Ray_10e_IMTB_Chapter_05

Chapter 5 – Description of Behavior Through Numerical Representation Chapter Outline Measurement Scales of Measurement Nominal Measurement Ordinal Measurement Interval Measurement Ratio Measurement Identifying Scales of Measurement Measurement and Statistics Pictorial Description of Frequency Information Descriptive Statistics Measures of Central Tendency Measures of Variability Pictorial Presentations of Numerical Data Transforming Data Standard Scores Measure of Association Chapter Overview Behavior is described through measurement that exists at four basic levels or scales: nominal, ordinal, interval, and ratio. Statistics are used to help us make sense of data. A frequency distribution provides a pictorial representation of frequency information from an experiment. Distributions of scores may take several shapes, such as a bimodal distribution or skewed distribution. Mean, median, and mode are commonly used measures of central tendency. The choice of which one to use depends on the distribution. Variability can be measured by range and standard deviation. Numerical data can be represented in line and bar graphs. Data can be transformed from one scale to another and to meet statistical assumptions. Standard scores such as z scores allow comparison of scores to other scores. Association between variables can be depicted in a scatter diagram and with correlation coefficients, which may be positive or negative. Correlation does not show causal relationships between variables, but does have value in prediction. Chapter Objectives 1. Measurement theory requires us to ask what two questions when doing research? 2. Compare and contrast the different scales of measurement. 3. Cite two ways to display data. CHAPTER 5 78

4. What three measures of central tendency are discussed in the text and what are their characteristics? 5. Discuss the concept of variability. Give examples of three different measures of variability and describe their functions. 6. Why is it common to transform data? Relate transformation of data to the concept of z-scores. 7. What is correlation? How does it aid us in understanding the relationship between variables? 8. How would you define a positive correlation? How would you define a negative correlation? 9. What can you say about the effect of one variable on the other in correlational studies? 10. What information is derived from squaring the correlation coefficient? How does this relate to variability accounted for? 11. Why is it important to assess underlying distribution before calculating correlations? Teaching Points There are several helpful points to make regarding scales of measurement. First, each subsequent scale builds upon the previous scale. Second, to make reasonable conclusions, one must be sensitive to the use of statistics for a particular scale. Third, use the acronym NOIR, the French word for black, as a cue for the scales of measurement. The chapter makes the excellent point that “statistics do not know and do not care where your numbers come from.” For instance, you could calculate a mean from nominal level data, but the result would be meaningless. Students might recognize this as an example of “garbage in garbage out.” Students often do not possess good graphing skills and have difficulty remembering what variables go on which axes. It would be helpful to give them practice at graphing by using one or more of the activities described below. Remind students that a “0” in a dataset counts in determining the number of scores in calculating the mean. Many students think too much about the mean (and central tendency in general) and not enough about variability. It is useful to demonstrate both mathematically and graphically that two or more data sets can have identical means but be very different in terms of how dispersed the values are in each set. CHAPTER 5 79

The Wadsworth’s Statistics Workshop site contains links to several sections relevant to students after reading the chapter. For example, Scale of Measurement discusses the four levels of measurement of variables in psychological research, their abstract number properties, and the types of statistics that are permitted for each variable. The tutorial ends with a summary and students can test their knowledge in identifying scales of measurement by doing the practice exercises. Another workshop entitled Central Tendency and Variability provides information on how to summarize various kinds of data. Students are led through an example that exercises their skill in calculating measures of central tendency and variability. The workshop segment z Scores elaborates on the value of the z score and provides a step-by-step process of calculating z. Finally, the section on Correlation describes the types of correlations (positive, negative, and zero) and introduces the formula for the Pearson’s correlation coefficient. As with each segment of the workshop site, this segment concludes with a brief quiz. Internet Resources Measurement theory (ftp://ftp.sas.com/pub/neural/measurement.faq) This site provides additional information on measurement theory, scales of measurement, and transformations. The Best and Worst of Statistical Graphics (www.math.yorku.ca/SCS/Gallery/) This site has some interesting examples of poorly conceptualized and created attempts at representing data visually. More resources on statistics (http://www.statsoft.com/textbook/) Elementary statistical concepts are explained concisely and effectively. Of particular interest is the statistical glossary. Datasets (lib.stat.cmu.edu/datasets) (http://www.umass.edu/statdata/) If you need real data to illustrate statistical concepts, these sites have many datasets from a variety of disciplines. The second site groups the datasets by what statistics would be appropriate to analyze the data. Statistics links (www.mccombs.utexas.edu/faculty/jonathan.koehler/links/statistics.asp) This site contains many links to interesting articles and activities related to statistics. Some of the topics addressed by the sites include games, gambling, sports, correlation, and comprehensive statistics sites. Suggested Readings Huff, D., & Geis, I. (1994). How to lie with statistics. New York: W. W. Norton. This amusing book is easy reading full of actual examples of statistical misuses. CHAPTER 5 81

Research Activities for Students The goals of the research activities are to: (1) relate Chapter 5 on an applied learning dimension, and (2) get you involved in research. 1. Physical Activity and Mood: A Correlational Study. For one week record the information using the following scales: Rate your level of physical activity on the following scale: 1. Very physically active 2. Moderately physically active 3. Somewhat physically active 4. Not very physically active 5. Very physically inactive Rate your level of mood on the following scale: 1. Very good mood 2. Relatively good mood 3. Neutral 4. Somewhat of a bad mood 5. Very bad mood Date Time of day Physical Activity Rating Mood Rating 1. 2. 3. 4. 5. 6. 7. DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 82

a. State your research hypotheses. b. Organize the data obtained from questions 2 and 3 into a scatterplot. The abscissa should be labeled “Degree of Knowledge” (values 1-5). The ordinate should be labeled “Degree of Influence” (values 1-5). c. Calculate a Pearson product moment correlation coefficient between for the data obtained from questions 1 and 2, questions 1 and 3, and questions 2 and 3. d. Interpret the results by explaining what type of relationships, if any, are supported by the data. Discuss inferences about causation. 3. Misleading with the Average. Entry level salary data was collected on small sample of college graduates who earned degrees in psychology and biology. Find the mean, median, and mode for each group. If you wanted to persuade someone that they should definitely pursue a degree in psychology, which measure of central tendency would you use? If you wanted to emphasize that it really didn’t matter what they majored in, which measure of central tendency would you use? Entry Level Salaries for Graduates in: Psychology Biology $30,000 $30,000 $36,000 $36,000 $40,000 $40,000 $42,000 $42,000 $149,000 $49,000 4. Misleading with Statistics. Use online references such as PsycINFO and PubMed to find articles about how statistics can lie. Explain this effect. Also see the suggested reading list at the end of this chapter. 5. Assess Your Self Esteem. Complete the following questions. Identify each item as either a nominal, ordinal, interval, or ratio number. Social Security Number _____ Male (01) _____ Female (02) _____ Year of Birth _____ Age _____ Class Standing: Freshman (1) _____ Sophomore (2) _____ Junior (3) _____ Senior (4) _____ Exact time you started this test? ______ DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 84

Rosenberg Self Esteem Scale This form asks questions about self esteem. Answer the following questions honestly using the response scale below. There are no right and wrong answers. 1 Strongly agree 2 Agree 3 Disagree 4 Strongly disagree _____ 1. I feel that I am a person of worth, at least on an equal basis with others. _____ 2. I feel that I have a number of good qualities. _____ 3. *All in all, I'm inclined to feel that I am a failure. _____ 4. I am able to do things as well as most other people. _____ 5. *I feel that I do not have much to be proud of. _____ 6. I take a positive attitude toward myself. _____ 7. On the whole, I am satisfied with myself. _____ 8. *I wish I could have more respect for myself. _____ 9. *I certainly feel useless at times. _____10. *At times I think I am no good at all. *Reverse score these items CHAPTER 5 85

ANS: B PTS: 1 REF: Scales of Measurement 7. If a scale has equal intervals between values and a single underlying quantitative dimension, then it is a(n) ____ scale of measurement. a. ordinal. b. nominal. c. ratio. d. interval. ANS: D PTS: 1 REF: Scales of Measurement MSC: WWW 8. Unlike the Fahrenheit and Celsius scales, the Kelvin temperature scale's lowest possible value is zero. Hence, the Kelvin scale can be considered to be ____. a. ordinal. b. nominal. c. ratio. d. interval. ANS: C PTS: 1 REF: Scales of Measurement 9. If a scale has equal intervals between values, a single underlying quantitative dimension, and an absolute zero, then it is a(n) ____ scale of measurement. a. ordinal. b. nominal. c. ratio. d. interval. ANS: C PTS: 1 REF: Scales of Measurement 10. When constructing a distribution of the number of people favoring each of five different political candidates, you should use: a. a frequency polygon. b. a bar graph. c. a line graph. d. either a frequency graph or a bar graph. ANS: B PTS: 1 REF: Pictorial Description of Frequency Information 11. When constructing a distribution of the number of people who obtained each score on a test, you should use: a. a frequency polygon. b. a bar graph. c. a line graph. d. either a frequency polygon or a bar graph. ANS: D PTS: 1 REF: Pictorial Description of Frequency Information MSC: WWW 12. When a variable is continuous, it is appropriate to use: CHAPTER 5 87

a. a frequency polygon. b. a bar graph. c. a line graph. d. either a frequency polygon or a bar graph. ANS: C PTS: 1 REF: Pictorial Description of Frequency Information 13. When a frequency distribution is constructed, the ____ appears on the y -axis. a. number of individuals who make each response b. responses of the participants c. independent variable d. dependent variable ANS: A PTS: 1 REF: Pictorial Description of Frequency Information 14. When a frequency distribution is constructed, the ____ appears on the x -axis. a. number of individuals who make each response b. responses of the participants c. independent variable d. dependent variable ANS: B PTS: 1 REF: Pictorial Description of Frequency Information 15. If most of the scores in a distribution are clustered near the middle, the distribution is: a. skewed. b. a frequency polygon. c. normal. d. bimodal. ANS: C PTS: 1 REF: Pictorial Description of Frequency Information 16. If scores in a distribution are clustered towards the left end of the distribution, the distribution is: a. negatively skewed. b. positively skewed. c. not skewed. d. irregularly skewed. ANS: B PTS: 1 REF: Pictorial Description of Frequency Information 17. A distribution with two peaks is said to be: a. asymmetrical. b. symmetrical. c. binormal. d. bimodal. ANS: D PTS: 1 REF: Pictorial Description of Frequency Information DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 88

24. What is the median of the following scores: 9, 5, 4, 8, 10, 12? a. 6 b. 7 c. 8 d. 9.5 ANS: C PTS: 1 REF: Descriptive Statistics 25. If you were told that the median of a distribution was greater than the mean, what would that tell you about the shape of the distribution? a. It was negatively skewed. b. It was positively skewed. c. It was normal. d. It would tell you nothing about the shape of the distribution. ANS: A PTS: 1 REF: Descriptive Statistics MSC: WWW 26. Under what conditions would the median, mode, and mean be the same value? a. when the distribution is positively skewed b. when the distribution is bimodal c. when the distribution is normal d. when the distribution is negatively skewed ANS: C PTS: 1 REF: Descriptive Statistics 27. Which of the following is not a measure of variability? a. frequency b. range c. variance d. standard deviation ANS: A PTS: 1 REF: Descriptive Statistics 28. The measure of variability that reflects the difference between the largest and smallest scores in a distribution is the: a. standard deviation. b. sum of squares. c. variance. d. range. ANS: D PTS: 1 REF: Descriptive Statistics 29. The average of the squared deviations from the mean is called: a. variance. b. sum of squares. c. range. d. standard deviation. ANS: A PTS: 1 REF: Descriptive Statistics 30. If two sets of data have the same mean and the same range, they will also have: a. the same variance. b. the same standard deviation. c. the same sum of squares. DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 90

d. none of these ANS: D PTS: 1 REF: Descriptive Statistics 31. If a group of scores are tightly clustered around the mean, they will have a: a. small median. b. small variance. c. small N. d. all of these ANS: B PTS: 1 REF: Descriptive Statistics 32. Professor Jones has given a statistics test to her class. After grading the tests she reports to the class that the mean score was a 99 and the standard deviation was 0. The class will be very happy because: a. Only a few students failed the test. b. Most people did moderately well on the test. c. Everyone got a 99. d. Most people scored around 99. ANS: C PTS: 1 REF: Descriptive Statistics MSC: WWW 33. SS/N is called the: a. sum of all squares. b. standard deviation. c. variance. d. mean. ANS: C PTS: 1 REF: Descriptive Statistics 34. The square root of the variance is called the: a. squared deviation. b. average squared deviation. c. sum of the squared deviations. d. standard deviation. ANS: D PTS: 1 REF: Descriptive Statistics 35. When graphing measures of central tendency from experimental data, you place the ____ on the x -axis. a. independent variable b. dependent variable c. measure of the central tendency d. responses of the research participants ANS: A PTS: 1 REF: Pictorial Presentations of Numerical Data CHAPTER 5 91

42. In a set of scores with a mean of 100 and a standard deviation of 5, what is the z score associated with a raw score of 95? a.  5 b. 5 c.  1 d. 1 ANS: C PTS: 1 REF: Standard Scores 43. If your z score on a test is 0, your score is: a. the worst in the class. b. average. c. above average. d. below average. ANS: B PTS: 1 REF: Standard Scores 44. You have a study partner who says that she has an IQ of 130. Given that the mean IQ is 100 and the standard deviation is 15, what would you conclude about your partner? a. She's pretty smart. b. She's above average in intelligence. c. Her intelligence z score is 2.00. d. All of these. ANS: D PTS: 1 REF: Standard Scores MSC: WWW 45. A transformation that allows all scores to be positive with a mean of 50 and a standard deviation of 10 is called a: a. T score. b. z score. c. F score. d. r score. ANS: A PTS: 1 REF: Standard Scores 46. The correlation coefficient is a value between: a.  1.0 and +1.0. b. the mean and standard deviation. c. low score and high score. d. .0 and +1.0. ANS: A PTS: 1 REF: Measure of Association CHAPTER 5 93

47. One of the most common correlation coefficients is called: a. range. b. t score. c. Pearson product moment correlation coefficient. d. standard correlation. ANS: C PTS: 1 REF: Box 5.1 48. If the correlation between two variables is  .80, then: a. low scores on one variable are associated with high scores on the other variable. b. low scores one variable are associated with low scores on the other variable. c. high scores on one variable are associated with high scores on the other variable. d. there is little relation between the two variables. ANS: A PTS: 1 REF: Measure of Association 49. When you construct a scatter diagram, the variable that you plot on the y -axis is: a. the independent variable. b. the dependent variable. c. either variable. d. the frequency of response. ANS: C PTS: 1 REF: Measure of Association 50. The appropriate graph for a correlation is a: a. frequency polygon. b. scatter diagram. c. line graph. d. bar graph. ANS: B PTS: 1 REF: Measure of Association 51. A scatter diagram that contains data points which are tightly clustered on a path that goes from the bottom left to the upper right is likely to depict a ____ correlation. a. weak negative b. strong positive c. weak positive d. strong negative ANS: B PTS: 1 REF: Measure of Association 52. If the correlation between age and memory is  .30, how much of the variance in memory is accounted for by age? a. .09% b. .9% c. 9% d. 90% ANS: C PTS: 1 REF: Measure of Association 53. If two variables are not related in any way, the coefficient of correlation must be: a.  1 b. 1 c. 0 DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 94

ANS: Answer not provided. PTS: 1 REF: Scales of Measurement 2. What does it mean to say, "statistics do not know and do not care where numbers come from"? ANS: Answer not provided. PTS: 1 REF: Measurement and Statistics 3. Draw and describe a bimodal distribution, positively skewed distribution, and negatively skewed distribution. ANS: Answer not provided. PTS: 1 REF: Pictorial Description of Frequency Information 4. What is the most commonly used measure of central tendency? Under what conditions is it inappropriate to use this measure? Why? ANS: Answer not provided. PTS: 1 REF: Descriptive Statistics 5. Why is the variance a preferred measure of variability to the range? ANS: Answer not provided. PTS: 1 REF: Descriptive Statistics 6. Show how two distributions with the same mean can have different measures of variability. In which of the two distributions is the mean a "better" descriptor? Why? ANS: Answer not provided. PTS: 1 REF: Descriptive Statistics 7. Describe why it might be appropriate to transform data. ANS: Answer not provided. PTS: 1 REF: Transforming Data 8. Graph the following results of an experiment: Mean weight loss for individuals on Diet A was 20 lbs., on Diet B was 30 lbs., and on Diet C was 40 lbs. ANS: Answer not provided. PTS: 1 REF: Pictorial Presentations of Numerical Data 9. How will the shape of a frequency distribution change as variability changes? ANS: Answer not provided. PTS: 1 REF: Pictorial Presentations of Numerical Data 10. What advantage do standard scores give a researcher? ANS: Answer not provided. PTS: 1 REF: Standard Scores DESCRIPTION OF BEHAVIOR THROUGH NUMERICAL REPRESENTATION 96

11. Suppose that you were told that the mean score on this test was 72 with a standard deviation of 4 and that your z -score was  .5. What was your actual score on this test? ANS: Answer not provided. PTS: 1 REF: Measure of Association 12. Suppose the correlation between depression and memory loss was  .50. Draw a rough scatter diagram of this relation. How would you interpret this correlation? How much of the variance in memory may be accounted for by depression? How much is accounted for by something other than depression? ANS: Answer not provided. PTS: 1 REF: Measure of Association 13. Why would a researcher select regression to analyze data? ANS: Answer not provided. PTS: 1 REF: Measure of Association CHAPTER 5 97