5.8: Social Categorization and Stereotyping
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Learning Objectives
- Describe the fundamental process of social categorization and its influence on thoughts, feelings, and behavior.
- Define stereotypes and describe the ways that stereotypes are measured.
- Review the ways that stereotypes influence our behavior.
Thinking about others in terms of their group memberships is known as social categorization— the natural cognitive process by which we place individuals into social groups. Social categorization occurs when we think of someone as a man (versus a woman), an old person (versus a young person), a Black person (versus an Asian or White person), and so on (Allport, 1954/1979). Just as we categorize objects into different types, so we categorize people according to their social group memberships. Once we do so, we begin to respond to those people more as members of a social group than as individuals.
Imagine for a moment that two college students, John and Sarah, are talking at a table in the student union at your college or university. At this point, we would probably not consider them to be acting as group members, but rather as two individuals. John is expressing his opinions, and Sarah is expressing hers. Imagine, however, that as the conversation continues, Sarah brings up an assignment that she is completing for her women’s studies class. It turns out that John does not think there should be a women’s studies program at the college, and he tells Sarah so. He argues that if there is a women’s studies program, then there should be a men’s studies program too. Furthermore, he argues that women are getting too many breaks in job hiring and that qualified men are the targets of discrimination. Sarah feels quite the contrary—arguing that women have been the targets of sexism for many, many years and even now do not have the same access to high-paying jobs that men do.
You can see that an interaction that began at individual level, as two individuals conversing, has now turned to the group level, in which John has begun to consider himself as a man, and Sarah has begun to consider herself as a woman. In short, Sarah is now arguing her points not so much for herself as she is as a representative of one of her ingroups—namely, women—and John is acting as a representative of one of his ingroups—namely, men. Sarah feels that her positions are correct, and she believes they are true not only for her but for women in general. And the same is true of John. You can see that these social categorizations may create some potential for misperception, and perhaps even hostility. And John and Sarah may even change their opinions about each other, forgetting that they really like each other as individuals, because they are now responding more as group members with opposing views.
Imagine now that while John and Sarah are still talking, some students from another college, each wearing the hats and jackets of that school, show up in the student union. The presence of these outsiders might change the direction of social categorization entirely, leading both John and Sarah to think of themselves as students at their own college. And this social categorization might lead them to become more aware of the positive characteristics of their college (the excellent basketball team, lovely campus, and intelligent students) in comparison with the characteristics of the other school. Now, rather than perceiving themselves as members of two different groups (men versus women), John and Sarah might suddenly perceive themselves as members of the same social category (students at their college).
Perhaps this example will help you see the flexibility of social categorization. We sometimes think of our relationships with others at the individual level and sometimes at the group level. And which groups we use in social categorization can change over time and in different situations. I think you would agree that you are more likely to categorize yourself as a member of your college or university when your basketball or football team has just won a really important game, or at your commencement day ceremony, than you would on a normal evening out with your family. In these cases, your membership as a university student is simply more salient and important than it is every day, and you are more likely to categorize yourself accordingly.
Spontaneous Social Categorization
Social categorization occurs spontaneously, without much thought on our part (Crisp & Hewstone, 2007). Shelley Taylor and her colleagues (Taylor, Fiske, Etcoff, & Ruderman, 1978) showed their research participants a slide and tape presentation of three male and three female college students who had supposedly participated in a discussion group. During the presentation, each member of the discussion group made a suggestion about how to advertise a college play. The statements were controlled so that across all the research participants, the statements made by the men and the women were of equal length and quality. Furthermore, one half of the participants were told that when the presentation was over, they would be asked to remember which person had made which suggestion, whereas the other half of the participants were told merely to observe the interaction without attending to anything in particular.
After they had viewed all the statements made by the individuals in the discussion group, the research participants were given a memory test (this was entirely unexpected for the participants who had not been given memory instructions). The participants were shown the list of all the statements that had been made, along with the pictures of each of the discussion group members, and were asked to indicate who had made each of the statements. The research participants were not very good at this task, and yet when they made mistakes, these errors were very systematic.
As you can see in Table 12.1, the mistakes were such that the statements that had actually been made by a man were more frequently wrongly attributed to another man in the group than to another woman, and the statements actually made by a woman were more frequently attributed to other women in the group than to a man. The participants evidently categorized the speakers by their gender, leading them to make more within-gender than across-gender confusions.
Interestingly, and suggesting that categorization is occurring all the time, the instructions that the participants had been given made absolutely no difference. There was just as much categorization for those who were not given any instructions as for those who were told to remember who said what. Other research using this technique has found that we spontaneously categorize each other on the basis of many other group memberships, including race, academic status (student versus teacher), social roles, and other social categories (Fiske, Haslam, & Fiske, 1991; Stangor, Lynch, Duan, & Glass, 1992).
| Instructions | Within race errors | Between race errors |
|---|---|---|
| Memory | 5.78 | 4.29 |
| No memory | 6.57 | 4.36 |
| Taylor, Fiske, Etcoff, and Ruderman (1978) demonstrated that people categorized others spontaneously. Even without any instructions to categorize, people nevertheless confused others by their sex. |
The conclusion is simple, if perhaps obvious: Social categorization is occurring all around us all the time. Indeed, social categorization occurs so quickly that people may have difficulty not thinking about others in terms of their group memberships (see Figure 12.4).
The Benefits of Social Categorization
The tendency to categorize others is normally quite useful. In some cases, we categorize because doing so provides us with information about the characteristics of people who belong to certain social groups (Lee, Jussim, & McCauley, 1995). If you found yourself lost in a city, you might look for a police officer or a taxi driver to help you find your way. In this case, social categorization would probably be useful because a police officer or a taxi driver might be particularly likely to know the layout of the city streets. Of course, using social categories will only be informative to the extent that the stereotypes held by the individual about that category are accurate. If police officers were actually not that knowledgeable about the city layout, then using this categorization would not be informative.
It has been argued that there is a kernel of truth in most stereotypes, and this seems to be the case. There is a correlation between how group members perceive the stereotypes of their own groups and how people from other groups perceive those same stereotypes (Judd & Park, 1993; Swim, 1994). This truth may come in part from the roles that individuals play in society. For instance, the stereotypes (which are held by many people) that women are “nurturing” and that men are “dominant” may occur in part because, on average, men and women find themselves in different social roles within a culture (Eagly & Steffen, 1984). In most cultures, men are more likely to be in higher-status occupations, such as doctors and lawyers, whereas women are more likely to play the role of homemakers and child-care workers. In this sense, the stereotypes are at least partly true for many of the members of the social category, in terms of their actual behaviors. Because men are more likely to be leaders than are women, they may well be, on average, more dominant; and because women are more likely to take care of children, they may, on average, act in a more nurturing way than do men.
On the other hand, we sometimes categorize others not because it seems to provide more information about them but because we may not have the time (or the motivation) to do anything more thorough. Using our stereotypes to size up another person might simply make our life easier (Macrae, Bodenhausen, Milne, & Jetten, 1994). According to this approach, thinking about other people in terms of their social category memberships is a functional way of dealing with the world—things are complicated, and we reduce complexity by relying on our stereotypes.
The Negative Outcomes of Social Categorization
Although thinking about others in terms of their social category memberships has some potential benefits for the person who does the categorizing, categorizing others, rather than treating them as unique individuals with their own unique characteristics, has a wide variety of negative, and often very unfair, outcomes for those who are categorized.
One problem is that social categorization distorts our perceptions such that we tend to exaggerate the differences between people from different social groups while at the same time perceiving members of groups (and particularly outgroups) as more similar to each other than they actually are. This overgeneralization makes it more likely that we will think about and treat all members of a group the same way. Tajfel and Wilkes (1963) performed a simple experiment that provided a picture of the potential outcomes of categorization. As you can see in Figure 12.5, the experiment involved having research participants judge the length of six lines. In one of the experimental conditions, participants simply saw six lines, whereas in the other condition, the lines were systematically categorized into two groups—one comprising the three shorter lines and one comprising the three longer lines.
Lines C and D were seen as the same length in the noncategorized condition, but line C was perceived as longer than line D when the lines were categorized into two groups. From Tajfel (1970).
Tajfel found that the lines were perceived differently when they were categorized, such that the differences between the groups and the similarities within the groups were emphasized. Specifically, he found that although lines C and D (which are actually the same length) were perceived as equal in length when the lines were not categorized, line D was perceived as being significantly longer than line C in the condition in which the lines were categorized. In this case, categorization into two groups—the “short lines group” and the “long lines group”—produced a perceptual bias such that the two groups of lines were seen as more different than they really were.
Similar effects occur when we categorize other people. We tend to see people who belong to the same social group as more similar than they actually are, and we tend to judge people from different social groups as more different than they actually are. The tendency to see members of social groups as similar to each other is particularly strong for members of outgroups, resulting in outgroup homogeneity— the tendency to view members of outgroups as more similar to each other than we see members of ingroups (Linville, Salovey, & Fischer, 1986; Ostrom & Sedikides, 1992; Meissner & Brigham, 2001). I’m sure you’ve had this experience yourself, when you found yourself thinking or saying, “Oh, them, they’re all the same!”
Patricia Linville and Edward Jones (1980) gave research participants a list of trait terms and asked them to think about either members of their own group (e.g., Blacks) or members of another group (e.g., Whites) and to place the trait terms into piles that represented different types of people in the group. The results of these studies, as well as other studies like them, were clear: People perceive outgroups as more homogeneous than the ingroup. Just as White people used fewer piles of traits to describe Blacks than Whites, young people used fewer piles of traits to describe elderly people than they did young people, and students used fewer piles for members of other universities than they did for members of their own university.
Outgroup homogeneity occurs in part because we don’t have as much contact with outgroup members as we do with ingroup members, and the quality of interaction with outgroup members is often more superficial. This prevents us from really learning about the outgroup members as individuals, and as a result, we tend to be unaware of the differences among the group members. In addition to learning less about them because we see and interact with them less, we routinely categorize outgroup members, thus making them appear more cognitively similar (Haslam, Oakes, & Turner, 1996).
Once we begin to see the members of outgroups as more similar to each other than they actually are, it then becomes very easy to apply our stereotypes to the members of the groups without having to consider whether the characteristic is actually true of the particular individual. If men think that women are all alike, then they may also think that they all have the same characteristics—they’re all “emotional” and “weak.” And women may have similarly simplified beliefs about men (they’re “insensitive,” “unwilling to commit,” etc.). The outcome is that the stereotypes become linked to the group itself in a set of mental representations (Figure 12.6). The stereotypes are “pictures in our heads” of the social groups (Lippman, 1922). These beliefs just seem right and natural, even though they are frequently distorted overgeneralizations (Hirschfeld, 1996; Yzerbyt, Schadron, Leyens, & Rocher, 1994).
Our stereotypes and prejudices are learned through many different processes. This multiplicity of causes is unfortunate because it makes stereotypes and prejudices even more likely to form and harder to change. For one, we learn our stereotypes in part through our communications with parents and peers (Aboud & Doyle, 1996) and from the behaviors we see portrayed in the media (Brown, 1995). Even 5-year-old children have learned cultural norms about the appropriate activities and behaviors for boys and girls and also have developed stereotypes about age, race, and physical attractiveness (Bigler & Liben, 2006). And there is often good agreement about the stereotypes of social categories among the individuals within a given culture. In one study assessing stereotypes, Stephanie Madon and her colleagues (Madon et al., 2001) presented U.S. college students with a list of 84 trait terms and asked them to indicate for which groups each trait seemed appropriate (Figure 12.6). The participants tended to agree about what traits were true of which groups, and this was true even for groups of which the respondents were likely to never have met a single member (Arabs and Russians). Even today, there is good agreement about the stereotypes of members of many social groups, including men and women and a variety of ethnic groups.
| Americans | % | Blacks | % | Italians | % |
|---|---|---|---|---|---|
| Materialistic | 53.6 | Musical | 27.6 | Loyal to family ties | 62.7 |
| Lazy | 30.4 | Pleasure loving | 26 | Tradition loving | 47.5 |
| Individualistic | 28.6 | Loud | 20.7 | Passionate | 39 |
| Pleasure loving | 28 | Aggressive | 15.5 | Religious | 37.3 |
| Industrious | 23.2 | Artistic | 13.8 | Quick tempered | 35.6 |
| Germans | % | Jews | % | Chinese | % |
| Intelligent | 45.8 | Very religious | 52.5 | Intelligent | 60.3 |
| Industrious | 37.3 | Intelligent | 49.2 | Loyal to family ties | 41.4 |
| Nationalistic | 30.5 | Tradition loving | 32.2 | Reserved | 36.2 |
| Scientifically minded | 27.1 | Shrewd | 30.5 | Industrious | 32.8 |
| Methodical | 20.3 | Loyal to family ties | 28.8 | Tradition loving | 31 |
Once they become established, stereotypes (like any other cognitive representation) tend to persevere. We begin to respond to members of stereotyped categories as if we already knew what they were like. Yaacov Trope and Eric Thompson (1997) found that individuals addressed fewer questions to members of categories about which they had strong stereotypes (as if they already knew what these people were like) and that the questions they did ask were likely to confirm the stereotypes they already had.
In other cases, stereotypes are maintained because information that confirms our stereotypes is better remembered than information that disconfirms them. When we see members of social groups perform behaviors, we tend to better remember information that confirms our stereotypes than we remember information that disconfirms our stereotypes (Fyock & Stangor, 1994). If we believe that women are bad drivers and we see a woman driving poorly, then we tend to remember it, but when we see a woman who drives particularly well, we tend to forget it. This is of course another example of the general principle of assimilation—we tend to perceive the world in ways that make it fit our existing beliefs more easily than we change our beliefs to fit the reality around us.
And stereotypes become difficult to change because they are so important to us—they become an integral and important part of our everyday lives in our culture. Stereotypes are frequently expressed on TV, in movies, and in chat rooms and blogs, and we learn a lot of our beliefs from these sources. Our friends also tend to hold beliefs similar to ours, and we talk about these beliefs when we get together with them (Schaller & Conway, 1999). In short, stereotypes and prejudice are powerful largely because they are important social norms that are part of our culture (Guimond, 2000).
Because they are so highly cognitively accessible, and because they seem so “right,” our stereotypes easily influence our judgments of and responses to those we have categorized. The social psychologist John Bargh once described stereotypes as “cognitive monsters” because their activation was so powerful and because the activated beliefs had such insidious influences on social judgment (Bargh, 1999). Making things even more difficult, stereotypes are strongest for the people who are in most need of change—the people who are most prejudiced (Lepore & Brown, 1997).
Because stereotypes and prejudice often operate out of our awareness, and also because people are frequently unwilling to admit that they hold them, social psychologists have developed methods for assessing them indirectly. In the next section we will consider two of these approaches—the bogus pipeline procedure and the Implicit Association Test (IAT)
Research Focus
Measuring Stereotypes Indirectly
One difficulty in measuring stereotypes and prejudice is that people may not tell the truth about their beliefs. Most people do not want to admit—either to themselves or to others—that they hold stereotypes or that they are prejudiced toward some social groups. To get around this problem, social psychologists make use of a number of techniques that help them measure these beliefs more subtly and indirectly.
One indirect approach to assessing prejudice is called the bogus pipeline procedure (Jones & Sigall, 1971). In this procedure, the experimenter first convinces the participants that he or he has access to their “true” beliefs, for instance, by getting access to a questionnaire that they completed at a prior experimental session. Once the participants are convinced that the researcher is able to assess their “true” attitudes, it is expected that they will be more honest in answering the rest of the questions they are asked because they want to be sure that the researcher does not catch them lying. The bogus pipeline procedure suggests that people may frequently mask their negative beliefs in public—people express more prejudice when they are in the bogus pipeline than they do when they are asked the same questions more directly.
Other indirect measures of prejudice are also frequently used in social psychological research, for instance—assessing nonverbal behaviors such as speech errors or physical closeness. One common measure involves asking participants to take a seat on a chair near a person from a different racial or ethnic group and measuring how far away the person sits (Sechrist & Stangor, 2001; Word, Zanna, & Cooper, 1974). People who sit farther away are assumed to be more prejudiced toward the members of the group.
Because our stereotypes are activated spontaneously when we think about members of different social groups, it is possible to use reaction-time measures to assess this activation and thus to learn about people’s stereotypes and prejudices. In these procedures, participants are asked to make a series of judgments about pictures or descriptions of social groups and then to answer questions as quickly as they can, but without making mistakes. The speed of these responses is used to determine an individual’s stereotypes or prejudice.
The most popular reaction-time implicit measure of prejudice—the Implicit Association Test (IAT) —is frequently used to assess stereotypes and prejudice (Nosek, Greenwald, & Banaji, 2007). In the IAT, participants are asked to classify stimuli that they view on a computer screen into one of two categories by pressing one of two computer keys, one with their left hand and one with their right hand. Furthermore, the categories are arranged such that the responses to be answered with the left and right buttons either “fit with” (match) the stereotype or do not “fit with” (mismatch) the stereotype. For instance, in one version of the IAT, participants are shown pictures of men and women and also shown words related to gender stereotypes (e.g., strong, leader, or powerful for men and nurturing, emotional, or weak for women). Then the participants categorize the photos (“Is this picture a picture of a man or a woman?”) and answer questions about the stereotypes (“Is this the word strong ?) by pressing either the Yes button or the No button using either their left hand or their right hand.
When the responses are arranged on the screen in a “matching” way, such that the male category and the “strong” category are on the same side of the screen (e.g., on the right side), participants can do the task very quickly and they make few mistakes. It’s just easier, because the stereotypes are matched or associated with the pictures in a way that makes sense. But when the images are arranged such that the women and the strong categories are on the same side, whereas the men and the weak categories are on the other side, most participants make more errors and respond more slowly. The basic assumption is that if two concepts are associated or linked, they will be responded to more quickly if they are classified using the same, rather than different, keys.
Implicit association procedures such as the IAT show that even participants who claim that they are not prejudiced do seem to hold cultural stereotypes about social groups. Even Black people themselves respond more quickly to positive words that are associated with White rather than Black faces on the IAT, suggesting that they have subtle racial prejudice toward Blacks.
Because they hold these beliefs, it is possible—although not guaranteed—that they may use them when responding to other people, creating a subtle and unconscious type of discrimination. Although the meaning of the IAT has been debated (Tetlock & Mitchell, 2008), research using implicit measures does suggest that—whether we know it or not, and even though we may try to control them when we can—our stereotypes and prejudices are easily activated when we see members of different social categories (Barden, Maddux, Petty, & Brewer, 2004).
Do you hold implicit prejudices? Try the IAT yourself, here: https://implicit.harvard.edu/implicit
Although in some cases the stereotypes that are used to make judgments might actually be true of the individual being judged, in many other cases they are not. Stereotyping is problematic when the stereotypes we hold about a social group are inaccurate overall, and particularly when they do not apply to the individual who is being judged (Stangor, 1995). Stereotyping others is simply unfair. Even if many women are more emotional than are most men, not all are, and it is not right to judge any one woman as if she is.
In the end, stereotypes become self-fulfilling prophecies, such that our expectations about the group members make the stereotypes come true (Snyder, Tanke, & Berscheid, 1977; Word, Zanna, & Cooper, 1974). Once we believe that men make better leaders than women, we tend to behave toward men in ways that makes it easier for them to lead. And we behave toward women in ways that makes it more difficult for them to lead. The result? Men find it easier to excel in leadership positions, whereas women have to work hard to overcome the false beliefs about their lack of leadership abilities (Phelan & Rudman, 2010). And self-fulfilling prophecies are ubiquitous—even teachers’ expectations about their students’ academic abilities can influence the students’ school performance (Jussim, Robustelli, & Cain, 2009).
Of course, you may think that you personally do not behave in these ways, and you may not. But research has found that stereotypes are often used out of our awareness, which makes it very difficult for us to correct for them. Even when we think we are being completely fair, we may nevertheless be using our stereotypes to condone discrimination (Chen & Bargh, 1999). And when we are distracted or under time pressure, these tendencies become even more powerful (Stangor & Duan, 1991).
Furthermore, attempting to prevent our stereotype from coloring our reactions to others takes effort. We experience more negative affect (particularly anxiety) when we are with members of other groups than we do when we are with people from our own groups, and we need to use more cognitive resources to control our behavior because of our anxiety about revealing our stereotypes or prejudices (Butz & Plant, 2006; Richeson & Shelton, 2003). When we know that we need to control our expectations so that we do not unintentionally stereotype the other person, we may try to do so—but doing so takes effort and may frequently fail (Macrae, Bodenhausen, Milne, & Jetten, 1994).
Social Psychology in the Public Interest
Stereotype Threat
Our stereotypes influence not only our judgments of others but also our beliefs about ourselves, and even our own performance on important tasks. In some cases, these beliefs may be positive, and they have the effect of making us feel more confident and thus better able to perform tasks. Because Asian students are aware of the stereotype that “Asians are good at math,” reminding them of this fact before they take a difficult math test can improve their performance on the test (Walton & Cohen, 2003). On the other hand, sometimes these beliefs are negative, and they create negative self-fulfilling prophecies such that we perform more poorly just because of our knowledge about the stereotypes.
One of the long-standing puzzles in the area of academic performance concerns why Black students perform more poorly on standardized tests, receive lower grades, and are less likely to remain in school in comparison with White students, even when other factors such as family income, parents’ education, and other relevant variables are controlled. Claude Steele and Joshua Aronson (1995) tested the hypothesis that these differences might be due to the activation of negative stereotypes. Because Black students are aware of the (inaccurate) stereotype that “Blacks are intellectually inferior to Whites,” this stereotype might create a negative expectation, which might interfere with their performance on intellectual tests through fear of confirming that stereotype.
In support of this hypothesis, Steele and Aronson’s research revealed that Black college students performed worse (in comparison with their prior test scores) on math questions taken from the Graduate Record Examination (GRE) when the test was described to them as being “diagnostic of their mathematical ability” (and thus when the stereotype was relevant) but that their performance was not influenced when the same questions were framed as “an exercise in problem solving.” And in another study, Steele and Aronson found that when Black students were asked to indicate their race before they took a math test (again activating the stereotype), they performed more poorly than they had on prior exams, whereas the scores of White students were not affected by first indicating their race.
Steele and Aronson argued that thinking about negative stereotypes that are relevant to a task that one is performing creates stereotype threat— performance decrements that are caused by the knowledge of cultural stereotypes . That is, they argued that the negative impact of race on standardized tests may be caused, at least in part, by the performance situation itself. Because the threat is “in the air,” Black students may be negatively influenced by it.
Research has found that the experience of stereotype threat can help explain a wide variety of performance decrements among those who are targeted by negative stereotypes. For instance, when a math task is described as diagnostic of intelligence, Latinos and particularly Latinas perform more poorly than do Whites (Gonzales, Blanton, & Williams, 2002). Similarly, when stereotypes are activated, children with low socioeconomic status perform more poorly in math than do those with high socioeconomic status, and psychology students perform more poorly than do natural science students (Brown, Croizet, Bohner, Fournet, & Payne, 2003). Even groups who typically enjoy advantaged social status can be made to experience stereotype threat. White men performed more poorly on a math test when they were told that their performance would be compared with that of Asian men (Aronson, Lustina, Good, Keough, & Steele, 1999), and Whites performed more poorly than Blacks on a sport-related task when it was described to them as measuring their natural athletic ability (Stone, 2002).
 In these situations, there is a discrepancy between our positive concept of our skills and abilities and the negative stereotypes suggesting poor performance. When our stereotypes lead us to be believe that we are likely to perform poorly on a task, we experience a feeling of unease and status threat.
Research has found that stereotype threat is caused by both cognitive and affective factors. On the cognitive side, individuals who are experiencing stereotype threat show an impairment in cognitive processing that is caused by increased vigilance toward the environment and attempts to suppress their stereotypical thoughts. On the affective side, stereotype threat creates stress as well as a variety of affective responses including anxiety (Schmader, Johns, & Forbes, 2008).
Stereotype threat is not, however, absolute—we can get past it if we try. What is important is to reduce the self-concern that is engaged when we consider the relevant negative stereotypes. Manipulations that affirm positive characteristics about oneself or one’s group are successful at reducing stereotype threat (Alter, Aronson, Darley, Rodriguez, & Ruble, 2010; Greenberg et al., 2003; McIntyre, Paulson, & Lord, 2003). In fact, just knowing that stereotype threat exists and may influence performance can help alleviate its negative impact (Johns, Schmader, & Martens, 2005)
Key Takeaways
- Beliefs about the characteristics of the groups and the members of those groups are known as stereotypes.
- Prejudice refers to an unjustifiable negative attitude toward an outgroup.
- Stereotypes and prejudice may create discrimination.
- Stereotyping and prejudice begin from social categorization—the natural cognitive process by which we place individuals into social groups.
- Social categorization influences our perceptions of groups—for instance, the perception of outgroup homogeneity.
- Once our stereotypes and prejudices become established, they are difficult to change and may lead to self-fulfilling prophecies, such that our expectations about the group members make the stereotypes come true.
- Stereotypes may influence our performance on important tasks through stereotype threat.
Exercises and Critical Thinking
- Look again at the pictures in Figure 12.2, and consider your thoughts and feelings about each person. What are your stereotypes and prejudices about them? Do you think your stereotypes are accurate?
- On which (if any) social categories do you categorize others? Why do you (or don’t you) categorize? Is your behavior fair or unfair to the people you are categorizing?
- Think of a task that one of the social groups to which you belong is considered to be particularly good (or poor) at. Do you think the cultural stereotypes about your group have ever influenced your performance on a task?
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6.2: Catalase
Cells produce hydrogen peroxide (H 2 O 2 ) as a toxic by-product of normal cellular reactions. The enzyme catalase quickly breaks down hydrogen peroxide into water and oxygen. In other words, catalase protects cells from the toxic effects of hydrogen peroxide. All aerobic cells produce catalase. One molecule of catalase enzyme may work on 40 million molecules of hydrogen peroxide per second!
If you have used hydrogen peroxide to clean a cut, you have seen catalase in action. When hydrogen peroxide comes in contact with the cut, it reacts with the catalase enzyme in the damaged cells to produce oxygen foam. When the catalase reaction is conducted in a test-tube, the oxygen gas bubbles. The height of the foam is an indication of the amount of catalase activity present.
Investigation of catalase activity in plant and animal tissues
Materials
- 5 test-tubes
- wax pencil
- millimeter ruler
- 4 blended tissue extracts (may include apple, potato, onion, Baker's yeast, beef steak or/and beef liver)
- distilled water
- hydrogen peroxide (H2O2) solution (3%)
Hypothesis
State a hypothesis concerning the relative abilities of the various tissue types to catalyze the conversion of hydrogen peroxide to water and oxygen.
Procedure
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Label tubes 1 – 5. Use a wax pencil to draw a line 1 cm from the bottom of
each tube. Draw an additional line at the 2 cm mark - Add the appropriate extract to each tube to the level of the 1 cm line.
- Add H2O2 to the 2 cm line in each tube. Do not contaminate the dropper. If extract comes into contact with the dropper, discard the dropper.
- Measure the height of the foam (in millimeters) and record the data.
- Wash the test-tubes. Rinse well with water and drain.
Results
Prepare a bar graph of catalase activity. Include labels.
Analysis
- Did the results of the experiment support the hypothesis? Explain.
- Write the reaction. Use molecular formulas. Identify substrate, enzyme and products.
- What type of gas is released in the bubbles?
- What happens to catalase itself in the reaction – is it used up? destroyed? unchanged?
- The experimental variable (independent variable) is the aspect that varies between the experimental groups. What is the experimental variable in this experiment?
- Controlled variables are those conditions that are kept constant between experimental groups. The controlled variables insure that only one experimental variable is tested per experiment. What are 3 controlled variables in this experiment?
- The dependent variable changes in response to the experimental variable. The dependent variable is what is physically measured to assess the experimental outcome. What is the dependent variable in this experiment?
The effect of temperature on the activity of catalase
Materials
- 3 clean test-tubes
- hydrogen peroxide solution
- wax pencil
- extract of liver or yeast
- millimeter ruler
- crushed ice
- water bath at 70 o C
- thermometer
Hypothesis
State a hypothesis concerning the relative action of catalase at various temperatures. Refer to section 6.1
Procedure
- Label tubes 1- 3. Use a wax pencil to draw a line 1cm from the bottom of each tube. Draw another line at the 2 cm mark.
- Add extract to the 1 cm line. Note that extract type is a controlled variable in this experiment. Do not add hydrogen peroxide until AFTER step 3.
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Incubate the tubes at the following temperatures for 5 minutes
Tube 1 on ice (record the temperature of the ice)
Tube 2 at room temperature (record the temperature)
Tube 3 at 70 o C - After 5 minutes, return the tubes to the lab bench. Add H 2 O 2 (hydrogen peroxide) to the 2 cm line. Use ice-cold H 2 O 2 for the tube on ice.
- Measure the height of the foam on each tube. Record.
- Wash the test-tubes with water and drain to remove the excess water
Results
Prepare a line graph of catalase activity as influenced by temperature.
Analysis
- Explain the results shown in the line graph. Did the results support the hypothesis?
- Identify the experimental variable in the experiment.
- List 2 controlled variables in the experiment.
- What is the dependent variable in the experiment?


14.2: Meteors
By the end of this section, you will be able to:
- Explain what a meteor is and why it is visible in the night sky
- Describe the origins of meteor showers
As we saw in Comets and Asteroids: Debris of the Solar System, the ices in comets evaporate when they get close to the Sun, together spraying millions of tons of rock and dust into the inner solar system. There is also dust from asteroids that have collided and broken up. Earth is surrounded by this material. As each of the larger dust or rock particles enters Earth’s atmosphere, it creates a brief fiery trail; this is often called a shooting star , but it is properly known as a meteor .
Observing Meteors
Meteors are created as tiny solid particles that enter Earth’s atmosphere from interplanetary space. Since the particles move at speeds of many kilometers per second, friction with the air vaporizes them at altitudes between 80 and 130 kilometers. The resulting flashes of light fade out within a few seconds. These “shooting stars” got their name because at night their luminous vapors look like stars moving rapidly across the sky. To be visible, a meteor must be within about 200 kilometers of the observer. On a typical dark, moonless night, an alert observer can see half a dozen meteors per hour. These sporadic meteors —those not associated with a meteor shower (explained in the next section)—are random occurrences. Over the entire Earth, the total number of meteors bright enough to be visible totals about 25 million per day.
The typical meteor is produced by a particle with a mass of less than 1 gram—no larger than a pea. How can we see such a small particle? The light you see comes from the much larger region of heated, glowing gas surrounding this little grain of interplanetary material. Because of its high speed, the energy in a pea-sized meteor is as great as that of an artillery shell fired on Earth, but this energy is dispersed high in Earth’s atmosphere. (When these tiny projectiles hit an airless body like the Moon, they do make small craters and generally pulverize the surface.)
If a particle the size of a golf ball strikes our atmosphere, it produces a much brighter trail called a fireball (Figure 14.2). A piece as large as a bowling ball has a fair chance of surviving its fiery entry if its approach speed is not too high. The total mass of meteoric material entering Earth’s atmosphere is estimated to be about 100 tons per day (which seems like a lot if you imagine it all falling in one place, but remember it is spread out all over our planet’s surface).
While it is difficult to capture images of fireballs and other meteors with still photography, it’s easy to capture the movement of these objects on video. The American Meteor Society maintains a website on which their members can share such videos.
Meteor Showers
Many—perhaps most—of the meteors that strike Earth are associated with specific comets . Some of these periodic comets still return to our view; others have long ago fallen apart, leaving only a trail of dust behind them. The dust particles from a given comet retain approximately the orbit of their parent, continuing to move together through space but spreading out over the orbit with time. When Earth, in its travels around the Sun, crosses such a dust stream, we see a sudden burst of meteor activity that usually lasts several hours; such an event is called a meteor shower .
The dust particles and pebbles that produce meteor showers are moving together in space before they encounter Earth. Thus, as we look up at the atmosphere, their parallel paths seem to come toward us from a place in the sky called the radiant . This is the direction in space from which the meteor stream seems to be diverging, just as long railroad tracks seem to diverge from a single spot on the horizon (Figure 14.3). Meteor showers are often designated by the constellation in which this radiant is located: for example, the Perseid meteor shower has its radiant in the constellation of Perseus. But you are likely to see shower meteors anywhere in the sky, not just in the constellation of the radiant. The characteristics of some of the more famous meteor showers are summarized in Table 14.1.
| Shower Name | Date of Maximum | Associated Parent Object | Comet’s Period (years) |
|---|---|---|---|
| Quadrantid | January 3–4 | 2003EH (asteroid) | — |
| Lyrid | April 22 | Comet Thatcher | 415 |
| Eta Aquarid | May 4–5 | Comet Halley | 76 |
| Delta Aquarid | July 29–30 | Comet Machholz | — |
| Perseid | August 11–12 | Comet Swift-Tuttle | 133 |
| Orionid | October 20–21 | Comet Halley | 76 |
| Southern Taurid | October 31 | Comet Encke | 3 |
| Leonid | November 16–17 | Comet Tempel-Tuttle | 33 |
| Geminid | December 13 | Phaethon (asteroid) | 1.4 |
The meteoric dust is not always evenly distributed along the orbit of the comet, so during some years more meteors are seen when Earth intersects the dust stream, and in other years fewer. For example, a very clumpy distribution is associated with the Leonid meteors, which in 1833 and again in 1866 (after an interval of 33 years—the period of the comet) yielded the most spectacular showers (sometimes called meteor storms ) ever recorded (Figure 14.4). During the Leonid storm on November 17, 1866, up to a hundred meteors were observed per second in some locations. The Leonid shower of 2001 was not this intense, but it peaked at nearly a thousand meteors per hour—one every few seconds—observable from any dark viewing site.
The most dependable annual meteor display is the Perseid shower, which appears each year for about three nights near August 11. In the absence of bright moonlight, you can see one meteor every few minutes during a typical Perseid shower. Astronomers estimate that the total combined mass of the particles in the Perseid swarm is nearly a billion tons; the comet that gave rise to the particles in that swarm, called Swift-Tuttle, must originally have had at least that much mass. However, if its initial mass were comparable to the mass measured for Comet Halley, then Swift-Tuttle would have contained several hundred billion tons, suggesting that only a very small fraction of the original cometary material survives in the meteor stream.
No shower meteor has ever survived its flight through the atmosphere and been recovered for laboratory analysis. However, there are other ways to investigate the nature of these particles and thereby gain additional insight into the comets from which they are derived. Analysis of the flight paths of meteors shows that most of them are very light or porous, with densities typically less than 1.0 g/cm 3 . If you placed a fist-sized lump of meteor material on a table in Earth’s gravity, it might well fall apart under its own weight.
Such light particles break up very easily in the atmosphere, accounting for the failure of even relatively large shower meteors to reach the ground. Comet dust is apparently fluffy, rather inconsequential stuff. NASA’s Stardust mission used a special substance, called aerogel, to collect these particles. We can also infer this from the tiny comet particles recovered in Earth’s atmosphere with high-flying aircraft (see Figure 13.19). This fluff, by its very nature, cannot reach Earth’s surface intact. However, more substantial fragments from asteroids do make it into our laboratories, as we will see in the next section.
Observing a meteor shower is one of the easiest and most enjoyable astronomy activities for beginners (Figure 14.5). The best thing about it is that you don’t need a telescope or binoculars—in fact, they would positively get in your way. What you do need is a site far from city lights, with an unobstructed view of as much sky as possible. While the short bright lines in the sky made by individual meteors could, in theory, be traced back to a radiant point (as shown in Figure 14.3), the quick blips of light that represent the end of the meteor could happen anywhere above you.
The key to observing meteor showers is not to restrict your field of view, but to lie back and scan the sky alertly. Try to select a good shower (see the list in Table 14.1) and a night when the Moon will not be bright at the time you are observing. The Moon, street lights, vehicle headlights, bright flashlights, and cell phone and tablet screens will all get in the way of your seeing the faint meteor streaks.
You will see more meteors after midnight, when you are on the hemisphere of Earth that faces forward—in the direction of Earth’s revolution around the Sun. Before midnight, you are observing from the “back side” of Earth, and the only meteors you see will be those that traveled fast enough to catch up with Earth’s orbital motion.
When you’ve gotten away from all the lights, give your eyes about 15 minutes to get “dark adapted”—that is, for the pupils of your eyes to open up as much as possible. (This adaptation is the same thing that happens in a dark movie theater. When you first enter, you can’t see a thing, but eventually, as your pupils open wider, you can see pretty clearly by the faint light of the screen—and notice all that spilled popcorn on the floor.)
Seasoned meteor observers find a hill or open field and make sure to bring warm clothing, a blanket, and a thermos of hot coffee or chocolate with them. (It’s also nice to take along someone with whom you enjoy sitting in the dark.) Don’t expect to see fireworks or a laser show: meteor showers are subtle phenomena, best approached with a patience that reflects the fact that some of the dust you are watching burn up may first have been gathered into its parent comet more than 4.5 billion years ago, as the solar system was just forming.


2.5: Competitive Advantage and Strategy
What is Competitive Advantage?
The concept of competitive advantage came into popular usage in the 1980s as an attempt to identify and define the strategic goals of the firm. In his now famous work Competitive Advantage: Creating and Sustaining Superior Performance , Michael Porter showed that the goal of all firms is to achieve a competitive advantage in relation to their rivals . This they do either by selling at a lower cost/price or by differentiating their product from those of their rivals (Porter 1985).
Text Source: Reinert, Kenneth A., and Ramkishen S. Rajan, editors. “Competitive Advantage.” The Princeton Encyclopedia of the World Economy , by Nigel S. Grimwade, 1st ed., Princeton University Press, 2010. Credo Reference , search.credoreference.com/articles/Qm9va0FydGljbGU6MzE0MDIx. Accessed 1 Sept. 2023.
Take some time to review the following graphic that depicts Porter's ideas:
Image Source: "Porter’s Generic Strategies: Differentiation, Cost Leadership and Focus," B2U (Website)
In addition to the strategies identified within the above graphic, a fifth strategy called 'best-cost' will be introduced in the 2:57 video that follows:
Where do you think the 'best-cost' strategy would be visible if it were included on the above graphic?


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12.12.4: Procedure- Checking the Right Dose
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Wash your hands.
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Read the container label on the medication bottle.
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Check the dose on the label against the dose for that medication listed on the Care Plan.
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If they are the same proceed to hand the bottle to the patient and/or assist with administration as trained and ordered on the Care Plan.
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If they are not the same, STOP! Do not proceed further.
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Contact a supervisor for further instructions.
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Explain the reason for this to the patient. Incorrect dose information could lead to the patient taking too much or not enough of a medication.
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Record and report any problem about the medication in question to a supervisor.


1. This course is designed to provide the student with information regarding developmental changes as they occur during the aging process.
Students who complete this course will gain knowledge regarding developmental changes as they occur during the aging process. This will be assessed through Chapter Discussions (1-15), and Module Recaps and Examinations (1-6).
2. This course is designed to enable the student to evaluate current issues related to the aging process.
Students who complete this course will be able to evaluate current issues related to the aging process. This will be assessed through Written Assignment Reviews and Discussions (1-6), and the Term Project (Parts 1-3).
Contributors and Attributions
CC licensed content, Original
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Course Objectives.
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: Kelly, Jessica.
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: Herkimer College.
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: ATD Course.
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2.8: Measures of the Spread of the Data
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An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean.
The standard deviation
- provides a numerical measure of the overall amount of variation in a data set, and
- can be used to determine whether a particular data value is close to or far from the mean.
The standard deviation provides a measure of the overall variation in a data set
The standard deviation is always positive or zero. The standard deviation is small when the data are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying the amount of time customers wait in line at the checkout at supermarket A and supermarket B . the average wait time at both supermarkets is five minutes. At supermarket A , the standard deviation for the wait time is two minutes; at supermarket B the standard deviation for the wait time is four minutes.
Because supermarket B has a higher standard deviation, we know that there is more variation in the wait times at supermarket B . Overall, wait times at supermarket B are more spread out from the average; wait times at supermarket A are more concentrated near the average.
The standard deviation can be used to determine whether a data value is close to or far from the mean.
Suppose that Rosa and Binh both shop at supermarket A . Rosa waits at the checkout counter for seven minutes and Binh waits for one minute. At supermarket A , the mean waiting time is five minutes and the standard deviation is two minutes. The standard deviation can be used to determine whether a data value is close to or far from the mean.
Rosa waits for seven minutes:
- Seven is two minutes longer than the average of five; two minutes is equal to one standard deviation.
- Rosa's wait time of seven minutes is two minutes longer than the average of five minutes.
- Rosa's wait time of seven minutes is one standard deviation above the average of five minutes.
Binh waits for one minute.
- One is four minutes less than the average of five; four minutes is equal to two standard deviations.
- Binh's wait time of one minute is four minutes less than the average of five minutes.
- Binh's wait time of one minute is two standard deviations below the average of five minutes.
- A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if it is more than two standard deviations away is more of an approximate "rule of thumb" than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is further away than two standard deviations. (You will learn more about this in later chapters.)
The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is one standard deviation to the right of five because \(5 + (1)(2) = 7\).
If one were also part of the data set, then one is two standard deviations to the left of five because \(5 + (-2)(2) = 1\).
- In general, a value = mean + (#ofSTDEV)(standard deviation)
- where #ofSTDEVs = the number of standard deviations
- #ofSTDEV does not need to be an integer
- One is two standard deviations less than the mean of five because: \(1 = 5 + (-2)(2)\).
The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population.
- sample: \[x = \bar{x} + \text{(#ofSTDEV)(s)}\]
- Population: \[x = \mu + \text{(#ofSTDEV)(s)}\]
The lower case letter s represents the sample standard deviation and the Greek letter \(\sigma\) (sigma, lower case) represents the population standard deviation.
The symbol \(\bar{x}\) is the sample mean and the Greek symbol \(\mu\) is the population mean.
Calculating the Standard Deviation
If \(x\) is a number, then the difference "\(x\) – mean" is called its deviation . In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols a deviation is \(x - \mu\). For sample data, in symbols a deviation is \(x - \bar{x}\).
The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lower case letter s represents the sample standard deviation and the Greek letter \(\sigma\) (sigma, lower case) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of \(\sigma\).
To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations (the \(x - \bar{x}\) values for a sample, or the \(x - \mu\) values for a population). The symbol \(\sigma^{2}\) represents the population variance; the population standard deviation \(\sigma\) is the square root of the population variance. The symbol \(s^{2}\) represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.
If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by \(N\), the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n – 1 , one less than the number of items in the sample.
Formulas for the Sample Standard Deviation
\[s = \sqrt{\dfrac{\sum(x-\bar{x})^{2}}{n-1}} \label{eq1}\]
or
\[s = \sqrt{\dfrac{\sum f (x-\bar{x})^{2}}{n-1}} \label{eq2}\]
For the sample standard deviation, the denominator is \(n - 1\), that is the sample size MINUS 1.
Formulas for the Population Standard Deviation
\[\sigma = \sqrt{\dfrac{\sum(x-\mu)^{2}}{N}} \label{eq3} \]
or
\[\sigma = \sqrt{\dfrac{\sum f (x-\mu)^{2}}{N}} \label{eq4}\]
For the population standard deviation, the denominator is \(N\), the number of items in the population.
In Equations \ref{eq2} and \ref{eq4}, \(f\) represents the frequency with which a value appears. For example, if a value appears once, \(f\) is one. If a value appears three times in the data set or population, \(f\) is three.
Sampling Variability of a Statistic
The statistic of a sampling distribution was discussed in Section 2.6. How much the statistic varies from one sample to another is known as the sampling variability of a statistic . You typically measure the sampling variability of a statistic by its standard error.
The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in Chapter 7. The notation for the standard error of the mean is \(\dfrac{\sigma}{\sqrt{n}}\) where \(\sigma\) is the standard deviation of the population and \(n\) is the size of the sample.
In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83, 83+, 84+ calculator, you need to select the appropriate standard deviation \(\sigma_{x}\) or \(s_{x}\) from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. (The calculator instructions appear at the end of this example.)
Example \(\PageIndex{1}\)
In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of n = 20 fifth grade students. The ages are rounded to the nearest half year:
9; 9.5; 9.5; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5;
\[\bar{x} = \dfrac{9+9.5(2)+10(4)+10.5(4)+11(6)+11.5(3)}{20} = 10.525 \nonumber\]
The average age is 10.53 years, rounded to two places.
The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating s .
| Data | Freq. | Deviations | Deviations 2 | (Freq.)( Deviations 2 ) |
|---|---|---|---|---|
| x | f | (x – \(\bar{x}\)) | (x – \(\bar{x}\))2 | (f)(x – \(\bar{x}\))2 |
| 9 | 1 | 9 – 10.525 = –1.525 | (–1.525)2 = 2.325625 | 1 × 2.325625 = 2.325625 |
| 9.5 | 2 | 9.5 – 10.525 = –1.025 | (–1.025) 2 = 1.050625 | 2 × 1.050625 = 2.101250 |
| 10 | 4 | 10 – 10.525 = –0.525 | (–0.525) 2 = 0.275625 | 4 × 0.275625 = 1.1025 |
| 10.5 | 4 | 10.5 – 10.525 = –0.025 | (–0.025) 2 = 0.000625 | 4 × 0.000625 = 0.0025 |
| 11 | 6 | 11 – 10.525 = 0.475 | (0.475) 2 = 0.225625 | 6 × 0.225625 = 1.35375 |
| 11.5 | 3 | 11.5 – 10.525 = 0.975 | (0.975) 2 = 0.950625 | 3 × 0.950625 = 2.851875 |
| The total is 9.7375 |
The sample variance, \(s^{2}\), is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1):
\[s^{2} = \dfrac{9.7375}{20-1} = 0.5125 \nonumber\]
The sample standard deviation s is equal to the square root of the sample variance:
\[s = \sqrt{0.5125} = 0.715891 \nonumber\]
and this is rounded to two decimal places, \(s = 0.72\).
Typically, you do the calculation for the standard deviation on your calculator or computer . The intermediate results are not rounded. This is done for accuracy.
- For the following problems, recall that value = mean + (#ofSTDEVs)(standard deviation) . Verify the mean and standard deviation or a calculator or computer.
- For a sample: \(x\) = \(\bar{x}\) + (#ofSTDEVs)( s )
- For a population: \(x\) = \(\mu\) + (#ofSTDEVs)\(\sigma\)
- For this example, use x = \(\bar{x}\) + (#ofSTDEVs)( s ) because the data is from a sample
- Verify the mean and standard deviation on your calculator or computer.
- Find the value that is one standard deviation above the mean. Find (\(\bar{x}\) + 1s).
- Find the value that is two standard deviations below the mean. Find (\(\bar{x}\) – 2s).
- Find the values that are 1.5 standard deviations from (below and above) the mean.
Solution
-
- Clear lists L1 and L2. Press STAT 4:ClrList. Enter 2nd 1 for L1, the comma (,), and 2nd 2 for L2.
- Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down.
- Put the data values (9, 9.5, 10, 10.5, 11, 11.5) into list L1 and the frequencies (1, 2, 4, 4, 6, 3) into list L2. Use the arrow keys to move around.
- Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER.
- \(\bar{x}\) = 10.525
- Use Sx because this is sample data (not a population): Sx=0.715891
- (\(\bar{x} + 1s) = 10.53 + (1)(0.72) = 11.25\)
- \((\bar{x} - 2s) = 10.53 – (2)(0.72) = 9.09\)
-
- \((\bar{x} - 1.5s) = 10.53 – (1.5)(0.72) = 9.45\)
- \((\bar{x} + 1.5s) = 10.53 + (1.5)(0.72) = 11.61\)
Exercise 2.8.1
On a baseball team, the ages of each of the players are as follows:
21; 21; 22; 23; 24; 24; 25; 25; 28; 29; 29; 31; 32; 33; 33; 34; 35; 36; 36; 36; 36; 38; 38; 38; 40
Use your calculator or computer to find the mean and standard deviation. Then find the value that is two standard deviations above the mean.
Answer
\(\mu\) = 30.68
\(s = 6.09\)
(\(\bar{x} + 2s = 30.68 + (2)(6.09) = 42.86\).
Explanation of the standard deviation calculation shown in the table
The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11 which is indicated by the deviations 0.97 and 0.47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero . (For Example \(\PageIndex{1}\), there are \(n = 20\) deviations.) So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation.
The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.
Notice that instead of dividing by \(n = 20\), the calculation divided by \(n - 1 = 20 - 1 = 19\) because the data is a sample. For the sample variance, we divide by the sample size minus one (\(n - 1\)). Why not divide by \(n\)? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by (\(n - 1\)) gives a better estimate of the population variance.
Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number which measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic.
The standard deviation, \(s\) or \(\sigma\), is either zero or larger than zero. When the standard deviation is zero, there is no spread; that is, all the data values are equal to each other. The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make \(s\) or \(\sigma\) very large.
The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better "feel" for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data . Display your data in a histogram or a box plot.
Example \(\PageIndex{2}\)
Use the following data (first exam scores) from Susan Dean's spring pre-calculus class:
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100
- Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places.
-
Calculate the following to one decimal place using a TI-83+ or TI-84 calculator:
- The sample mean
- The sample standard deviation
- The median
- The first quartile
- The third quartile
- IQR
- Construct a box plot and a histogram on the same set of axes. Make comments about the box plot, the histogram, and the chart.
Answer
- See Table
-
- The sample mean = 73.5
- The sample standard deviation = 17.9
- The median = 73
- The first quartile = 61
- The third quartile = 90
- IQR = 90 – 61 = 29
- The \(x\)-axis goes from 32.5 to 100.5; \(y\)-axis goes from -2.4 to 15 for the histogram. The number of intervals is five, so the width of an interval is (\(100.5 - 32.5\)) divided by five, is equal to 13.6. Endpoints of the intervals are as follows: the starting point is 32.5, \(32.5 + 13.6 = 46.1\), \(46.1 + 13.6 = 59.7\), \(59.7 + 13.6 = 73.3\), \(73.3 + 13.6 = 86.9\), \(86.9 + 13.6 = 100.5 =\) the ending value; No data values fall on an interval boundary.
The long left whisker in the box plot is reflected in the left side of the histogram. The spread of the exam scores in the lower 50% is greater (\(73 - 33 = 40\)) than the spread in the upper 50% (\(100 - 73 = 27\)). The histogram, box plot, and chart all reflect this. There are a substantial number of A and B grades (80s, 90s, and 100). The histogram clearly shows this. The box plot shows us that the middle 50% of the exam scores ( IQR = 29) are Ds, Cs, and Bs. The box plot also shows us that the lower 25% of the exam scores are Ds and Fs.
| Data | Frequency | Relative Frequency | Cumulative Relative Frequency |
|---|---|---|---|
| 33 | 1 | 0.032 | 0.032 |
| 42 | 1 | 0.032 | 0.064 |
| 49 | 2 | 0.065 | 0.129 |
| 53 | 1 | 0.032 | 0.161 |
| 55 | 2 | 0.065 | 0.226 |
| 61 | 1 | 0.032 | 0.258 |
| 63 | 1 | 0.032 | 0.29 |
| 67 | 1 | 0.032 | 0.322 |
| 68 | 2 | 0.065 | 0.387 |
| 69 | 2 | 0.065 | 0.452 |
| 72 | 1 | 0.032 | 0.484 |
| 73 | 1 | 0.032 | 0.516 |
| 74 | 1 | 0.032 | 0.548 |
| 78 | 1 | 0.032 | 0.580 |
| 80 | 1 | 0.032 | 0.612 |
| 83 | 1 | 0.032 | 0.644 |
| 88 | 3 | 0.097 | 0.741 |
| 90 | 1 | 0.032 | 0.773 |
| 92 | 1 | 0.032 | 0.805 |
| 94 | 4 | 0.129 | 0.934 |
| 96 | 1 | 0.032 | 0.966 |
| 100 | 1 | 0.032 | 0.998 (Why isn't this value 1?) |
Exercise \(\PageIndex{2}\)
The following data show the different types of pet food stores in the area carry.
6; 6; 6; 6; 7; 7; 7; 7; 7; 8; 9; 9; 9; 9; 10; 10; 10; 10; 10; 11; 11; 11; 11; 12; 12; 12; 12; 12; 12;
Calculate the sample mean and the sample standard deviation to one decimal place using a TI-83+ or TI-84 calculator.
Answer
\(\mu = 9.3\) and \(s = 2.2\)
Standard deviation of Grouped Frequency Tables
Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula:
\[\text{Mean of Frequency Table} = \dfrac{\sum fm}{\sum f}\]
where \(f\) interval frequencies and \(m =\) interval midpoints.
Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how “unusual” individual data is compared to the mean.
Example \(\PageIndex{3}\)
Find the standard deviation for the data in Table \(\PageIndex{3}\).
| Class | Frequency, f | Midpoint, m | m 2 | \(\bar{x}\) | fm 2 | Standard Deviation |
|---|---|---|---|---|---|---|
| 0–2 | 1 | 1 | 1 | 7.58 | 1 | 3.5 |
| 3–5 | 6 | 4 | 16 | 7.58 | 96 | 3.5 |
| 6–8 | 10 | 7 | 49 | 7.58 | 490 | 3.5 |
| 9–11 | 7 | 10 | 100 | 7.58 | 700 | 3.5 |
| 12–14 | 0 | 13 | 169 | 7.58 | 0 | 3.5 |
| 15–17 | 2 | 16 | 256 | 7.58 | 512 | 3.5 |
For this data set, we have the mean, \(\bar{x}\) = 7.58 and the standard deviation, \(s_{x}\) = 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 – 3.5 – 3.5 = 0.58. While the formula for calculating the standard deviation is not complicated, \(s_{x} = \sqrt{\dfrac{f(m - \bar{x})^{2}}{n-1}}\) where \(s_{x}\) = sample standard deviation, \(\bar{x}\) = sample mean, the calculations are tedious. It is usually best to use technology when performing the calculations.
Find the standard deviation for the data from the previous example
| Class | 0-2 | 3-5 | 6-8 | 9–11 | 12–14 | 15–17 |
|---|---|---|---|---|---|---|
| Frequency, f | 1 | 6 | 10 | 7 | 0 | 2 |
First, press the STAT key and select 1:Edit
Input the midpoint values into L1 and the frequencies into L2
Select STAT , CALC , and 1: 1-Var Stats
Select 2 nd then 1 then , 2 nd then 2 Enter
You will see displayed both a population standard deviation, \(\sigma_{x}\), and the sample standard deviation, \(s_{x}\).
Comparing Values from Different Data Sets
The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.
- For each data value, calculate how many standard deviations away from its mean the value is.
- Use the formula: value = mean + (#ofSTDEVs)(standard deviation); solve for #ofSTDEVs.
- \(\text{#ofSTDEVs} = \dfrac{\text{value-mean}}{\text{standard deviation}}\)
- Compare the results of this calculation.
#ofSTDEVs is often called a " z -score"; we can use the symbol \(z\). In symbols, the formulas become:
| Sample | \(x = \bar{x} + zs\) | \(z = \dfrac{x - \bar{x}}{s}\) |
| Population | \(x = \mu + z\sigma\) | \(z = \dfrac{x - \mu}{\sigma}\) |
Example \(\PageIndex{4}\)
Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?
| Student | GPA | School Mean GPA | School Standard Deviation |
|---|---|---|---|
| John | 2.85 | 3.0 | 0.7 |
| Ali | 77 | 80 | 10 |
Answer
For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.
\[z = \text{#ofSTDEVs} = \left(\dfrac{\text{value-mean}}{\text{standard deviation}}\right) = \left(\dfrac{x + \mu}{\sigma}\right) \nonumber\]
For John,
\[z = \text{#ofSTDEVs} = \left(\dfrac{2.85-3.0}{0.7}\right) = -0.21 \nonumber\]
For Ali,
\[z = \text{#ofSTDEVs} = (\dfrac{77-80}{10}) = -0.3 \nonumber\]
John has the better GPA when compared to his school because his GPA is 0.21 standard deviations below his school's mean while Ali's GPA is 0.3 standard deviations below his school's mean.
John's z -score of –0.21 is higher than Ali's z -score of –0.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school.
Exercise \(\PageIndex{4}\)
Two swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the 50 meter freestyle when compared to her team. Which swimmer had the fastest time when compared to her team?
| Swimmer | Time (seconds) | Team Mean Time | Team Standard Deviation |
|---|---|---|---|
| Angie | 26.2 | 27.2 | 0.8 |
| Beth | 27.3 | 30.1 | 1.4 |
Answer
For Angie:
\[z = \left(\dfrac{26.2-27.2}{0.8}\right) = -1.25 \nonumber\]
For Beth:
\[z = \left(\dfrac{27.3-30.1}{1.4}\right) = -2 \nonumber\]
The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.
For ANY data set, no matter what the distribution of the data is:
- At least 75% of the data is within two standard deviations of the mean.
- At least 89% of the data is within three standard deviations of the mean.
- At least 95% of the data is within 4.5 standard deviations of the mean.
- This is known as Chebyshev's Rule.
For data having a distribution that is BELL-SHAPED and SYMMETRIC:
- Approximately 68% of the data is within one standard deviation of the mean.
- Approximately 95% of the data is within two standard deviations of the mean.
- More than 99% of the data is within three standard deviations of the mean.
- This is known as the Empirical Rule.
- It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.
References
- Data from Microsoft Bookshelf.
- King, Bill.“Graphically Speaking.” Institutional Research, Lake Tahoe Community College. Available online at www.ltcc.edu/web/about/institutional-research (accessed April 3, 2013).
Review
The standard deviation can help you calculate the spread of data. There are different equations to use if are calculating the standard deviation of a sample or of a population.
- The Standard Deviation allows us to compare individual data or classes to the data set mean numerically.
- \(s = \sqrt{\dfrac{\sum(x-\bar{x})^{2}}{n-1}}\) or \(s = \sqrt{\dfrac{\sum f (x-\bar{x})^{2}}{n-1}}\) is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, \(\mu\), and the formula \(\sigma = \sqrt{\dfrac{\sum(x-\mu)^{2}}{N}}\) or \(\sigma = \sqrt{\dfrac{\sum f (x-\mu)^{2}}{N}}\). ∑ f ( x − μ ) 2 N − − − − − − − − − √ .
Formula Review
\[s_{x} = \sqrt{\dfrac{\sum fm^{2}}{n} - \bar{x}^2}\]
where \(s_{x} =\text{sample standard deviation}\) and \(\bar{x} = \text{sample mean}\)
Use the following information to answer the next two exercises : The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles.
29; 37; 38; 40; 58; 67; 68; 69; 76; 86; 87; 95; 96; 96; 99; 106; 112; 127; 145; 150
Exercise 2.8.4
Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.
Answer
\(s\) = 34.5
Exercise 2.8.5
Find the value that is one standard deviation below the mean.
Exercise 2.8.6
Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team?
| Baseball Player | Batting Average | Team Batting Average | Team Standard Deviation |
|---|---|---|---|
| Fredo | 0.158 | 0.166 | 0.012 |
| Karl | 0.177 | 0.189 | 0.015 |
Answer
For Fredo:
\(z\) = \(\dfrac{0.158-0.166}{0.012}\) = –0.67
For Karl:
\(z\) = \(\dfrac{0.177-0.189}{0.015}\) = –0.8
Fredo’s z -score of –0.67 is higher than Karl’s z -score of –0.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.
Exercise 2.8.7
Use Table to find the value that is three standard deviations:
- above the mean
- below the mean
Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84 .
Exercise 2.8.5
Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.
-
Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 -
Daily Low Temperature Frequency 49.5–59.5 53 59.5–69.5 32 69.5–79.5 15 79.5–89.5 1 89.5–99.5 0 -
Points per Game Frequency 49.5–59.5 14 59.5–69.5 32 69.5–79.5 15 79.5–89.5 23 89.5–99.5 2
Answer
- \(s_{x} = \sqrt{\dfrac{\sum fm^{2}}{n} - \bar{x}^{2}} = \sqrt{\dfrac{193157.45}{30} - 79.5^{2}} = 10.88\)
- \(s_{x} = \sqrt{\dfrac{\sum fm^{2}}{n} - \bar{x}^{2}} = \sqrt{\dfrac{380945.3}{101} - 60.94^{2}} = 7.62\)
- \(s_{x} = \sqrt{\dfrac{\sum fm^{2}}{n} - \bar{x}^{2}} = \sqrt{\dfrac{440051.5}{86} - 70.66^{2}} = 11.14\)
Bringing It Together
Exercise 2.8.7
Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:
| # of movies | Frequency |
|---|---|
| 0 | 5 |
| 1 | 9 |
| 2 | 6 |
| 3 | 4 |
| 4 | 1 |
- Find the sample mean \(\bar{x}\).
- Find the approximate sample standard deviation, \(s\).
Answer
- 1.48
- 1.12
Exercise 2.8.8
Forty randomly selected students were asked the number of pairs of sneakers they owned. Let \(X =\) the number of pairs of sneakers owned. The results are as follows:
| \(X\) | Frequenc y |
|---|---|
| 1 | 2 |
| 2 | 5 |
| 3 | 8 |
| 4 | 12 |
| 5 | 12 |
| 6 | 0 |
| 7 | 1 |
- Find the sample mean \(\bar{x}\)
- Find the sample standard deviation, s
- Construct a histogram of the data.
- Complete the columns of the chart.
- Find the first quartile.
- Find the median.
- Find the third quartile.
- Construct a box plot of the data.
- What percent of the students owned at least five pairs?
- Find the 40 th percentile.
- Find the 90 th percentile.
- Construct a line graph of the data
- Construct a stemplot of the data
Exercise 2.8.9
Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year.
177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265
- Organize the data from smallest to largest value.
- Find the median.
- Find the first quartile.
- Find the third quartile.
- Construct a box plot of the data.
- The middle 50% of the weights are from _______ to _______.
- If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why?
- If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why?
-
Assume the population was the San Francisco 49ers. Find:
- the population mean, \(\mu\).
- the population standard deviation, \(\sigma\).
- the weight that is two standard deviations below the mean.
- When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard deviations above or below the mean was he?
- That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?
Answer
- 174; 177; 178; 184; 185; 185; 185; 185; 188; 190; 200; 205; 205; 206; 210; 210; 210; 212; 212; 215; 215; 220; 223; 228; 230; 232; 241; 241; 242; 245; 247; 250; 250; 259; 260; 260; 265; 265; 270; 272; 273; 275; 276; 278; 280; 280; 285; 285; 286; 290; 290; 295; 302
- 241
- 205.5
- 272.5
- 205.5, 272.5
- sample
- population
-
- 236.34
- 37.50
- 161.34
- 0.84 std. dev. below the mean
- Young
Exercise 2.8.10
One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The 12 change scores are as follows:
3; 8; –1; 2; 0; 5; –3; 1; –1; 6; 5; –2
- What is the mean change score?
- What is the standard deviation for this population?
- What is the median change score?
- Find the change score that is 2.2 standard deviations below the mean.
Exercise 2.8.11
Refer to Figure determine which of the following are true and which are false. Explain your solution to each part in complete sentences.
<figure >
</figure>
- The medians for all three graphs are the same.
- We cannot determine if any of the means for the three graphs is different.
- The standard deviation for graph b is larger than the standard deviation for graph a.
- We cannot determine if any of the third quartiles for the three graphs is different.
Answer
- True
- True
- True
- False
Exercise 2.8.12
In a recent issue of the IEEE Spectrum , 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. Let X = the length (in days) of an engineering conference.
- Organize the data in a chart.
- Find the median, the first quartile, and the third quartile.
- Find the 65 th percentile.
- Find the 10 th percentile.
- Construct a box plot of the data.
- The middle 50% of the conferences last from _______ days to _______ days.
- Calculate the sample mean of days of engineering conferences.
- Calculate the sample standard deviation of days of engineering conferences.
- Find the mode.
- If you were planning an engineering conference, which would you choose as the length of the conference: mean; median; or mode? Explain why you made that choice.
- Give two reasons why you think that three to five days seem to be popular lengths of engineering conferences.
Exercise 2.8.13
A survey of enrollment at 35 community colleges across the United States yielded the following figures:
6414; 1550; 2109; 9350; 21828; 4300; 5944; 5722; 2825; 2044; 5481; 5200; 5853; 2750; 10012; 6357; 27000; 9414; 7681; 3200; 17500; 9200; 7380; 18314; 6557; 13713; 17768; 7493; 2771; 2861; 1263; 7285; 28165; 5080; 11622
- Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."
- Construct a histogram of the data.
- If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?
- Calculate the sample mean.
- Calculate the sample standard deviation.
- A school with an enrollment of 8000 would be how many standard deviations away from the mean?
Answer
-
Enrollment Frequency 1000-5000 10 5000-10000 16 10000-15000 3 15000-20000 3 20000-25000 1 25000-30000 2 - Check student’s solution.
- mode
- 8628.74
- 6943.88
- –0.09
Use the following information to answer the next two exercises. \(X =\) the number of days per week that 100 clients use a particular exercise facility.
| \(x\) | Frequency |
|---|---|
| 0 | 3 |
| 1 | 12 |
| 2 | 33 |
| 3 | 28 |
| 4 | 11 |
| 5 | 9 |
| 6 | 4 |
Exercise 2.8.14
The 80 th percentile is _____
- 5
- 80
- 3
- 4
Exercise 2.8.15
The number that is 1.5 standard deviations BELOW the mean is approximately _____
- 0.7
- 4.8
- –2.8
- Cannot be determined
Answer
a
Exercise 2.8.16
Suppose that a publisher conducted a survey asking adult consumers the number of fiction paperback books they had purchased in the previous month. The results are summarized in the Table.
| # of books | Freq. | Rel. Freq. |
|---|---|---|
| 0 | 18 | |
| 1 | 24 | |
| 2 | 24 | |
| 3 | 22 | |
| 4 | 15 | |
| 5 | 10 | |
| 7 | 5 | |
| 9 | 1 |
- Are there any outliers in the data? Use an appropriate numerical test involving the IQR to identify outliers, if any, and clearly state your conclusion.
- If a data value is identified as an outlier, what should be done about it?
- Are any data values further than two standard deviations away from the mean? In some situations, statisticians may use this criteria to identify data values that are unusual, compared to the other data values. (Note that this criteria is most appropriate to use for data that is mound-shaped and symmetric, rather than for skewed data.)
- Do parts a and c of this problem give the same answer?
- Examine the shape of the data. Which part, a or c, of this question gives a more appropriate result for this data?
- Based on the shape of the data which is the most appropriate measure of center for this data: mean, median or mode?
Glossary
- Standard Deviation
- a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation.
Contributors and Attributions
- Variance
- mean of the squared deviations from the mean, or the square of the standard deviation; for a set of data, a deviation can be represented as \(x\) – \(\bar{x}\) where \(x\) is a value of the data and \(\bar{x}\) is the sample mean. The sample variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one.
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by
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