Statistic Worksheet Help

Chapter 1

Introduction

Welcome to PSYC3700: Statistics for Psychology! This course is designed to equip you with the essential skills needed to apply quantitative statistics to the study of human behavior. Throughout this course, you will gain practical experience using statistical software to compute descriptive statistics, construct graphs, test differences between groups, evaluate correlations, calculate effect sizes, and determine both practical and statistical significance.

This textbook, Learning Statistics with JASP: A Tutorial for Psychology Students and Other Beginners, was originally authored by Danielle J. Navarro, David R. Foxcroft, and Thomas J. Faulkenberry. It has been carefully edited to align with the curriculum of PSYC3700 at Capella University. The original version of the textbook can be accessed at https://tomfaulkenberry.github.io/JASPbook/. We hope this tailored version of the textbook will enhance your learning experience and help you succeed in your studies. Let’s embark on this exciting journey into the world of statistics together!

1.1 Why do we Learn Statistics?

To the surprise of many students, statistics is a fairly significant part of a psychological education. To the surprise of no one, statistics is very rarely the favorite part of one’s psychological education. After all, if you really loved the idea of doing statistics, you would probably be enrolled in a statistics class right now, not a psychology class. So, not surprisingly, there is usually a large proportion of the student base that isn’t happy about the fact that psychology has so much statistics in it. In view of this, I thought that the right place to start might be to answer some of the more common questions that people have about statistics.

A big part of this issue at hand relates to the very idea of statistics. What is it? What is it there for? And why are scientists so obsessed with it? These are all good questions when you think about it. So, let’s start with the last one. As a group, scientists seem to be bizarrely fixated on running statistical tests on everything. In fact, we use statistics so often that we sometimes forget to explain to people why we do. It’s a kind of article of faith among scientists – and especially social scientists – that your findings can’t be trusted until you’ve done some statistics. Rarely does someone take the time to answer one very simple question:

Why do we do statistics? Why don’t scientists just use common sense?

It’s a naive question in some ways, but most good questions are. There are a lot of good answers to it,1 but for my money, the best answer is a really simple one: we don’t trust ourselves enough. We worry that we’re human, and susceptible to all of the biases, temptations, and frailties that humans suffer from. Much of statistics is basically a safeguard. Using “common sense” to evaluate evidence means trusting gut instincts, relying on verbal arguments, and on using the raw power of human reason to come up with the right answer. Most scientists don’t think this approach is likely to work.

In fact, come to think of it, this sounds a lot like a psychological question to me, and since we are in a psychology department, it seems like a good idea to dig a little deeper here. Is it really plausible to think that this “common sense” approach is very trustworthy? Verbal arguments have to be constructed in language,

1Including the suggestion that common sense is in short supply among scientists.

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and all languages have biases – some things are harder to say than others, and not necessarily because they’re false (e.g., quantum electrodynamics is a good theory, but hard to explain in words). The instincts of our “gut” aren’t designed to solve scientific problems, they’re designed to handle day-to-day inferences – and given that biological evolution is slower than cultural change, we should say that they’re designed to solve the day-to-day problems for a different world than the one we live in. Most fundamentally, reasoning sensibly requires people to engage in “induction.” Making wise guesses and going beyond the immediate evidence of the senses to make generalizations about the world. If you think that you can do that without being influenced by various distractions, well, I have a bridge in London I’d like to sell you. As the next section shows, we can’t even solve “deductive” problems (ones where no guessing is required) without being influenced by our pre-existing biases.

1.1.1 The Curse of Belief Bias

People are mostly pretty smart. We’re certainly smarter than the other species that we share the planet with (though many people might disagree). Our minds are quite amazing things, and we seem to be capable of the most incredible feats of thought and reason. That doesn’t make us perfect though. And among the many things that psychologists have shown over the years is that we really do find it hard to be neutral, to evaluate evidence impartially and without being swayed by pre-existing biases. A good example of this is the belief bias effect in logical reasoning: if you ask people to decide whether a particular argument is logically valid (i.e., conclusion would be true if the premises were true), we tend to be influenced by the believability of the conclusion, even when we shouldn’t. For instance, here’s a valid argument where the conclusion is believable:

All cigarettes are expensive (Premise 1) Some addictive things are inexpensive (Premise 2) Therefore, some addictive things are not cigarettes (Conclusion)

And here’s a valid argument where the conclusion is not believable:

All addictive things are expensive (Premise 1) Some cigarettes are inexpensive (Premise 2) Therefore, some cigarettes are not addictive (Conclusion)

The logical structure of argument #2 is identical to the structure of argument #1, and they’re both valid. However, in the second argument, there are good reasons to think that premise 1 is incorrect, and as a result it’s probably the case that the conclusion is also incorrect. But that’s entirely irrelevant to the topic at hand; an argument is deductively valid if the conclusion is a logical consequence of the premises. That is, a valid argument doesn’t have to involve true statements.

On the other hand, here’s an invalid argument that has a believable conclusion:

All addictive things are expensive (Premise 1) Some cigarettes are inexpensive (Premise 2) Therefore, some addictive things are not cigarettes (Conclusion)

And finally, an invalid argument with an unbelievable conclusion:

All cigarettes are expensive (Premise 1) Some addictive things are inexpensive (Premise 2) Therefore, some cigarettes are not addictive (Conclusion)

Now, suppose that people really are perfectly able to set aside their pre-existing biases about what is true and what isn’t, and purely evaluate an argument on its logical merits. We’d expect 100% of people to say that the valid arguments are valid, and 0% of people to say that the invalid arguments are valid. So, if you ran an experiment looking at this, you’d expect to see data like this:

conclusion feels true conclusion feels false

argument is valid 100% say “valid” 100% say “valid”

argument is invalid 0% say “valid” 0% say “valid”

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If the psychological data looked like this (or even a good approximation to this), we might feel safe in just trusting our gut instincts. That is, it’d be perfectly okay just to let scientists evaluate data based on their common sense, and not bother with all this murky statistics stuff. However, you all have taken psych classes, and by now you probably know where this is going.

In a classic study, Evans (1983) ran an experiment looking at exactly this. What they found is that when pre-existing biases (i.e., beliefs) were in agreement with the structure of the data, everything went the way you’d hope:

conclusion feels true conclusion feels false

argument is valid 92% say “valid”

argument is invalid 8% say “valid”

Not perfect, but that’s pretty good. But look what happens when our intuitive feelings about the truth of the conclusion run against the logical structure of the argument:

conclusion feels true conclusion feels false

argument is valid 92% say “valid” 46% say “valid”

argument is invalid 92% say “valid” 8% say “valid”

Oh dear, that’s not as good. Apparently, when people are presented with a strong argument that contradicts our pre-existing beliefs, we find it pretty hard to even perceive it to be a strong argument (people only did so 46% of the time). Even worse, when people are presented with a weak argument that agrees with our pre-existing biases, almost no-one can see that the argument is weak (people got that one wrong 92% of the time!).2

If you think about it, it’s not as if these data are horribly damning. Overall, people did do better than chance at compensating for their prior biases, since about 60% of people’s judgments were correct (you’d expect 50% by chance). Even so, if you were a professional “evaluator of evidence” and someone came along and offered you a magic tool that improves your chances of making the right decision from 60% to (say) 95%, you’d probably jump at it, right? Of course you would. Thankfully, we actually do have a tool that can do this. But it’s not magic, it’s statistics. So that’s reason #1 why scientists love statistics. It’s just too easy for us to “believe what we want to believe.” So instead, if we want to “believe in the data,” we’re going to need a bit of help to keep our personal biases under control. That’s what statistics does, it helps keep us honest.

1.1.2 The Cautionary Tale of Simpson’s Paradox

Simpson’s Paradox occurs when a trend that appears in several different groups of data disappears or reverses when the groups are combined. This paradox highlights the importance of considering the context and structure of data before drawing conclusions. It often arises in situations where data is aggregated across different groups, leading to misleading or counterintuitive results.

Let’s look at one frequently cited example of Simpson’s paradox. In 1973, the University of California, Berkeley had some worries about the admissions of students into their postgraduate courses. Specifically, the thing that caused the problem was that the gender breakdown of their admissions, which is shown in Table 1.1.

Table 1.1: Berkeley Application Counts and Admissions Rates by Gender Number of applicants Percent admitted

Men 8442 44% Women 4321 35%

2In my more cynical moments I feel like this fact alone explains 95% of what I read on the internet.

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Given this, they were worried about being sued! Given that there were nearly 13,000 applicants, a difference of 9% in admission rates between men and women was just way too big to be a coincidence. Pretty compelling data, right? And if I were to say to you that these data actually reflect a weak bias in favor of women, you probably wouldn’t believe me.

Oddly, it’s actually sort of true. When people started looking more carefully at the admissions data they told a rather different story (Bickel et al., 1975). Specifically, when they looked at it on a department by department basis, it turned out that most of the departments actually had a slightly higher success rate for women than for men. Table 1.2 shows the admission figures for the six largest departments (with the names of the departments removed for privacy reasons).

Table 1.2: Berkeley Application Counts and Admissions Rates by Department and Gender Men Women

Department Applicants Percent admitted Applicants Percent admitted

A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7%

Remarkably, most departments had a higher rate of admissions for women than for men! Yet, the overall rate of admission across the university for women was lower. How can this be? How can both of these statements be true at the same time?

Here’s what’s going on. Firstly, notice that the departments are not equal to one another in terms of their admission percentages: some departments (e.g., A, B) tended to admit a high percentage of the qualified applicants, whereas others (e.g., F) tended to reject most of the candidates, even if they were high quality. So, among the six departments shown above, notice that department A is the most generous, followed by B, C, D, E and F in that order. Next, notice that women and men tended to apply to different departments. If we rank the departments in terms of the total number of applicants who identified as men, we get A>B>D>C>F>E (the “easy” departments are in bold). On the whole, men tended to apply to the departments that had high admission rates. Now compare this to how the applicants who identified as women distributed themselves. Ranking the departments in terms of the total number of women applicants produces a quite different ordering C>E>D>F>A>B. In other words, what these data seem to be suggesting is that women tended to apply to “harder” departments. And in fact, if we look at Figure 1.1 we see that this trend is systematic, and quite striking.

This effect is known as Simpson’s paradox. It’s not common, but it does happen in real life, and most people are very surprised by it when they first encounter it, and many people refuse to even believe that it’s real. It is very real. And while there are lots of very subtle statistical lessons buried in there, I want to use it to make a much more important point: doing research is hard, and there are lots of subtle, counter-intuitive traps lying in wait for the unwary. That’s reason #2 why scientists love statistics, and why we teach research methods. Because science is hard, and the truth is sometimes cunningly hidden in the nooks and crannies of complicated data.

Before leaving this topic entirely, I want to point out something else really critical that is often overlooked. Statistics only solves part of the problem. Remember that we started all this with the concern that Berkeley’s admissions processes might be unfairly biased against women. When we looked at the “aggregated” data, it did seem like the university was discriminating against women, but when we “disaggregate” and looked at the individual behavior of all the departments, it turned out that the actual departments were, if anything, slightly biased in favor of women. The gender bias in total admissions was caused by the fact that women tended to self-select for harder departments. From a legal perspective, that would probably put the university in the clear. Postgraduate admissions are determined at the level of the individual department, and there are good reasons to do that. At the level of individual departments, the decisions are more or less unbiased (the weak bias in favor of women at that level is small, and not consistent across departments). Since the

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Figure 1.1: The Berkeley 1973 college admissions data. This figure plots the admission rate for the 85 departments that had at least one female applicant, as a function of the percentage of applicants that were female. The plot is a redrawing of Figure 1 from Bickel (1975). Circles plot departments with more than 40 applicants; the area of the circle is proportional to the total number of applicants. The crosses plot departments with fewer than 40 applicants.

university can’t dictate which departments people choose to apply to, and the decision making takes place at the level of the department, it can hardly be held accountable for any biases that those choices produce.

That was the basis for my somewhat glib remarks earlier, but that’s not exactly the whole story, is it? After all, if we’re interested in this from a more sociological and psychological perspective, we might want to ask why there are such strong gender differences in applications. Why do men tend to apply to engineering more often than women, and why is this reversed for the English department? And why is it the case that the departments that tend to have more women apply tend to have lower overall admission rates than those departments that have more men apply? Might this not still reflect a gender bias, even though every single department is itself unbiased? It might. Suppose, hypothetically, that men preferred to apply to “hard sciences” and women prefer “humanities.” And suppose further that the reason for why the humanities departments have low admission rates is because the government doesn’t want to fund the humanities (Ph.D. places, for instance, are often tied to government funded research projects). Does that constitute a gender bias? Or just an unenlightened view of the value of the humanities? What if someone at a high level in the government cut the humanities funds because they felt that the humanities are “useless chick stuff”. That seems pretty blatantly gender biased. None of this falls within the purview of statistics, but it matters to the research project. If you’re interested in the overall structural effects of subtle gender biases, then you probably want to look at both the aggregated and disaggregated data. If you’re interested in the decision making process at Berkeley itself then you’re probably only interested in the disaggregated data.

In short there are a lot of critical questions that you can’t answer with statistics, but the answers to those questions will have a huge impact on how you analyze and interpret data. And this is the reason why you should always think of statistics as a tool to help you learn about your data. No more and no less. It’s a powerful tool to that end, but there’s no substitute for careful thought.

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1.2 Statistics in Psychology

I hope that the discussion above helped explain why science in general is so focused on statistics. But I’m guessing that you have a lot more questions about what role statistics plays in psychology. So, here’s my attempt to answer a few of them...

• Why does psychology have so much statistics?

To be perfectly honest, there’s a few different reasons, some of which are better than others. The most important reason is that psychology is a statistical science. What I mean by that is that the “things” that we study are people. Real, complicated, gloriously messy, infuriatingly perverse people. The “things” of physics include objects like electrons, and while there are all sorts of complexities that arise in physics, electrons don’t have minds of their own. They don’t have opinions, they don’t differ from each other in weird and arbitrary ways, they don’t get bored in the middle of an experiment, and they don’t get angry at the experimenter and then deliberately try to sabotage the data set (not that I’ve ever done that!). At a fundamental level psychology is harder than physics.3

Basically, we teach statistics to you as psychologists because you need to be better at statistics than physicists. There’s actually a saying used sometimes in physics, to the effect that “if your experiment needs statistics, you should have done a better experiment.” They have the luxury of being able to say that because their objects of study are pathetically simple in comparison to the vast mess that confronts social scientists. And it’s not just psychology. Most social sciences are desperately reliant on statistics. Not because we’re bad experimenters, but because we’ve picked a harder problem to solve. We teach you statistics because you really, really need it.

• Can’t someone else do the statistics?

To some extent, but not completely. It’s true that you don’t need to become a fully trained statistician just to do psychology, but you do need to reach a certain level of statistical competence. In my view, there’s three reasons that every psychological researcher ought to be able to do basic statistics:

– Firstly, there’s the fundamental reason: statistics is deeply intertwined with research design. If you want to be good at designing, or even just interpreting, psychological studies, you need to at the very least understand the basics of statistics.

– Secondly, if you want to be good at the psychological side of the research, then you need to be able to understand the psychological literature, right? But almost every paper in the psycholog- ical literature reports the results of statistical analyses. So if you really want to understand the psychology, you need to be able to understand what other people did with their data. And that means understanding a certain amount of statistics.

– Thirdly, there’s a big practical problem with being dependent on other people to do all your statistics: statistical analysis is expensive. If you ever get bored and want to look up how much the Australian government4 charges for university fees, you’ll notice something interesting: statistics is designated as a “national priority” category, and so the fees are much, much lower than for any other area of study. This is because there’s a massive shortage of statisticians out there. So, from your perspective as a psychological researcher, the laws of supply and demand aren’t exactly on your side here! As a result, in almost any real life situation where you want to do psychological research, the cruel facts will be that you don’t have enough money to afford a statistician. So, the economics of the situation mean that you have to be pretty self-sufficient.

Note that a lot of these reasons generalize beyond researchers. If you want to be a practicing psychologist and stay on top of the field, it helps to be able to read the scientific literature, which relies heavily on statistics.

3Which might explain why physics is just a teensy bit further advanced as a science than we are. 4The first author of this textbook, Danielle Narvarro, is from Australia, thus the references to Australia throughout the

book.

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• I don’t care about jobs, research, or clinical work. Do I need statistics?

Okay, now you’re just messing with me. Still, I think it should matter to you too. Statistics should matter to you in the same way that statistics should matter to everyone. We live in the 21st century, and data are everywhere. Frankly, given the world in which we live these days, a basic knowledge of statistics is close to a survival tool! Which is the topic of the next section.

1.2.1 Statistics in Everyday Life

“We are drowning in information, but we are starved for knowledge” – Various authors, original probably John Naisbitt

When I started writing up my lecture notes I took the 20 most recent news articles posted to the ABC news website. Of those 20 articles, it turned out that 8 of them involved a discussion of something that I would call a statistical topic and 6 of those made a mistake. The most common error, if you’re curious, was failing to report baseline data (e.g., the article mentions that 5% of people in situation X have some characteristic Y, but doesn’t say how common the characteristic is for everyone else!). The point I’m trying to make here isn’t that journalists are bad at statistics (though many are), it’s that a basic knowledge of statistics is very helpful for trying to figure out when someone else is either making a mistake or even lying to you. In fact, one of the biggest things that a knowledge of statistics does to you is cause you to get angry at the newspaper or the internet on a far more frequent basis.

1.3 Conclusion

In this chapter, we explored the fundamental role that statistics plays in the field of psychology. From understanding why scientists rely so heavily on statistical methods to examining specific phenomena like Simpson’s Paradox, we’ve seen that statistics is crucial for making sense of data.

Statistics helps us navigate the inherent biases and limitations of human reasoning, providing a safeguard against some of our own cognitive shortcomings. It allows us to move beyond gut instincts and verbal arguments, enabling us to draw more reliable and objective conclusions from our research.

The practical aspects of conducting psychological research necessitate a solid grasp of statistics. Whether it’s designing experiments, interpreting results, or staying current with the latest literature, statistical com- petence is essential for any aspiring psychologist.

In summary, while statistics may not be the most beloved part of a psychological education, it is un- deniably one of the most important. It equips us with the tools to critically evaluate evidence, uncover hidden patterns, and ultimately contribute to the advancement of psychological science. As you continue your journey in PSYC 3700, remember that mastering statistics is not just about passing a course—it’s about becoming a better psychologist.

1.4 Reference

Bickel, P. J., Hammel, E., A., & O’Connell, J. W. (1975). Sex bias in graduate admissions: Data from Berkeley. Science, 187, 398-404. https://homepage.stat.uiowa.edu/~mbognar/1030/Bickel-Berkeley.pdf

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