Interpreting and Using Statistics in Psychological Research. Andrew N. Christopher
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to draw conclusions about the world (i.e., the population). At this point and throughout the remainder of the book, we want to learn how to interpret and use statistics correctly.
P for Population, P for Parameter
S for Sample, S for Statistic
A nice heuristic to remember these concepts
Parameter: number that expresses a value in the population.
Statistic: number that expresses a value in the sample.
Descriptive Statistics Versus Inferential Statistics
There are two different broad categories of statistics, and each one has a specific purpose. With descriptive statistics, we are organizing and summarizing a body of information. We will discuss descriptive statistical tools at length in Chapters 2 through 5. Descriptive statistics are useful for learning about the characteristics of our sample. For instance, Pam Mueller and Daniel Oppenheimer (2014) conducted a study to learn whether college students (the population) retain more information when taking notes in class with a computer or with a pen and paper. They sampled college students and measured their performance on test questions based on which note-taking method they used. They calculated certain statistics, such as those we will encounter in the next four chapters, to compare the performance of students using the two different note-taking methods in their sample.
Although descriptive statistics are vital to research, we tend to be more interested in drawing conclusions about a population based on information from our sample. And, again, most always we cannot collect information from everyone in our population, so we rely on our sample to make inferences about the population. In that situation, we use inferential statistics. The purpose of inferential statistics is to draw an inference about conditions that exist in a population by studying a sample drawn from the population. We will learn about different inferential statistical tools in Chapters 6 through 14. Mueller and Oppenheimer (2014) were interested in learning about how note-taking was related to academic performance in a population of college students. Not being able to test all college students, they sampled students from Princeton University and UCLA. From this sample, they drew conclusions about college students and the effects of how they take notes on test performance.
Descriptive statistics: quantitative procedures that are used to organize and summarize (describe) information about a sample.
Inferential statistics: quantitative procedures that are used to learn if we can draw conclusions (inferences) about a population based on a sample.
Sampling Error
Of course, samples are not always (and hardly ever are) perfect representations of populations. The extent to which a sample doesn’t reflect the population is called sampling error. Sampling error occurs when there is a difference between the characteristics of the population and the characteristics of the sample. Let’s consider an example of sampling error. Let’s again consider Mueller and Oppenheimer’s (2014) research. They drew their sample of college students from two schools, Princeton and UCLA. From this sample, they wanted to learn about the population of college students. Let’s list some parameters for the population of college students in the United States (National Center for Education Statistics, 2014). These parameters for three variables appear in the top portion of Figure 1.5. How well does the sample that Mueller and Oppenheimer (2014) used map onto the population? The statistics for the same variables appear in the lower portion of Figure 1.5.
Sampling error: discrepancy between characteristics of the population and characteristics of the sample.
As you can see, there are some discrepancies between the population parameters and the sample statistics. For example, the sample did not contain any students from two-year schools. Does such sample error make Mueller and Oppenheimer’s (2014) work pointless? Absolutely not. We just need to remember the sample characteristics when drawing conclusions from this research. For instance, it might be a worthwhile idea to conduct this study again using a sample of students from two-year colleges, as no such students were included in this sample. Indeed, learning about the world around us through research is a process that can never be accomplished in a single research study. Indeed, one could argue correctly that the population in this research was in fact college students at four-year universities. Would the results of this research apply to my school, which is a four-year liberal arts college that serves only undergraduate students? Would the results apply to your school? Without conducting the research with a sample of students from my school and a sample of students from your school, we cannot know.
Figure 1.5 Population Parameter and Sample Statistics in Mueller and Oppenheimer’s (2014) Research
To take another example of sampling error, suppose the average SAT score at your college or university is 1700 (if your college or university required the ACT, suppose that the average ACT score is 22). The population in this instance is students at your school. Now, if you take two samples of 10 students each, do you think each sample will have an average SAT of 1700 (or average ACT of 22)? Probably not; in fact, I bet neither sample will have precisely that average. One sample may have an average SAT of 1684 (20 ACT), and the second sample may have an average SAT of 1733 (23 ACT). The discrepancy is the sampling error.
You may often hear on the news about public opinion surveys that various polling groups (e.g., the Wall Street Journal or CNN) conduct on different topics (e.g., feelings about the economy or Congressional approval ratings). To conduct such surveys, the polling agencies do not (and realistically cannot) ask every member of the population for their input. So, they sample the population of interest. As you know, we now have to consider sampling error. That is what is meant when you hear about this “margin of error” the news is (or should be) reporting. If the president has a 54% approval rating, there should be a margin of error reported. That 54% is based on the sample, so there will be some variability around that number in the larger population. A 54% approval rating with a “plus or minus 3% margin of error” means that the presidential approval rating in the population is between 51% and 57%.
Learning Check
1 Explain the difference between a parameter and a statistic.A: A parameter is a population value, whereas a statistic is a sample value.
2 Explain the difference between descriptive statistics and inferential statistics.A: Descriptive statistics organize and summarize information about a sample. They are the first step toward using inferential statistics, which are procedures used to learn whether we can make conclusions about a population based on data from a sample.
3 Why is it the case that in almost any research study, there will be some degree of sampling error?A: Unless everyone in the population is included in the sample, there will be some discrepancy between the population characteristics and the sample characteristics.
Notes
1. If you’ve ever suffered a serious fall in your home, you may well fear this event more than a terrorist attack. However, if you are basing this relative fear on your experience and not on statistical information, you are still making use of the availability heuristic.
2. The expression “A broken clock is correct twice each day” is making use of the law of small numbers. At two times out of the 1,440 possible times (60 minutes × 24 hours) each day, the broken clock will tell the correct time. Of course, the other 1,438 times, the clock is incorrect, but if you looked at it those