people. The dot marks the location of the sample mean, and the line shows the 95% confidence interval. In total 47 of the 50 confidence intervals do contain the true mean (i.e., 100), but the three intervals marked with asterisks do not. The lower graph (panel b) shows a similar simulation, but this time we simulate replications of an experiment that measures the IQs of 25 people.

The critical difference here is that the Bayesian claim makes a probability statement about the population mean (i.e., it refers to our uncertainty about the population mean), which is not allowed under the frequentist interpretation of probability because you can’t “replicate” a population! In the frequentist claim, the population mean is fixed and no probabilistic claims can be made about it. Confidence intervals, however, are repeatable so we can replicate experiments. Therefore a frequentist is allowed to talk about the probability that the confidence interval (a random variable) contains the true mean; but is not allowed to talk about the probability that the true population mean (not a repeatable event) falls within the confidence interval.

I know that this seems a little pedantic, but it does matter. It matters because the difference in interpretation leads to a difference in the mathematics. There is a Bayesian alternative to confidence intervals, known as credible intervals. In most situations credible intervals are quite similar to confidence intervals, but in other cases they are drastically different. If you are interested in learning more about Bayesian statistics, I can recommend Danielle Navarro’s book chapter about it in Learning statistics with R, or have a look at John Kruschke’s “Doing Bayesian Data Analysis”.

5.7 That’s it for this week

Yay, rejoice. That wasn’t too bad, I hope! Like last week, there are additional practice materials for this week on SOWISO. The materials on SOWISO have a slightly different take on estimation and sampling theory, so have a look at them if want (or need) a different point of view.

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17. Mix of Matthew Crump & Danielle Navarro. Code for figures from Emily Kothe’s Bookdown adaptation of Learning statistics with R↩︎

18. If you run the code above on your computer the results are going to be different due to the (pseudo) random nature of the rnorm() function↩︎

19. As usual, I’m being a bit sloppy here. The central limit theorem is a bit more general than this section implies. Like most introductory stats texts, I’ve discussed one situation where the central limit theorem holds: when you’re taking an average across lots of independent events drawn from the same distribution. However, the central limit theorem is much broader than this. There’s a whole class of things called “\(U\)-statistics” for instance, all of which satisfy the central limit theorem and therefore become normally distributed for large sample sizes. The mean is one such statistic, but it’s not the only one.↩︎

20. Please note that if you were actually interested in this question, you would need to be a lot more careful than I’m being here. You can’t just compare IQ scores in Whyalla to Port Pirie and assume that any differences are due to lead poisoning. Even if it were true that the only differences between the two towns corresponded to the different refineries (and it isn’t, not by a long shot), you need to account for the fact that people already believe that lead pollution causes cognitive deficits: if you recall back to Chapter 3, this means that there are different demand effects for the Port Pirie sample than for the Whyalla sample. In other words, you might end up with an illusory group difference in your data, caused by the fact that people think that there is a real difference. I find it pretty implausible to think that the locals wouldn’t be well aware of what you were trying to do if a bunch of researchers turned up in Port Pirie with lab coats and IQ tests, and even less plausible to think that a lot of people wouldn’t be pretty resentful of you for doing it. Those people won’t be as co-operative in the tests. Other people in Port Pirie might be more motivated to do well because they don’t want their home town to look bad. The motivational effects that would apply in Whyalla are likely to be weaker, because people don’t have any concept of “iron ore poisoning” in the same way that they have a concept for “lead poisoning.” Psychology is hard.↩︎

21. I should note that I’m hiding something here. Unbiasedness is a desirable characteristic for an estimator, but there are other things that matter besides bias. However, it’s beyond the scope of this book to discuss this in any detail. I just want to draw your attention to the fact that there’s some hidden complexity here.↩︎

22. Okay, I’m hiding something else here. In a bizarre and counterintuitive twist, since \(\hat\sigma^2\) is an unbiased estimator of \(\sigma^2\), you’d assume that taking the square root would be fine, and \(\hat\sigma\) would be an unbiased estimator of \(\sigma\). Right? Weirdly, it’s not. There’s actually a subtle, tiny bias in \(\hat\sigma\). This is just bizarre: \(\hat\sigma^2\) is and unbiased estimate of the population variance \(\sigma^2\), but when you take the square root, it turns out that \(\hat\sigma\) is a biased estimator of the population standard deviation \(\sigma\). Weird, weird, weird, right? So, why is \(\hat\sigma\) biased? The technical answer is “because non-linear transformations (e.g., the square root) don’t commute with expectation,” but that just sounds like gibberish to everyone who hasn’t taken a course in mathematical statistics. Fortunately, it doesn’t matter for practical purposes. The bias is small, and in real life everyone uses \(\hat\sigma\) and it works just fine. Sometimes mathematics is just annoying.↩︎