In Section 2-6, you learned that two phenomena (events) are correlated when they change together. For example, major depression is correlated with gender (i.e., whether one is male or female): about two times as many females than males develop major depression at some point during their lives. Calvete and Cardeñoso (2005) stated that this gender difference appears by early adolescence: “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 results such as these, however, we need to be aware of two issues. First, finding a correlation between two events tells us nothing about what is causing the correlation. In the case of the gender-depression example, there are many possible causes that we would need to examine before we could explain why males are less likely than females to be diagnosed with depression. Second, finding a correlation between two events indicates only what is true on average in a large group of people. If we want to develop a prediction or explanation regarding a particular individual in that group, the correlation is unlikely to help us. For example, if we know that Ethan and Sophia each experienced a very stressful life event, the depression-gender correlation does not allow us to predict that Sophia will become depressed but Ethan will not.
In this section, we’ll look at what can and cannot be inferred from group averages. The issue of what can and cannot be concluded about causes based on the finding of a correlation will be examined in more depth in a future section.
The Mean of a Sample of Observations
The word statistics refers to the analysis of numerical data. We would use statistics, for example, if we wanted to know which letters in English words are most likely to follow the letter “e.” Our statistical analysis, in this case, would involve counting the number of times each of the 26 letters in the English alphabet follows an e in words (the most frequently occurring letters after an e are r, s, n, d). In order to perform a statistical analysis, researchers first select a sample of events to observe. A sample is a subset of events taken from a large group of those events. For example, if a researcher wants to know which letters are most likely to follow the letter e, she can’t look at all English words because there are too many. Instead, she would choose a subset of words taken from all English words containing at least one e. (We’ll look at the issue of how to select samples in a future section.)
An important type of statistical analysis in the behavioral sciences is the calculation of group averages. The most commonly used average in research is the mean, which is calculated by:
- summing all the individual observations (measurements) in a sample;
- dividing this sum by the total number of measurements.
For example, let’s say that we want to calculate the mean of the test scores of 10 students who just took a 100-question test. The scores are: 70, 52, 90, 96, 46, 36, 78, 88, 66, and 98. The sum of these ten scores is 720. Dividing 720 by the number of scores (10), we get 72. Thus, the mean test score is 72 out of 100, or 72% (a C). From this analysis, we can conclude that, on average, students in the class answered 72% of the test questions correctly. Another way of saying this is that students had a tendency to receive a test score of 72%. Based on the mean score alone, of course, we cannot infer that a particular student earned a 72% on the test. In fact, none of the students in the sample earned the mean score, and only three students fell within six points or less (66, 70, and 78).
Although the mean of a group usually does not allow us to say anything about individuals in that group, people who are untrained in statistics sometimes make such inferences. For example, here is a passage from an article providing information about different styles of nonverbal communication in men and women — information taken from research that looked at differences between the mean scores of men and women on measures of nonverbal communication:
In non-verbal behavior women will nod their heads to show that they are listening. Men leave the conversation thinking that a head nod means agreement and will be surprised to find out that the woman didn’t agree at all. When a woman is speaking to a man, he will not say anything and will use neutral body language to show that he is listening; a woman will interpret that as the man being bored or not understanding what she is saying. This can lead the woman to become very uncomfortable and repeat what she is saying or ask the man each time if he understands what she is saying. The man then interprets that as insecurity, … which then leads him to think she is not assertive or confident [enough] to be a leader. Women … use more direct eye contact in conversation to create relationship and connection while many men take that as a challenge to their power or position. Women will also approach a man from the front while men often approach from the side at an angle, which is how each of them tends to stand or sit when talking to others. Men interpret the face-to-face [interaction] as too personal or aggressive, and women will interpret [side-to-side talking as an indication that] he is not being upfront or even hiding something from her. (Lieberman, undated; some sentences were altered to correct the grammar)
The way in which the author of this passage summarized the research on gender differences leads to the unmistakeable conclusion that, when communicating, all women use one set of nonverbal behaviors and all men use a different set of nonverbal behaviors. But this is incorrect (and the author probably didn’t intend to do this). Finding a difference between the mean of a group of men and the mean of a group of women does not allow us to conclude anything about a particular man or woman. Some men, for example, nod their heads in conversations to show that they are listening (a “female behavior”); and some women think that these head nods mean agreement (a “male behavior”). The research results tell us that women are more likely to use one set of nonverbal behaviors when communicating and men are more likely to use a different set of nonverbal behaviors.
This point is easily understood by looking at differences in the average heights of males and females. 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 mean heights of Americans from 1960 to 2002 separated by age, race, ethnicity, and sex. Table 1 shows the average heights for non-hispanic white males and females between the ages of 20 and 39 years. The men are, on average, 5 inches taller than the women.
70″ (5′ 10″)
65″ (5′ 5″)
Table 1. Means, variances, and sample sizes in a group of non-Hispanic white Americans between the ages of 20 and 39 years
Based on your everyday observations, you know that some men are shorter than the female mean in Table 1, and that some women are taller than the male mean in Table 1. In other words, even though Table 1 shows that the mean height of men is five inches more than the mean height of women, it isn’t true that all men are taller than all women. These statistics therefore, tell us nothing about the height of a particular man or woman. Although we can predict that the next man we see probably will be closer to 70 inches than 65 inches, and that this man probably will be taller than the next woman we see, we won’t know for sure until we actually see them.
Figure 1 shows in graphical form the points made in the previous paragraph (the graph is based on fictional data that are consistent with what would be observed in real life). Each red bar indicates the number of females of a particular height — heights that range from 58 inches (4’8″) to 72 inches (6’0″). The height with the largest number of women (the tallest red bar) is also the mean for all women. The blue bars indicate the same for men. The graph shows what you already know from everyday experience: individual males and females vary around their respective means; and the heights of men and women overlap (i.e., some women are taller than many men, and some men are shorter than many women).
The Variance of a Sample of Observations
Measures of psychological characteristics, such as IQ scores measuring intelligence, almost always show individual differences: each individual differs not only from other individuals but also from the group average. Figure 1, for instance, shows that the heights of most males and most females differ from their group means. The overall “spread” of measurements is estimated by a statistic called the variance, which may be defined as the degree to which individuals differ from the group average. A simple measure of variance is the range of scores. It is calculated by subtracting the lowest score in a group from the highest score. For example, the ten test scores listed earlier in this section — 70, 52, 90, 96, 46, 36, 78, 88, 66, and 98 — show a large range: subtracting 36 (the lowest score) from 98 (the highest score) gives a range of 62. Tests that do a good job at distinguishing students who know the material well from students who don’t should have a large range (assuming, of course, that differences in knowledge of the course material actually exist in the group of students).
Because individuals within a group usually differ from each other, results involving group means should be described in one or more of the following ways:
- “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 ….”
The words tendency, on average, and more likely than, indicate that at least some individuals in the group differ from the group’s mean. Let’s look at one more gender-difference example just to make sure you understand this. You probably have heard about research showing that men, on average, desire to have more sexual partners than women do; and that men, on average, are more sexually promiscuous than women are. Many people conclude from these results that (a) virtually all men want more sexual partners than do women, and (b) virtually all men are more sexually active than are women. But these conclusions don’t follow from the results and, in fact, they aren’t true. When researchers looked at the scores of men and women on measures of sexual behavior and sexual attitudes, they found that the scores of a large percentage of men are equal to or below the mean scores for females; and that the scores of a large percentage of women are equal to or above the mean scores for males. Based on analyses of studies of sexual behaviors and sexual attitudes published between 1993 and 2007, Petersen and Hyde (2010) concluded that “small gender differences for the majority of sexual behaviors and attitudes suggest that men and women are more similar than they are different in terms of sexuality” (p. 36). Petersen and Hyde warned that misinterpretations of the results of this research reinforces stereotypical thinking about men and women, and encourages people to judge the sexual behaviors of men and women by different standards.
Group Means and the Study of Causes
Students often take psychology because they are curious about the causes of mental events and behavior, such as the cause of a family member’s mental disorder. Let’s say, for example, that an instructor is asked by a student why the student’s mother developed bipolar disorder. The instructor’s response should include the words, “I don’t know”; but this should be only a part of her response. She then should describe what research has shown to be possible or likely causes, such as abnormal genes, stressful life events, disturbances of biochemical activity in the brain (see this article for more information). With these findings in mind, the instructor might speculate that the student’s mother inherited genes that, in conjunction with (a) highly stressful life events and (b) unknown environmental factors — such as, perhaps, viral infections — led to the development bipolar disorder. The instructor also should tell the student that certain medications can reduce or eliminate the symptoms of bipolar disorder and that his mother probably will need to take the medications for at least several years. The instructor’s answer is based on the results of research consisting of differences between group means, as well as correlations among various factors (correlations, as you learned, reflect the average degree of association between two or more things). Because averages can’t be used to infer with certainty what is true for an individual, the instructor can’t (or shouldn’t) give an answer that is anything but tentative.
If averages tell us nothing definite about individuals, then why do researchers spend so much time calculating them? Let’s try to answer this question by looking at a couple of examples. The difference in the average heights of men and women must be due to causal factors important for determining height; and these factors must differ (on average) between men and women. Researchers would begin to identify possible causes by relying on what research has shown to be the major causes of physical development factors such as growth hormone, genes, nutrition, etc. Thus, knowing that there is a gender difference in height is important because it tells researchers that factors that determine height differ, on average, between men and women; and by searching for these factors, they will eventually develop a better understanding of the causes of height.
Another example is the gender difference involving rates of depression. The discovery that women are twice as likely as men to be diagnosed with depression led researchers to begin to search for factors that differ, on average, between men and women — factors that other research has suggested might be important for causing depression (e.g., hormone levels, sex chromosomes, physical activity, cultural expectations, etc.). If researchers find gender differences in some of these possible causal factors, this should lead to better theories of depression, which eventually should help in developing better treatments.
These two examples show that researchers are interested in group means because a difference between the means of groups indicates the influence of causal factors that further research may be able to identify. Thus, even though averages tell us little about what is going on with individuals, they can help us to identify causal factors.
How much sleep do you need?
Let’s review what you have learned in this section by trying to answer the following question: how much sleep do you need per night? On average, young adults get about 7-8 hours of sleep per night (e.g., Steptoe, Peacey, & Wardle, 2006). 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.
The answer confuses the average amount of sleep that adults get per night with the average amount of sleep that:
- adults need per night.
- you get per night.
- you need per night.
In other words, the average number of hours that adults actually sleep per night is not necessarily the same as the average number of hours that adults actually need to sleep per night. Furthermore, the average number of hours that adults actually need to sleep per night tells us little about the number of hours that you need to sleep per night. Thus, instead of referring to statistics and graphs, Moorcroft (1993) answered the question of how much sleep each person needs 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)
Similar advice was given more recently on the web site of the Sleep Disorders Center at the University of Maryland’s Medical Center:
While the average normal amount of sleep is around 7.5 hours per night, there are some people who do just fine on 5 hours per night, and some who require as much as 9 hours per night. The key is to find the right amount for you. The best way to tell is by seeing how you function during the day. For example, if after 6 hours of sleep you feel refreshed in the morning and awake during your daylight hours, then you don’t need more than that.
In other words, go to sleep when you feel tired and see what time you wake up in the morning. If you want to see how much sleep you need, make sure that you haven’t had too much caffeine or other stimulants during the day; don’t use alcohol or any sedating medications before bed; and don’t use an alarm to wake up in the morning. Go to sleep when you feel tired in the evening and, the next morning, get up when you feel rested. Repeat this procedure for several nights (just to make sure that you sleep about the same number of hours each night). If you don’t feel overly tired during the day and you seem to function well, then this is the amount of sleep you need.
In a study that included over 110, 000 adult Americans of all ages and racial/ethnic backgrounds, Krueger and Friedman (2009) found that 63% reported that they sleep 7 or 8 hours per night, 28% reported that they sleep 6 or fewer hours per night, and 9% reported that they sleep 9 or more hours per night. These results suggest that most adults probably need about 7 or 8 hours of sleep per night. Adults who sleep 6 or fewer hours per night either need less sleep than the average or are unable to get as much sleep as they need due to a variety of causes (e.g., illness, having young children, alcohol/drug abuse, etc.). Healthy adults who sleep 9 or more hours probably need more sleep than the average; but others are sleeping more than they would normally due to a variety of causes (e.g., pregnancy, physical or mental illness, etc.).
Many factors affect the amount of sleep you need per night, some of which are obvious, such as engaging in a lot of physical or mental activity during the day. Other factors may not be as obvious, such as the effects of genes you inherited from your parents. For example, He, Jones, et al. (2009; see Parker-Pope, 2009, for a summary) observed the sleep duration of a mother and daughter who shared a particular gene variant known to be linked to sleep. Both women needed only about six hours of sleep per night, about two hours shorter than average.
Figure 2 presents the percentage of people reporting sleep of different lengths, ranging from about 3 to about 11 hours per night.
In the next section, you will learn about the stages of sleep. Keep in mind that the discussion is based on studies reporting averages for young adult (people aged 20 to 40 years). You might find that your sleep is different. Just as with height, you should expect that most people will deviate to varying extents from these averages (in psychology, a statistical deviation is simply a difference from an average). If your sleep deviates from the average, 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 and mental disorders will be dealt with in other sections.
Study Questions for Section 2-7
- 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
- What is the range of the set of IQ scores in Question #1?
- Does anyone in the group have an IQ score equal to the mean score?
- On average, young adults sleep about eight hours per night. Given this finding, how much sleep should you get per night?
- There is a gender difference in scores on tests of math ability: males, on average, score higher than do females. If you were in charge of hiring an accountant and had two equally qualified candidates, Tanya and Tony, who should you hire given the gender difference in math ability?
- There is a gender difference in scores on tests of verbal ability: females, on average, score higher than do males. Given this gender difference, will Peter or Patricia get a lower grade in Critical Reading 101?
- Researchers have found that the following factors probably are causes of schizophrenia: genes, viral infections during fetal development, abnormal brain activity, damage to certain structures in the brain, birth complications, biological changes at puberty, and stressful life events. Amir has just developed symptoms of schizophrenia. What caused these symptoms to appear?
- What is the best way to figure out how much sleep you need per night?
- If you want to know how much sleep you need per night, why is it not useful to look at how much sleep, on average, people you age need per night?
- How much sleep, on average, do adults get per night?
- Should a person who needs only 4 hours of sleep per night be worried? Why or why not?
Practice Quiz for Section 2-7
Click HERE to find the details about the articles, books, etc., referred to in this section.