As discussed in Section 2-5, when two variables are correlated, this means that they change together, on average, in a large group of individuals. For example, major depression is correlated with sexual identity[∂]: compared to males, about twice as many females develop the disorder during their lives. Calvete and Cardeñoso (2005) summarized research demonstrating that this sex difference appears by adolescence:

[R]ecent research suggests that gender differences in psychological problems are evident during childhood and adolescence, prior to the acquisition of adult social roles. ... For instance, gender differences in depression begin to emerge at age 14 ..., and during the period from ages 15 to 18 the female rate of depression rises to double the prevalence rate for males. (p. 179)

When interpreting these results, there are two caveats[∂] of which we must be aware. First, the finding of this correlation tells us nothing about its cause, although it could be that there is something about women (biologically, psychologically, socially, and/or culturally) that makes them more susceptible than men are to developing major depression. Second, finding that women are about twice as likely as men to develop major depression does not mean that a particular group of depressed people will have twice as many female members as male members. A group of depressed people will vary around the ratio of two depressed females for every depressed male. Thus, when interpreting a correlation, we must be careful not to draw conclusions about the following:

• The nature of the causal relationship between the correlated variables. In a later section, you will learn about two problems — the directionality and third-variable problems — that limit what we can infer from a single correlation.
• The physical or psychological characteristics of individuals. Correlations represent what is true, on average, in a group of individuals. We cannot tell from an average what is true about particular individuals (see below).

The Mean of a Sample of Observations

The average of a sample of observations typically is calculated as the mean, which is the summation of individual measurements divided by the number of measurements. For example, let's say that, in a class of ten students, the following test scores were obtained on a test with 100 questions: 70, 52, 90, 96, 46, 36, 78, 88, 66, and 98. Adding these ten scores, we get 720. Dividing 720 by the number of scores (10), we get 72. Thus, the mean test score is 72 (a C): the "average person" in the class (who doesn't actually exist) answered 72% of the test questions correctly. Saying this another way, students had a tendency to receive a score of 72% on the test. Based on this information alone, however, a student cannot infer what his or her test score was. In fact, no student received a test score of 72% in the sample of ten students, and only three students were close (66, 70, and 78).

Nevertheless, many people with little or no training in research methods and statistics misunderstand and misinterpret the mean of a sample of observations. This becomes obvious when interpreting the results of research on social-group differences, such as studies of gender differences in psychological characteristics. For example, Bryan (1997) reported on research investigating gender differences among first-graders in their approaches to solving arithmetic problems:

Results showed that by January of their first grade year, gender differences existed ... in the way that the children approached problem solving, not in the number of problems the students solved correctly. In both individual and group settings girls were more likely to use overt methods — counting on counters or counting on fingers — to solve the problems. Boys were more likely to use retrieval — relying on memorized answers — in both individual and group settings.

Over the course of the school year, boys were also more likely to increase their attempts to use retrieval even if they were not successful. Girls, however, seemed to be more concerned with being right and used backup strategies of counting on counters and counting on fingers.

It is probable that many nonspecialists reading this report, especially those who show a strong reliance on gender stereotypes[∂] when thinking about gender differences, would conclude that boys approach math problems in one way and girls approach math problems in a very different way. But, as phrases such as "more likely" imply, gender-difference studies investigate differences between the mean of a group of males and the mean of a group of females, not differences among individual males and individual females. Thus, finding that there is a difference between the means of a group of boys and a group of girls does not allow us to conclude that a particular boy or a particular girl has one or the other characteristic. In short, some boys will be very concerned with being correct and use overt methods of counting (just like the average girl) and some girls will be relatively unconcerned with being correct and rely on memorized answers (just like the average boy).

This point is easily understood by looking at an example of a gender difference in an easily observable physical characteristic, such as height. In October, 2004, the National Center for Health Statistics of the Centers for Disease Control and Prevention published a report (Ogden, Fryar, Carroll, & Fiegal, 2004) that included the average heights of Americans from 1960 to 2002 by age, race ethnicity, and sex. Table 1 shows the average heights (rounded to the nearest whole number) for non-hispanic white males and females between the ages of 20 and 39, inclusive[∂], during the years 1999 to 2002 (for other groups and time periods, please see the report). The men are, on average, 5 inches taller than the women.

Table 1. Means, variances, and sample sizes in a group of non-Hispanic white Americans between the ages of 20 and 39 years

But the finding that, in this group, men are five inches taller than women, on average, obviously does not mean that a particular man selected from this group is five inches taller than a particular woman; or even that he is taller than the woman by any amount. Furthermore, these statistics tell us nothing specific about what the height will be of the next non-Hispanic white man or woman between 20 and 39 years of age that we see. Although we can predict that the man probably will be closer to 70 inches than 65 inches, and that the woman probably will be closer to 65 inches than 70 inches, and that the man is likely to be taller than the woman, we will not know any of this for certain until we measure their heights. This is because, as can be seen in Figure 1, individual males and females vary around their respective means. The graph shows clearly that many women are taller than many men (indicated by the overlapping portion of the two distributions). In fact, some men are shorter than the mean height of women; and some women are taller than the mean height of men.

Figure 1. The distribution of heights in a group of non-Hispanic white Americans between the ages of 20 and 39 years

The Variance of a Sample of Observations

Most individuals differ to varying extents from the means of their groups. The overall "spread" of measurements in a group generally is estimated by a statistic called the variance, which measures the degree to which individuals differ from the group mean (for a basic introduction to descriptive statistics[∂] such as the mean and variance, click on this link). The variance is important for understanding the average degree to which individuals differ from the mean. As you can see in Figure 1, the heights of individuals differ a great deal not only from the group mean, but also from each other. One very simple measure of variance is the difference between the lowest and highest scores — a measure called the range of scores. In the test-score example presented above, the lowest score was 36 and the highest score was 98 — a range of 62 — which is a very large spread of scores. Most school tests that discriminate well between those who know the material and those who don't should show a large range as long as there is a good deal of variance in how well students know the material.

Because individuals typically differ from each other, even when they are drawn from a relatively homogeneous[∂] group, we must use statements such as the following when reporting on group averages and group differences:

• "there was a tendency for members of this group to..."
• "on average, members of this group will..."
• "individuals from this group were more likely than individuals from the other group to ...."

Nevertheless, people often misunderstand what the italicized words mean. For example, when the claim is made that men, on average, desire to have more sexual partners than women do, or that men are more sexually active, on average, than women are, some people conclude that, compared to all women, all men want more sexual partners and are more sexually active (in other words, that men "think" with their genitals). But such inferences do not follow from the finding of a difference between group means. There always is a significant number of men who show little or no sexual activity (either alone or with partners) and a significant number of women who show a great deal of sexual activity. The main danger of misinterpreting mean differences between groups is that it encourages stereotypical thinking — a topic that will be discussed in another section of this text.

What Do Differences Between Group Means Tell Us?

Students often ask psychology instructors about the cause(s) of a family member's or friend's mental disorder, such as bipolar disorder (formerly called manic-depressive disorder). The most accurate answer instructors can give to such questions is, "I don't know." But then they should discuss what research has shown to be possible or likely causes, such as the correlations that have been found between genes and bipolar disorder, the finding that stressful events often precede the first manic[∂] or depressive[∂] episodes, or the research that points to disturbances of biochemical activity in the brain. With these findings in mind, the instructor then might speculate that the person inherited genes that predisposed him or her to develop bipolar disorder; that a stressful event(s) probably triggered the development of the disorder; and that it probably would be beneficial for the person to try medications that reduce the symptoms of bipolar disorder. This answer is based on discoveries of average differences between groups or average associations between variables. Because averages can't be used to infer anything specific about an individual, however, the results of these studies don't allow the instructor to give a definitive answer to the question about the cause(s) of bipolar disorder in the relative or friend.

If averages tell us nothing definite about individuals, then why do psychologists spend so much time calculating them? Perhaps an example will help to answer this question. Let's say that we calculated the mean scores on the first PSY 101 test for two groups of students with different instructors, Dr. Smith and Dr. Jones, and obtained the results shown in Table 2. Based on the mean scores, which instructor would you choose? Most of you probably chose Dr. Jones because his mean test score was higher. But, in doing this, you are ignoring the individual differences. That is, even though the average student received a lower test score in Dr. Smith's class, it may be that you would receive a much higher test score in her class, perhaps because her style of teaching fits better with your style of learning.

Table 2. A fictional example of the means and ranges of student test scores in two sections of PSY 101 taught by different instructors

Nevertheless, the difference in mean scores tells us that there is something about Dr. Smith's class that causes students, on average, to receive lower test scores. Because the study is correlational, however, we can't determine what the important factor(s) is(are). It may be any one or more of the following:

• the time of day the class is held
• the average intelligence of the students
• the teaching style of Dr. Smith
• the room in which the class is held
• the average motivation of the students
• the difficulty level of the test
• the textbook used
• the material covered
• the study aids provided
• the day on which the test was given,
• etc.

Differences between group means help researchers to discover and test possible causal factors. Knowing that there is a difference between the average heights of men and women tells us that there is something associated with gender that determines height. Perhaps it has something to do with differing hormone levels, genes on the Y chromosome, etc. Researchers can test their causal hypotheses by controlling for all possible causes except for the one they are looking at (a topic we will return to later), thereby eliminating other causal explanations.

How much sleep do you need?

As one last example of the difference between group averages and variances, let’s try to answer the following question: how much sleep do you need per night? On average, American adults get about 8 hours of sleep per night. So you might answer with confidence: “I need about 8 hours of sleep per night.” What's the problem with this answer? Actually, there are three problems with this answer. It confuses the average amount of sleep that adults get per night with the average amount of sleep that:

(1) adults need per night.
(2) you get per night.
(3) you need per night.

In other words, how much sleep that a group of adults actually get per night, on average, is not necessarily the amount that they need to get per night, on average. Furthermore, a group average tells us nothing specific about an individual member of that group. Thus, instead of referring to statistics and graphs, Moorcroft (1993) answered the question of how much sleep is needed by each person per night in this way:

Sleep as much as necessary so you do not feel tired the next day,”.... This is probably not the kind of answer you expected, but it is the best answer. While 7 1/2 hours per night is the average amount of sleep required for young adults [this claim is questionable: as just stated, this is the average amount that they get per night, not necessarily what they need], there are wide individual differences.... Some people do well with 6 hours or less per night, while others cannot live comfortably with less than 10 or 11 hours of sleep (Webb, 1975). Only you can determine how much sleep you require, since only you know when you feel good the next day. (p. 94)

So, go to sleep at a certain time each night for a few nights in a row and see what time you wake up in the morning (without an alarm clock). This is how much time you need for sleeping. For most of you, this will be between seven and nine hours, but some of you will be outside of this range (see Figure 2). According to various reports, the least amount of sleep per night needed by a person was the case of an elderly woman who needed only one hour. Another person needed only three hours per night. But almost all of us need much more than this. If you need more than the average amount of sleep per night, try to arrange your daily schedule so that you can get that much sleep.

Figure 2. The percentage of adults who sleep from 3.5 to 10.5 hours per night (from Moorcroft, 1993, p. 96)

Genes probably are involved, to some extent, in how much sleep we need per night. He, Jones, et al. (2009) found a mother and daughter who shared a particular gene variant known to be associated with sleep. They both also required only about six hours of sleep per night, a couple of hours shorter than the average (see Parker-Pope, 2009). The fact that they both had this rare gene variant and also needed less sleep suggests that the gene may be a primary cause of their reduced need for sleep.

Study Questions for Section 2-7

1. What is the mean of the following set of IQ scores?
   95, 102, 99, 106, 93, 112, 105, 115, 80, 103,
   88, 98, 94, 110, 120, 97, 90, 101, 85, 107
2. What is the range of the set of IQ scores?
3. Did anyone in the group get the average IQ score for the group?
4. On average, young adults sleep an average of about eight hours per night. Given this finding, how much sleep should you get per night?
5. If you were trying to hire an accountant and had to decide between two equally qualified candidates, Tanya and Tony, whom should you hire given that males do better than females in math, on average?
6. On average, females have better verbal abilities than males. Who will get a lower grade in ENG 101: Peter or Patricia?
7. Research has shown that the following factors probably are causes of schizophrenia: genes, viral infections during fetal development, abnormal activity in the frontal lobes, damaged structures in the limbic system, birth complications, biological changes at puberty, abnormal activity in the temporal lobes, and stressful events that trigger the first psychotic episode. Amir has just developed symptoms of schizophrenia. What caused his schizophrenia?
8. What is the best way to figure out how much sleep you need per night?
9. Why are means of the amount of sleep people get per night not useful for determining how much sleep you need per night?
10. About how much sleep do adults get per night?
11. Should a person who needs only 4 hours of sleep per night be worried? Why or why not?

In the next section, you will learn about the characteristics of several stages of sleep. Keep in mind that what is being described there is accurate for the average young adult (people aged 20 to 40 years). You might find that you do not fit this average very well. Just as with height, it is to be expected that most people will deviate to varying extents from these averages (in psychology, a statistical deviation is simply a difference from some average). If you do deviate, this does not mean that you are abnormal in the sense that something is wrong with you. The very difficult issue of what constitutes abnormal behavior will be dealt with in other sections of this course.

Go to Quiz 2-7 questions

Go to Readings Section 2-8