Lecture 6: Samples, Populations, and Distributions
= = In this lecture we’ll learn about samples and populations and how samples and populations can be described with distributions.
Dr Lincoln Colling (University of Sussex)
November 02 2020
Table of Contents
Samples, populations, and distributions
This lecture is made up of two parts. Part one will introduce the concepts of populations and samples and how they are related to one another. In Part Two, we’ll learn about distributions and about how natural processes can give rise to particular kinds of distributions. In particular, we’ll learn about how natural phenomena can give rise to the normal distribution, and we will lay some of the groundwork for understanding the sampling distribution, which will be introduced in a later lecture. Finally, at the end of Part Two, we’ll try to put everything together, and we’ll look at the relationship between samples, populations, and distributions.
Samples and populations
One of the key things that we want to do with statistics is to make inferences about populations from the information we get from samples. That is, we often want to make a judgement, or draw a conclusion, about an aspect of the population when all we have access to is a sample.
We’ll get to more formal definitions of populations and samples shortly, but first, let’s make things more concrete by introducing an example.
Let’s say you’re interested in the average height of people in the UK. The “easy” way to find an answer to this question is to measure all the people in the UK and then work out the average height. Doing this will give you the exact answer to your question. But if you can’t measure everyone in the UK, then what do you do?
One option is to select a smaller group, or subset, of people from the UK. You can then measure the height of people in this group, and then try to use this information to figure out plausible values for the average height of people in the UK.
In this example, the group (or groups) you’re making claims about is the population. You want to claims about the average height of people in the UK. And the sample is a subset of this population—the smaller group of people that you were eventually able to measure.
It’s important to note that there isn’t a single population. What counts as the population will depend on the claim you’re making. For example, let’s say I’m interested in testing the claim, “Do people in East Sussex show an interference effect on the Stroop task?”. Here the population would be people in East Sussex. If, however, I want to make claims about people in general, then the population might be all living humans. The sample is always going to be a subset of the population.
Researchers in psychology aren’t always explicit about which population they’re making inferences about. One might assume it to be something like "humans in general". But if this is the case, then the samples that psychological scientists study should be a subset of humans in general. However, several researchers have noted that the samples used in the vast majority of psychology research tend to be rather WEIRD. That is, the samples used in most published psychological science are drawn from Western, Educated, Industrialized, Rich, and Democratic (WEIRD) societies (Henrich, Heine, and Norenzayan 2010). The preponderance of WEIRD samples implies that researchers either assume that there is little variation across different population groups when it comes to the phenomena they’re studying, or that these samples are as representative of humans in general as any other group.
But is this true? Answering this question is difficult because it would plausibly be conditional on the nature of the research question or phenomenon being studied. For example, it seems plausible that there would be little variation across human populations when it comes to phenomena like low-level auditory and visual processing. However, it also seems plausible to expect to see more variation when it comes to phenomena like moral reasoning, where cultural practices may play a more prominent role. However, it is an empirical question—that is, it is a question that can only be answered by looking at the actual data. Unfortunately, the data required to answer these questions are sparse. However, the data that does exist suggests that there is probably more variation than people would expect, even when it comes to low-level phenomena like visual perception(Henrich, Heine, and Norenzayan 2010). In the past few years, there has been some movement to try and make samples in psychological research more diverse, but there is still a long way to go and, therefore, this is an issue worth bearing in mind.
The relationship between samples and populations
Let’s assume that we have explicitly defined our population (for example, as all people in the UK) and we’ve collected a sample by taking measurements from a subset of this population. What is the relationship between this sample and the population from which it was drawn?
The sample should resemble the population in some way. Most often we’re interest in making inferences about averages—for example, average performance or average score on a measure or a test. In the example I introduced earlier, we were interested in average height. But we might also be interested in things a difference between two averages—for example, whether there is a difference in average depression levels before and after some intervention, or whether average response times are different between the two conditions of a Stroop task. Ideally then, the average height of our sample should resemble the average height of our population, or the averageresponsetimedifference in our experimental sample should resemble the average response time difference in our population. But if we don’t know the average of our population, then how will we know whether our sampleresembles it?
In short, we can’t know for sure. But we can think of a couple of things that will influence the relationship between our sample and the population. To figure out what these are, let’s do a thought experiment and think of some extreme cases.
First, consider the case where all the members of a population are identical. If this were the case, then our sample will have an identical average to the population. The height of one person would be the same as the average height of two people, which would be the same as the average height of 100 people, which would be same as the average height of the population because people only come in one height. But if the members of the population are all different from one another, then there is no guarantee that the sample’s average will resemble the population’s average.
The second extreme scenario is if our sample is very large. Let’s say that it is so large that it includes all the members of the population. If this were the case, then, by definition, our sample average would be identical to the population average. However, if our sample is smaller than the entire population, then once again, there is no guarantee that the sample’s average will resemble the population’s average.
Based on this reasoning, we can say that two things will influence whether your sample resembles your population. These are 1) the amount of variation in our population, and 2) the size of our sample.
Importantly, however, and barring the extreme cases above, for any particular sample we won’t know whether it resembles the population or not, because, remember, we don’t know the average of the population. Instead, we should think about these two factors as influencing how likely it is for samples to resemble the population. But what does this mean?
One way to think about this is in terms of repeatedly taking samples from the same population. For example, if we take a large sample from the population—large, but not so large as to include the entire population—then we can’t say that our particular sample will resemble the population. But if we take many samples (of that size), then we can say that on average those samples will be closer to the population than would be the case for a collection of smaller samples.
The same reasoning applies to variation in the population. If there is less variation in the population, then the samples drawn from that population will tend to be closer to each other and closer to the population average. But again, we won’t be able to say whether a particular sample has an average that is close to the population average.
Of course, sample size and population variation exert their influence together. If we want our sample averages to be close to the population average, then we need samples that are big enough, but what counts as big enough will depend on the population variation. Therefore, knowing whether our sample is big enough depends on knowing the population variation. Unfortunately, we don’t know this; however, there is a way to estimate it. But that’s a topic for another lecture.
The example in Box 1 allows you to explore the relationship between samples and the population. You can make adjustments see to how the relationship is influenced by sample size and variation in the population.
In this example, the population average is 170. On the left side of Box 1, there are frequency plots or histograms of nine different samples that have been drawn from the population. The average of each sample is shown with a red vertical line. You can make two adjustments using the options on the right. You can change the size of samples, and you can change how similar the members of the population are to each other. Click Start to start drawing samples from the population. Make adjustments to the sample size and the population variation and look at the averages of the samples. Some settings produce samples that look very different from each other, and some settings produce samples that look very similar to each to each other.
In the lecture next week, you’ll learn more about how to calculate the average and the spread of a sample.
Figure 1: The relationship between the population and samples.
The second half of this lecture is about distributions. However, before we start talking about distributions, let’s do another simple thought experiment to try and understand how distributions come about.
The binomial distribution
In our thought experiment, we’ll take a coin, and we’ll flip it. When we flip a coin, one of two outcomes is possible. Either the coin will land showing heads, or it will land showing tails. We can say that there are two possible events or two possible sequences of events (sequences will make more sense when we add more coins) than can happen when we flip a coin.
Now let’s think of each of the sequences and count up how many heads are showing in each. In the first sequence, where the coin lands showing tails, no heads occur. In the second sequence, where the coin lands showing heads, there is one head. Therefore, we have one sequence that produces 0 heads, and one sequence that produces one head. And those are all the possible sequences.
But now let’s make it more complicated. Let’s flip two coins. Now there’s a greater number of possible sequences. We can list them:
The first coin shows heads, and so does the second (HH),
The first coin shows heads and the second shows tails (HT)
The first coin shows tails and the second shows heads (TH)
and the first coins shows tails and the second shows tails (TT)
Therefore, there are four possible sequences. Let’s count up the number of sequences that lead to 0 heads, one head, two heads, etc. If we do this, we’ll see that one sequence leads to 0 heads (TT). Two sequences lead to 1 head (HT, and TH). And one sequence leads to 2 heads (HH).
Let’s now add more coins. Things will get trickier from here because the number of sequences rapidly goes up. With three coins, there would be eight possible sequences, and with four coins, there would be 16 possible sequences. Figuring out the number sequences, and the nature of the sequences (whether they produce 0 heads, one head, two heads, etc.) quickly becomes difficult. To make things easier, we’ll draw a plot. First, we’ll draw a plot to trace out the sequences. We’ll use different coloured dots to indicate heads and tails. We can do this in the form of the branching tree diagram shown in Box 2. Once we’ve visualised the sequences, it’s easy to count up how many sequences result in 0 heads, one head, two heads etc. We can put our counts on another plot. For this, we’ll make a frequency plot or histogram. On the x-axis, we’ll have the number of heads. And on the y-axis, we’ll have the count of how many sequences result in that number of heads. This frequency plot is also shown in Box 2.
Figure 2: Sequences of coin flips (top) and the number of sequences producing a particular number of heads.
You can adjust the slider in Box 2 to change the number of coins you want to flip. Increasing the number of coins increases the number of possible sequences, and it changes the number of ways of getting one head, two heads, three heads and so on changes (however, there’s always only one way to get 0 heads and one way to get all heads). Notice that as you adjust the slider and add more and more coins, the frequency plot takes on a characteristic shape. You can mathematically model the shape of this plot using a binomial distribution.
In our coin flipping example, we created this shape by counting up the number of sequences that produced various quantities of heads. But if we look around at natural processes, we’ll see that this shape occurs often.
One natural process that gives rise to this shape is the “bean machine”1. In a bean machine, small steel balls fall from the top of the device to the bottom of the device. On their way down, they bump into pegs. When one of the balls hits a peg, it has a roughly equal chance of bouncing off to the left or the right, not unlike a coin which has a roughly equal chance of landing heads up or tails up. At the bottom of the device are equally-spaced bins for collecting the balls. If enough balls are dropped into the device, then the distribution of balls across the bins will start to take on the shape of the binomial distribution. Very few balls will be at the very edges, because this would require the balls to bounce left or right every time.
Similarly, very few sequences of coin flips result in large numbers of heads or large numbers of tails. The greatest number of balls are seen in the bins near the middle. The balls that land here have bounced left and right a roughly equal number of times. Again a similar pattern can be seen with the coin flips.
In Box 3, I’ve included a computer simulation of a bean machine. Press the Start button to display the bean machine and watch the balls drop. Press Replay to drop more balls.
Figure 3: The bean machine showing beans falling and collecting the a bell shape.
The normal distribution
The shape seen in the binomial distribution is also seen in another distribution called the normal distribution. There are two key differences between the normal distribution and the binomial distribution.
The binomial distribution is bounded. That means that one end represents 0 heads and the other end represents all heads. That is, the distribution can only range from 0 to n (where n is the number of coins that have been flipped)—it is bounded at 0 and n. The normal distribution, however, ranges from negative infinity to positive infinity. Additionally, for the binomial distribution, the steps along the x-axis are discrete. That is, you can have 0 heads, one head, two heads and so on, but you can have anything in between—for example, it’s not possible to have sequences of coin flips that results in 1.5 heads. In contrast, the normal distribution is continuous.
The normal distribution is a mathematical abstraction, but we can use it as a model of real-life frequency distributions. That is, we can use it as a model of populations that are produced by certain kinds of natural processes. Because normal distributions are unbounded and continuous, nothing, in reality, is normally distributed. For example, it’s impossible to have infinity or negative infinity of anything. This is what is meant by an abstraction. But natural processes can give rise to frequency distributions that look a lot like normal distributions, which means that normal distributions can be used as a model of these processes.
Describing the normal distribution
Before we see how natural processes can give rise to the normal distribution, let us take a look at one. Box 4 shows an example of a normal distribution. Two parameters can be used to describe the normal distribution. The location parameter denoted μ describes where the distribution is centred. The scale parameter denoted σ describes the width of the distribution. When you learn how to represent populations with distributions, then these two parameters will correspond to the population average and population variation, respectively.
Figure 4: An example normal distribition. It is described by two parameters. The μ parameter controls where it is centred and the σ parameter controls how wide it is.
Processes giving rise to normal distributions
To see how natural processes can give rise to the normal distribution, let us take a look at the simple simulation in Box 5. The simulation in Box 5 is a model of a dice game. In this dice game, each player rolls a dice a certain number of times and then adds up the values of all the rolls. For example, if a player rolled the dice three times and they got 1, 4, and 3, then their score would be 8 (1 + 4 + 3 = 8). We can plot a frequency distribution of how many players got each of the possible scores. If you have enough players and enough dice rolls, then this frequency distribution will begin to look a lot like normal distribution (See Box 4).
Click Roll! to see it in action. Try adjusting the number of dice rolls and the number of players, and you’ll see how the distribution looks more and more like a normal distribution. (For now, we’ll just ignore the boxes that say Add and Multiply, but we will discuss those later).
Figure 5: A dice game gives rise to a normal distribtion when a players score is calcualted by adding the values from all their rolls. Multiplying the values does not give rise to the normal distribution
Although this is a dice game, we can imagine other processes that might work similarly. For example, a developmental process might work similarly. Let’s say that we have a developmental process like height. At each point in time some value can be added on to the person’s current height—that is, at each time point a person can grow by some amount just like players scores can increase by some amount on each dice roll. If you have enough people, and time points, then the distribution of height will start to look like a normal distribution. The applies for any process that develops or grows by adding bits to its current value.
In the dice game, a person’s score can increase by either 1, 2, 3, 4, 5, or 6 after each roll. And if the dice is balanced an increase of 1 will be no more common than an increase of 6, or 5, or any other number in-between. But even if we had unbalanced dice, so people were more likely to roll a 6 than a 1, then a normal distribution would still appear. A normal distribution appearing isn’t dependent on the numbers that are added at each round. Rather, the important thing is that numbers are added (or subtracted, because subtraction is just the addition of a negative number). If instead of adding we multiplied the numbers, then a normal distribution would not appear.
Processes that don’t produce normal distributions
We won’t cover other distribution shapes in much detail, but let us take a look at an example of a process that doesn’t produce a normal distribution. To do this, we’ll just modify the dice game in Box 5.
In Box 5, click the option that says Multiply. Doing so changes the rules of the dice game so that a player’s score is determined by multiplying together the values of their dice rolls. For example, under the new rules, if a player rolled 1, 4, 3 then their score would be 12 (1 × 4 × 3 = 12). Now try clicking Roll! to see the shape of the distribution. This new distribution has an extreme skew. The vast majority of players have fairly low scores, but a tiny minority of players have extremely high scores. When you have a process that grows by multiplying, then you’ll get a distribution that looks like this.
In psychology, we won’t study many processes that grow like this, but some processes that grow like this will be very familiar to you. Think about something like wealth. Wealth tends to grow by multiplying. For example, earning interest or a return on investment of 10% means that the new value of the investment is 1.10 times the original value. This feature of wealth growth explains why, for example, in the UK, the top 10% of people control more wealth than the bottom 50%.
Describing deviations from the normal distribution
When you clicked the Multiply button, the dice game produced a distribution that was skew. Skew is a technical term that describes one way in which a distribution can deviate from a normal distribution. The normal distribution is symmetrical, but a skew distribution is not. A left-skewed distribution has a longer left tail, and a right-skewed distribution has a longer right tail. Use Box 6 to explore skewness.
Figure 6: An example of a skew distribution. Negative, or left skew, means the left tail is longer. Positive, or right skew, means the right tail is longer.
Apart from skew, deviations from the normal distribution can occur when a distribution either has fatter or skinnier tails than the normal distribution. The tailedness of a distribution is given by its kurtosis. The kurtosis of a distribution is often specified with reference to the normal distribution. In this case, what is being reported is excess kurtosis. A distribution with positive excess kurtosis has a higher kurtosis value than the normal distribution, and a distribution with negative excess kurtosis has a lower kurtosis value than the normal distribution. In your research methods courses, you probably won’t come across many distributions that have negative excess kurtosis. However, the distribution that describes dice rolls is one such distribution, and this will be discussed briefly later in this course. You will encounter distributions with positive excess kurtosis more often. In particular, the t-distributions, a distribution with positive excess kurtosis, will be used in several of the statistical procedures that you will learn. In Box 7, you can explore excess kurtosis. When excess kurtosis is set to 0, then the figure displays a normal distribution. Distributions will no excess kurtosis are called mesokurtotic. When excess kurtosis is negative, the figure displays a thin tailed or platykurtotic distribution. And when excess kurtosis is positive, the figure displays a fat-tailed or leptokurtotic distribution.
Figure 7: An example of excess kurtosis. Positive excess kutoris results in fat tails (exteme values are more common) and negative excess kurtosis results in skinny tails (extreme values are less common)
Distributions and samples
Now that we’ve covered distributions, we can return to samples and see how the two are related to one another. We saw in the dice game (see Box 5) that when we produce values by adding, then those values will start to look like a normal distribution as long as we add up enough values.
Importantly, there’s nothing normally distributed about the numbers we’re. For example, a balanced dice will show the values 1, 2, 3, 4, 5, and 6 with roughly equal frequency. But when you produce values but adding up the result of lots of dice rolls, then those values will be normally distributed.
Now let’s try to relate this to the idea of samples. Let’s say that we have some unknown population, and we can measure members of that population. That is, we can take a sample. And let’s further say that when we’ve measured all the members of our sample, we’ll just add up all those measurements to arrive at some final value. Now let’s repeat that process. We’ll take another sample, sampling the same number of members as before, and add up all the values to arrive at some final value. We can repeat this process over and over until we have lots of sums (created by adding up all the values in the sample). How will these values be distributed?
This situation is exactly analogous to the dice game. When we measure a member of the population, then that is analogous to rolling the dice. Adding the values in the sample to arrive at the final value is analogous to adding up all the dice rolls to get to the player’s score. And the player’s score is equivalent to the sum of all the measurements in our sample. Since the players’ scores are normally distributed (provided we have enough dice rolls) the sample sums will also be normally distributed, provided we use a large enough sample size.
We can now add another mathematical twist to this story. Let’s say that before we add up all the values, we divide each value by the number of values (so if we have a sample size of 10, we’ll divide each value by 10). Now our sum is an average (a mean to be exact).
Now think about what this means. It means that when we take samples from a population, no matter how the population is distributed if our samples are big enough, then our sample averages will be normally distributed. This insight will underlie the concept of the sampling distribution, which you’ll learn about in a later lecture, and it is one of the most fundamental ideas in statistics.
The “bean machine” is sometimes also called the Galton board after it’s inventor, Francis Galton. In addition to inventing the Galton board, Galton was also an advocate of eugenics (a term which he invented). Because of Galton’s association with eugenics and racism, I prefer the term “bean machine”.↩︎