Experimental Design and Ethics Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as the influence of alcohol? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data. The purpose of an experiment is to investigate the relationship between two variables. When one variable causes change in another, we call the first variable the explanatory variable . The affected variable is called the response variable . In a randomized experiment, the researcher manipulates values of the explanatory variable and measures the resulting changes in the response variable. The different values of the explanatory variable are called treatments . An experimental unit is a single object or individual to be measured. You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements, choosing not to smoke. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention. Additional variables that can cloud a study are called lurking variables . In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments randomly, all of the potential lurking variables are spread equally among the groups. At this point, the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables. The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing drugs, researchers noted: Results showed that believing one had taken the substance resulted in [ performance ] times almost as fast as those associated with consuming the drug itself. In contrast, taking the drug without knowledge yielded no significant performance increment. McClung, M. Collins, D. “Because I know it will!”: placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology. 2007 Jun. 29(3):382-94. Web. April 30, 2013. When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group . This group is given a placebo treatment–an active treatment that cannot directly influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill which contains no actual medication, then the power of suggestion is no longer a factor. Blinding or masking in a randomized experiment preserves the power of suggestion. When a person involved in a research study is blinded, they do not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded. Researchers want to investigate whether taking aspirin regularly reduces the risk of heart attack. Four hundred people between the ages of 50 and 84 are recruited as participants. The people are divided randomly into two groups: one group will take aspirin, and the other group will take a placebo. Each person takes one pill each day for three years, but they don't know whether they are taking aspirin or the placebo. At the end of the study, researchers count the number of people in each group who have had heart attacks. Identify the following values for this study: population, sample, experimental units, explanatory variable, response variable, treatments. The population is people aged 50 to 84. The sample is the 400 people who participated. The experimental units are the individual people in the study. The explanatory variable is oral medication. The treatments are aspirin and a placebo. The response variable is whether a subject had a heart attack. Try It A study needs to be conducted of the effect of three medicines A, B, and C on the height of adults aged 30 to 45. 90 adults were selected randomly and divided into three equal groups. The first group was asked to take medicine A for 6 months. The second group was asked to take medicine B for 6 months. The third group was asked to take medicine C for 6 months. The average change in height in each group is calculated at the end of the study. Identify the following values for this study: population, sample, experimental units, explanatory variables, response variable, treatments. The population is adults aged 30 to 45. The sample is 90 adults that were selected randomly. The experimental units are the individual adults in the study. The explanatory variable is the medicines. The treatments are medicines A, B, and C. The response variable is the average change in height in the group. The Smell & Taste Treatment and Research Foundation conducted a study to investigate whether smell can affect learning. Subjects completed mazes multiple times while wearing masks. They completed the pencil and paper mazes three times wearing floral-scented masks, and three times with unscented masks. Participants were assigned at random to wear the floral mask during the first three trials or during the last three trials. For each trial, researchers recorded the time it took to complete the maze and the subject’s impression of the mask’s scent: positive, negative, or neutral. Describe the explanatory and response variables in this study. What are the treatments? Identify any lurking variables that could interfere with this study. Is it possible to use blinding in this study? The explanatory variable is scent, and the response variable is the time it takes to complete the maze. There are two treatments: a floral-scented mask and an unscented mask. All subjects experienced both treatments. The order of treatments was randomly assigned so there were no differences between the treatment groups. Random assignment eliminates the problem of lurking variables. Subjects will clearly know whether they can smell flowers or not, so subjects cannot be blinded in this study. Researchers timing the mazes can be blinded, though. The researcher who is observing a subject will not know which mask is being worn. Try It The Placebo Research Group conducted a study to find the extent of placebo effects. A group of men randomly selected were asked to take a test before and after taking a pill that induces a mild headache. The pill in half of the randomly selected men was replaced with a similar pill that has no effect. For each trial, researchers recorded the change in time men took to complete the tests before and after taking the pill. Describe the explanatory and response variable in this study. What are the treatments? Identify any lurking variables that could interfere with this study. Is it possible to use blinding in this study? The explanatory variable is the pill, and the response variable is the change in time taken between the two tests. There are two treatments: a pill that induces a mild headache and a pill that has no effect. All subjects have been given pills. The men for kind of pill intake are randomly selected. Random assignment eliminates the problem of lurking variables. Subjects do not know which pill they are taking, so they are blinded in this study. The researchers know which subject is taking which pill, so they cannot be blinded in this study. A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment? The explanatory variable is birth order. You cannot randomly assign a person’s birth order. Random assignment eliminates the impact of lurking variables. When you cannot assign subjects to treatment groups at random, there will be differences between the groups other than the explanatory variable. Try It You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes? Describe the explanatory and response variables in the study. What are the treatments? What should you consider when selecting participants? Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not? Identify any lurking variables that could interfere with this study. How can blinding be used in this study? Ethics The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do. An investigation of famous social psychologist, Diederik Stapel, has led to the retraction of his articles from some of the world’s top journals including Journal of Experimental Social Psychology, Social Psychology, Basic and Applied Social Psychology, British Journal of Social Psychology, and the magazine Science . Diederik Stapel is a former professor at Tilburg University in the Netherlands. An extensive investigation involving three universities where Stapel has worked concluded that the psychologist is guilty of fraud on a colossal scale. Falsified data taints over 55 papers he authored and 10 Ph.D. dissertations that he supervised. Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct sexy results that journals found attractive. “It was a quest for aesthetics, for beauty—instead of the truth,” he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud, like a junkie seeking a bigger and better high. Yudhijit Bhattacharjee, “The Mind of a Con Man,” Magazine, New York Times, April 26, 2013. Available online at: http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?src=dayp&_r=2& (accessed May 1, 2013). The committee investigating Stapel concluded that he is guilty of several practices including: creating datasets, which largely confirmed the prior expectations, altering data in existing datasets, changing measuring instruments without reporting the change, and misrepresenting the number of experimental subjects. Clearly, it is never acceptable to falsify data the way this researcher did. Sometimes, however, violations of ethics are not as easy to spot. Researchers have a responsibility to verify that proper methods are being followed. The report describing the investigation of Stapel’s fraud states that, “statistical flaws frequently revealed a lack of familiarity with elementary statistics.” “Flawed Science: The Fraudulent Research Practices of Social Psychologist Diederik Stapel,” Tillburg University, November 28, 2012, http://www.tilburguniversity.edu/upload/064a10cd-bce5-4385-b9ff-05b840caeae6_120695_Rapp_nov_2012_UK_web.pdf (accessed May 1, 2013). Many of Stapel’s co-authors should have spotted irregularities in his data. Unfortunately, they did not know very much about statistical analysis, and they simply trusted that he was collecting and reporting data properly. Many types of statistical fraud are difficult to spot. Some researchers simply stop collecting data once they have just enough to prove what they had hoped to prove. They don’t want to take the chance that a more extensive study would complicate their lives by producing data contradicting their hypothesis. Professional organizations, like the American Statistical Association, clearly define expectations for researchers. There are even laws in the federal code about the use of research data. When a statistical study uses human participants, as in medical studies, both ethics and the law dictate that researchers should be mindful of the safety of their research subjects. The U.S. Department of Health and Human Services oversees federal regulations of research studies with the aim of protecting participants. When a university or other research institution engages in research, it must ensure the safety of all human subjects. For this reason, research institutions establish oversight committees known as Institutional Review Boards (IRB) . All planned studies must be approved in advance by the IRB. Key protections that are mandated by law include the following: Risks to participants must be minimized and reasonable with respect to projected benefits. Participants must give informed consent . This means that the risks of participation must be clearly explained to the subjects of the study. Subjects must consent in writing, and researchers are required to keep documentation of their consent. Data collected from individuals must be guarded carefully to protect their privacy. These ideas may seem fundamental, but they can be very difficult to verify in practice. Is removing a participant’s name from the data record sufficient to protect privacy? Perhaps the person’s identity could be discovered from the data that remains. What happens if the study does not proceed as planned and risks arise that were not anticipated? When is informed consent really necessary? Suppose your doctor wants a blood sample to check your cholesterol level. Once the sample has been tested, you expect the lab to dispose of the remaining blood. At that point the blood becomes biological waste. Does a researcher have the right to take it for use in a study? It is important that students of statistics take time to consider the ethical questions that arise in statistical studies. How prevalent is fraud in statistical studies? You might be surprised—and disappointed. There is a website dedicated to cataloging retractions of study articles that have been proven fraudulent. A quick glance will show that the misuse of statistics is a bigger problem than most people realize. Vigilance against fraud requires knowledge. Learning the basic theory of statistics will empower you to analyze statistical studies critically. Describe the unethical behavior in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A researcher is collecting data in a community. The researcher selects a block where they are comfortable walking because they know many of the people living on the street. No one seems to be home at four houses on the route. They do not record the addresses and do not return at a later time to try to find residents at home. The researcher skips four houses on the route because they are running late for an appointment. When they get home, they fill in the forms by selecting random answers from other residents in the neighborhood. By selecting a convenient sample, the researcher is intentionally selecting a sample that could be biased. Claiming that this sample represents the community is misleading. The researcher needs to select areas in the community at random. Intentionally omitting relevant data will create bias in the sample. Suppose the researcher is gathering information about jobs and child care. By ignoring people who are not home, They may be missing data from working families that are relevant to her study. They need to make every effort to interview all members of the target sample. It is never acceptable to fake data. Even though the responses the researcher are “real” responses provided by other participants, the duplication is fraudulent and can create bias in the data. They researcher needs to work diligently to interview everyone on their route. Try It Describe the unethical behavior, if any, in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A study is commissioned to determine the favorite brand of fruit juice among teens in California. The survey is commissioned by the seller of a popular brand of apple juice. There are only two types of juice included in the study: apple juice and cranberry juice. Researchers allow participants to see the brand of juice as samples are poured for a taste test. Twenty-five percent of participants prefer Brand X, 33% prefer Brand Y and 42% have no preference between the two brands. Brand X references the study in a commercial saying “Most teens like Brand X as much as or more than Brand Y.” References “Vitamin E and Health,” Nutrition Source, Harvard School of Public Health, http://www.hsph.harvard.edu/nutritionsource/vitamin-e/ (accessed May 1, 2013). Stan Reents. “Don’t Underestimate the Power of Suggestion,” athleteinme.com, http://www.athleteinme.com/ArticleView.aspx?id=1053 (accessed May 1, 2013). Ankita Mehta. “Daily Dose of Aspiring Helps Reduce Heart Attacks: Study,” International Business Times, July 21, 2011. Also available online at http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacks-study-300443 (accessed May 1, 2013). The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html (accessed May 1, 2013). M.L. Jacskon et al., “Cognitive Components of Simulated Driving Performance: Sleep Loss effect and Predictors,” Accident Analysis and Prevention Journal, Jan no. 50 (2013), http://www.ncbi.nlm.nih.gov/pubmed/22721550 (accessed May 1, 2013). “Earthquake Information by Year,” U.S. Geological Survey. http://earthquake.usgs.gov/earthquakes/eqarchives/year/ (accessed May 1, 2013). “Fatality Analysis Report Systems (FARS) Encyclopedia,” National Highway Traffic and Safety Administration. http://www-fars.nhtsa.dot.gov/Main/index.aspx (accessed May 1, 2013). Data from www.businessweek.com (accessed May 1, 2013). Data from www.forbes.com (accessed May 1, 2013). “America’s Best Small Companies,” http://www.forbes.com/best-small-companies/list/ (accessed May 1, 2013). U.S. Department of Health and Human Services, Code of Federal Regulations Title 45 Public Welfare Department of Health and Human Services Part 46 Protection of Human Subjects revised January 15, 2009. Section 46.111:Criteria for IRB Approval of Research. “April 2013 Air Travel Consumer Report,” U.S. Department of Transportation, April 11 (2013), http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report (accessed May 1, 2013). Lori Alden, “Statistics can be Misleading,” econoclass.com, http://www.econoclass.com/misleadingstats.html (accessed May 1, 2013). Maria de los A. Medina, “Ethics in Statistics,” Based on “Building an Ethics Module for Business, Science, and Engineering Students” by Jose A. Cruz-Cruz and William Frey, Connexions, http://cnx.org/content/m15555/latest/ (accessed May 1, 2013). Chapter Review A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. To counter the influence of the power of suggestion in the study, participants in the control group receive an inactive placebo treatment that looks exactly like the active treatments, but cannot directly influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due to the influence of the explanatory variable. “An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.” (Andrew Gelman, “Open Data and Open Methods,” Ethics and Statistics, http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf (accessed May 1, 2013).) Ethical violations in statistics are not always easy to spot. Professional associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis. Design an experiment. Identify the explanatory and response variables. Describe the population being studied and the experimental units. Explain the treatments that will be used and how they will be assigned to the experimental units. Describe how blinding and placebos may be used to counter the power of suggestion. Discuss potential violations of the rule requiring informed consent. People in a correctional facility are offered good behavior credit in return for participation in a study. A research study is designed to investigate a new children’s allergy medication. Participants in a study are told that the new medication being tested is highly promising, but they are not told that only a small portion of participants will receive the new medication. Others will receive placebo treatments and traditional treatments. People who are incarcerated may not feel comfortable refusing participation, or may feel obligated to take advantage of the promised benefits. They may not feel truly free to refuse participation. Parents can provide consent on behalf of their children, but children are not competent to provide consent for themselves. All risks and benefits must be clearly outlined. Study participants must be informed of relevant aspects of the study in order to give appropriate consent. HOMEWORK How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation. Use key terms from this module to describe the design of this experiment. Explanatory variable: amount of sleep Response variable: performance measured in assigned tasks Treatments: normal sleep and 27 hours of total sleep deprivation Experimental Units: 19 professional drivers Lurking variables: none – all drivers participated in both treatments Random assignment: treatments were assigned in random order; this eliminated the effect of any “learning” that may take place during the first experimental session Control/Placebo: completing the experimental session under normal sleep conditions Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time An advertisement for Acme Investments displays the two graphs in to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected? As the graphs show, Acme consistently outperforms the Other Guys! The graph in shows the number of complaints for six different airlines as reported to the US Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints? You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading. Explanatory Variable the independent variable in an experiment; the value controlled by researchers Treatments different values or components of the explanatory variable applied in an experiment Response Variable the dependent variable in an experiment; the value that is measured for change at the end of an experiment Experimental Unit any individual or object to be measured Lurking Variable a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable Random Assignment the act of organizing experimental units into treatment groups using random methods Control Group a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups Informed Consent Any human subject in a research study must be cognizant of any risks or costs associated with the study. The subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits. Consent must be given freely by an informed, fit participant. Institutional Review Board a committee tasked with oversight of research programs that involve human subjects Placebo an inactive treatment that cannot directly affect the response variable, used to counter the power of suggestion Blinding not telling participants which treatment a subject is receiving Double-blind experiment an experiment in which both the subjects of an experiment and the researchers who work with the subjects are blinded

Math


Data Collection Experiment Data Collection Experiment Class Time: Names: Student Learning Outcomes The student will demonstrate the systematic sampling technique. The student will construct relative frequency tables. The student will interpret results and their differences from different data groupings. Movie Survey Ask five classmates from a different class how many movies they saw at the theater last month. Do not include rented movies. Record the data. In class, randomly pick one person. On the class list, mark that person’s name. Move down four names on the class list. Mark that person’s name. Continue doing this until you have marked 12 names. You may need to go back to the start of the list. For each marked name record the five data values. You now have a total of 60 data values. For each name marked, record the data. Order the Data Complete the two relative frequency tables below using your class data. Frequency of Number of Movies Viewed Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0 1 2 3 4 5 6 7+ Frequency of Number of Movies Viewed Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0–1 2–3 4–5 6–7+ Using the tables, find the percent of data that is at most two. Which table did you use and why? Using the tables, find the percent of data that is at most three. Which table did you use and why? Using the tables, find the percent of data that is more than two. Which table did you use and why? Using the tables, find the percent of data that is more than three. Which table did you use and why? Discussion Questions Is one of the tables “more correct” than the other? Why or why not? In general, how could you group the data differently? Are there any advantages to either way of grouping the data? Why did you switch between tables, if you did, when answering the question above?

Math


Sampling Experiment Sampling Experiment Class Time: Names: Student Learning Outcomes The student will demonstrate the simple random, systematic, stratified, and cluster sampling techniques. The student will explain the details of each procedure used. In this lab, you will be asked to pick several random samples of restaurants. In each case, describe your procedure briefly, including how you might have used the random number generator, and then list the restaurants in the sample you obtained. NOTE The following section contains restaurants stratified by city into columns and grouped horizontally by entree cost (clusters). Restaurants Used In Sample Restaurants Stratified by City and Entree Cost Entree Cost Under $10 $10 to under $15 $15 to under $20 Over $20 San Jose El Abuelo Taq, Pasta Mia, Emma’s Express, Bamboo Hut Emperor’s Guard, Creekside Inn Agenda, Gervais, Miro’s Blake’s, Eulipia, Hayes Mansion, Germania Palo Alto Senor Taco, Olive Garden, Taxi’s Ming’s, P.A. Joe’s, Stickney’s Scott’s Seafood, Poolside Grill, Fish Market Sundance Mine, Maddalena’s, Spago’s Los Gatos Mary’s Patio, Mount Everest, Sweet Pea’s, Andele Taqueria Lindsey’s, Willow Street Toll House Charter House, La Maison Du Cafe Mountain View Maharaja, New Ma’s, Thai-Rific, Garden Fresh Amber Indian, La Fiesta, Fiesta del Mar, Dawit Austin’s, Shiva’s, Mazeh Le Petit Bistro Cupertino Hobees, Hung Fu, Samrat, Panda Express Santa Barb. Grill, Mand. Gourmet, Bombay Oven, Kathmandu West Fontana’s, Blue Pheasant Hamasushi, Helios Sunnyvale Chekijababi, Taj India, Full Throttle, Tia Juana, Lemon Grass Pacific Fresh, Charley Brown’s, Cafe Cameroon, Faz, Aruba’s Lion & Compass, The Palace, Beau Sejour Santa Clara Rangoli, Armadillo Willy’s, Thai Pepper, Pasand Arthur’s, Katie’s Cafe, Pedro’s, La Galleria Birk’s, Truya Sushi, Valley Plaza Lakeside, Mariani’s A Simple Random Sample Pick a simple random sample of 15 restaurants. Describe your procedure. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ A Systematic Sample Pick a systematic sample of 15 restaurants. Describe your procedure. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ A Stratified Sample Pick a stratified sample , by city, of 20 restaurants. Use 25% of the restaurants from each stratum. Round to the nearest whole number. Describe your procedure. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ A Stratified Sample Pick a stratified sample , by entree cost, of 21 restaurants. Use 25% of the restaurants from each stratum. Round to the nearest whole number. Describe your procedure. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ 21. __________ A Cluster Sample Pick a cluster sample of restaurants from two cities. The number of restaurants will vary. Describe your procedure. Complete the table with your sample. 1. ________ 6. ________ 11. ________ 16. ________ 21. ________ 2. ________ 7. ________ 12. ________ 17. ________ 22. ________ 3. ________ 8. ________ 13. ________ 18. ________ 23. ________ 4. ________ 9. ________ 14. ________ 19. ________ 24. ________ 5. ________ 10. ________ 15. ________ 20. ________ 25. ________

Math


Introduction When you have large amounts of data, you will need to organize it in a way that makes sense. These ballots from an election are rolled together with similar ballots to keep them organized. (credit: modification of work “HELMAND PROVINCE, Afghanistan (Aug. 21, 2009) Voting ballots organized and arranged for counting by Afghan presidential election workers at a local school in the Nawa District.” by U.S. Marine Corps photo by Staff Sgt. William Greeson/ Wikimedia Commons, Public Domain) Chapter Objectives By the end of this chapter, the student should be able to: Display data graphically and interpret graphs: stemplots, histograms, and box plots. Recognize, describe, and calculate the measures of location of data: quartiles and percentiles. Recognize, describe, and calculate the measures of the center of data: mean, median, and mode. Recognize, describe, and calculate the measures of the spread of data: variance, standard deviation, and range. Once you have collected data, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data. In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called "Descriptive Statistics." You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs. A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Newspapers and the Internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then, more formal tools may be applied. Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. NOTE This book contains instructions for constructing a histogram and a box plot for the TI-83+ and TI-84 calculators. The Texas Instruments (TI) website provides additional instructions for using these calculators.

Math


Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs One simple graph, the stem-and-leaf graph or stemplot , comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit . For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem. For Professor Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest): 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 Stem-and-Leaf Graph Stem Leaf 3 3 4 2 9 9 5 3 5 5 6 1 3 7 8 8 9 9 7 2 3 4 8 8 0 3 8 8 8 9 0 2 4 4 4 4 6 10 0 The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% ( 8 31 ) were in the 90s or 100, a fairly high number of As. Try It For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest): 32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61 Construct a stem plot for the data. The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later. The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data: 1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3 Do the data seem to have any concentration of values? NOTE The leaves are to the right of the decimal. The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers. Stem Leaf 1 1 5 2 3 5 7 3 2 3 3 5 8 4 0 2 5 5 7 8 5 5 6 6 5 7 7 8 9 10 11 12 3 Try It The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers: 0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0 A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. and show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data. Presidential Ages at Inauguration President Age President Age President Age Washington 57 Lincoln 52 Hoover 54 J. Adams 61 A. Johnson 56 F. Roosevelt 51 Jefferson 57 Grant 46 Truman 60 Madison 57 Hayes 54 Eisenhower 62 Monroe 58 Garfield 49 Kennedy 43 J. Q. Adams 57 Arthur 51 L. Johnson 55 Jackson 61 Cleveland 47 Nixon 56 Van Buren 54 B. Harrison 55 Ford 61 W. H. Harrison 68 Cleveland 55 Carter 52 Tyler 51 McKinley 54 Reagan 69 Polk 49 T. Roosevelt 42 G.H.W. Bush 64 Taylor 64 Taft 51 Clinton 47 Fillmore 50 Wilson 56 G. W. Bush 54 Pierce 48 Harding 55 Obama 47 Buchanan 65 Coolidge 51 Presidential Age at Death President Age President Age President Age Washington 67 Lincoln 56 Hoover 90 J. Adams 90 A. Johnson 66 F. Roosevelt 63 Jefferson 83 Grant 63 Truman 88 Madison 85 Hayes 70 Eisenhower 78 Monroe 73 Garfield 49 Kennedy 46 J. Q. Adams 80 Arthur 56 L. Johnson 64 Jackson 78 Cleveland 71 Nixon 81 Van Buren 79 B. Harrison 67 Ford 93 W. H. Harrison 68 Cleveland 71 Reagan 93 Tyler 71 McKinley 58 Polk 53 T. Roosevelt 60 Taylor 65 Taft 72 Fillmore 74 Wilson 67 Pierce 64 Harding 57 Buchanan 77 Coolidge 60 Ages at Inauguration Ages at Death 9 9 8 7 7 7 6 3 2 4 6 9 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 2 1 1 1 1 1 0 5 3 6 6 7 7 8 9 8 5 4 4 2 1 1 1 0 6 0 0 3 3 4 4 5 6 7 7 7 8 7 0 1 1 1 2 3 4 7 8 8 9 8 0 1 3 5 8 9 0 0 3 3 The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stemand-leaf plot of these wins and losses. Losses Wins Season Losses Wins Season 34 48 1 41 41 22 34 48 2 39 43 23 46 36 3 44 38 24 46 36 4 39 43 25 36 46 5 25 57 26 47 35 6 40 42 27 51 31 7 36 46 28 53 29 8 26 56 29 51 31 9 32 50 30 41 41 10 19 31 31 36 46 11 54 28 32 32 50 12 57 25 33 51 31 13 49 33 34 40 42 14 47 35 35 39 43 15 54 28 36 42 40 16 69 13 37 48 34 17 56 26 38 32 50 18 52 30 39 25 57 19 45 37 40 32 50 20 35 47 41 30 52 21 29 53 42 Another type of graph that is useful for specific data values is a line graph . In the particular line graph shown in , the x -axis (horizontal axis) consists of data values and the y -axis (vertical axis) consists of frequency points . The frequency points are connected using line segments. In a survey, 40 parents were asked how many times per week a teenager must be reminded to do their chores. The results are shown in and in . Number of times teenager is reminded Frequency 0 2 1 5 2 8 3 14 4 7 5 4 Try It In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in . Construct a line graph. Number of times in shop Frequency 0 7 1 10 2 14 3 9 Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in has age groups represented on the x -axis and proportions on the y -axis . The percentage of U.S.-based TikTok users by age is shown in . Construct a bar graph using this data. Age groups Proportion (%) of TikTok users 10–19 32.5% 20–29 29.5% 30–39 16.4% 40–49 13.9% 50+ 7.1% Try It The population in Park City is made up of children, working-age adults, and retirees. shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Construct a bar graph showing the proportions. Age groups Number of people Proportion of population Children 67,059 19% Working-age adults 152,198 43% Retirees 131,662 38% The columns in show the projected data for the year 2030 for the number and percentages of high school graduates by geographic region in the United States. Create a bar graph for this data with the geographic region (qualitative data) on the x -axis and the percentage of high school data (quantitative data) on the y -axis. Region Number of Graduates Percentage of Graduates Northeast 517,720 16.1% Midwest 695,170 21.6% South 1,253,540 39.0% West 749,400 23.3% Try It Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district. District Registered voter population Overall city population 1 15.5% 19.4% 2 12.2% 15.6% 3 9.8% 9.0% 4 17.4% 18.5% 5 22.8% 20.7% 6 22.3% 16.8% References Doyle, Brandon. "TikTok Statistics – Updated Mar 2023." Wallaroo. March 21, 2023. https://wallaroomedia.com/blog/social-media/tiktok-statistics/. “Table 219.20. Public High School Graduates, by Region, State, and Jurisdiction: Selected Years, 1980-81 through 2030-31.” National Center for Education Statistics. Accessed May 24, 2023. https://nces.ed.gov/programs/digest/d21/tables/dt21_219.20.asp. “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013). Chapter Review A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends. That is, finding a general pattern in data sets including temperature, sales, employment, company profit or cost over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs). Bar graphs are especially useful when categorical data is being used. For each of the following data sets, create a stem plot and identify any outliers. The miles per gallon rating for 30 cars are shown below (lowest to highest). 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43 Stem Leaf 1 9 9 9 2 0 1 1 5 5 5 6 6 8 9 3 1 1 2 2 3 4 5 6 7 7 8 8 8 8 4 1 3 3 The height in feet of 25 trees is shown below (lowest to highest). 25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54 The data are the prices of different laptops at an electronics store. Round each value to the nearest ten. 249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610 Stem Leaf 2 5 5 6 7 7 8 3 0 0 1 2 3 3 5 5 5 7 7 9 4 1 6 9 5 6 7 7 6 1 The data are daily high temperatures in a town for one month. 61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95 For the next three exercises, use the data to construct a line graph. In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in . Number of times in store Frequency 1 4 2 10 3 16 4 6 5 4 In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in . Years since last purchase Frequency 0 2 1 8 2 13 3 22 4 16 5 9 Several children were asked how many TV shows they watch each day. The results of the survey are shown in . Number of TV Shows Frequency 0 12 1 18 2 36 3 7 4 2 The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students. Seasons Number of students Proportion of population Spring 8 24% Summer 9 26% Autumn 11 32% Winter 6 18% Using the data from Ms. Ramirez’s math class supplied in , construct a bar graph showing the percentages. David County has six high schools. Each school sent students to participate in a county-wide science competition. shows the percentage breakdown of competitors from each school, and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school. High School Science competition population Overall student population Alabaster 28.9% 8.6% Concordia 7.6% 23.2% Genoa 12.1% 15.0% Mocksville 18.5% 14.3% Tynneson 24.2% 10.1% West End 8.7% 28.8% Use the data from the David County science competition supplied in . Construct a bar graph that shows the county-wide population percentage of students at each school. Homework Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99 Construct a stem-and-leaf plot of the data. Are there any potential outliers? If so, which scores are they? Why do you consider them outliers? contains the percentages of adults who were never married in the 50 U.S. states and Washington, DC. State Percent (%) State Percent (%) State Percent (%) Alabama 32.2 Kentucky 31.3 North Dakota 27.2 Alaska 24.5 Louisiana 31.0 Ohio 29.2 Arizona 24.3 Maine 26.8 Oklahoma 30.4 Arkansas 30.1 Maryland 27.1 Oregon 26.8 California 24.0 Massachusetts 23.0 Pennsylvania 28.6 Colorado 21.0 Michigan 30.9 Rhode Island 25.5 Connecticut 22.5 Minnesota 24.8 South Carolina 31.5 Delaware 28.0 Mississippi 34.0 South Dakota 27.3 Washington, DC 22.2 Missouri 30.5 Tennessee 30.8 Florida 26.6 Montana 23.0 Texas 31.0 Georgia 29.6 Nebraska 26.9 Utah 22.5 Hawaii 22.7 Nevada 22.4 Vermont 23.2 Idaho 26.5 New Hampshire 25.0 Virginia 26.0 Illinois 28.2 New Jersey 23.8 Washington 25.5 Indiana 29.6 New Mexico 25.1 West Virginia 32.5 Iowa 28.4 New York 23.9 Wisconsin 26.3 Kansas 29.4 North Carolina 27.8 Wyoming 25.1 Use a random number generator to randomly pick eight states. Construct a bar graph of the rates of those eight states. Construct a bar graph for all the states beginning with the letter "A." Construct a bar graph for all the states beginning with the letter "M." Example solution for using the random number generator for the TI-84+ to generate a simple random sample of 8 states. Instructions are as follows. Number the entries in the table 1–51 (Includes Washington, DC; Numbered vertically) Press MATH Arrow over to PRB Press 5:randInt( Enter 51,1,8) Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}. Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

Math


Histograms, Frequency Polygons, and Time Series Graphs For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of 100 values or more. A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If: f = frequency n = total number of data values (or the sum of the individual frequencies), and RF = relative frequency, then: RF = f n For example, if three students in an English class of 40 students received from 90% to 100%, then, f = 3, n = 40, and RF = f n = 3 40 = 0.075. 7.5% of the students received 90–100%. 90–100% are quantitative measures. To construct a histogram , first decide how many bars or intervals , also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A convenient starting point is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is 6.1 and this is the smallest value, a convenient starting point is 6.05 (6.1 – 0.05 = 6.05). We say that 6.05 has more precision. If the value with the most decimal places is 2.23 and the lowest value is 1.5, a convenient starting point is 1.495 (1.5 – 0.005 = 1.495). If the value with the most decimal places is 3.234 and the lowest value is 1.0, a convenient starting point is 0.9995 (1.0 – 0.0005 = 0.9995). If all the data happen to be integers and the smallest value is two, then a convenient starting point is 1.5 (2 – 0.5 = 1.5). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data. The following data are the heights (in inches to the nearest half inch) of 100 semiprofessional soccer players. The heights are continuous data, since height is measured. 60; 60.5; 61; 61; 61.5 63.5; 63.5; 63.5 64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5 66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5 68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5 70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71 72; 72; 72; 72.5; 72.5; 73; 73.5 74 The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point. 60 – 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95. The largest value is 74, so 74 + 0.05 = 74.05 is the ending value. Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars. 74.05 - 59.95 = 14.1 14.1 ÷ 8 = 1.76 NOTE We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the number of bars or class intervals is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals. The boundaries are: 59.95 59.95 + 2 = 61.95 61.95 + 2 = 63.95 63.95 + 2 = 65.95 65.95 + 2 = 67.95 67.95 + 2 = 69.95 69.95 + 2 = 71.95 71.95 + 2 = 73.95 73.95 + 2 = 75.95 The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95. The following histogram displays the heights on the x -axis and relative frequency on the y -axis. Try It The following data are the shoe sizes of 50 students. The sizes are discrete data since shoe size is measured in whole and half units only. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars. 9; 9; 9.5; 9.5; 10; 10; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5; 10.5 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5; 11.5 12; 12; 12; 12; 12; 12; 12; 12.5; 12.5; 12.5; 12.5; 14 Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data , since books are counted. 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3 4; 4; 4; 4; 4; 4 5; 5; 5; 5; 5 6; 6 Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books. Because the data are integers, subtract 0.5 from 1, the smallest data value and add 0.5 to 6, the largest data value. Then the starting point is 0.5 and the ending value is 6.5. Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers 1, 2, 3, 4, 5, 6, and the starting point is 0.5, a width of one places the 1 in the middle of the interval from 0.5 to 1.5, the 2 in the middle of the interval from 1.5 to 2.5, the 3 in the middle of the interval from 2.5 to 3.5, the 4 in the middle of the interval from _______ to _______, the 5 in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ . 3.5 to 4.5 4.5 to 5.5 6 5.5 to 6.5 Calculate the number of bars as follows: 6.5 - 0.5 = 6 6 ÷ 1 = 6 where 1 is the width of a bar. Therefore, bars = 6. The following histogram displays the number of books on the x -axis and the frequency on the y -axis. Go to . There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for . Press Y=. Press CLEAR to delete any equations. Press STAT 1:EDIT . If L1 has data in it, arrow up into the name L1 , press CLEAR and then arrow down. If necessary, do the same for L2 . Into L1 , enter 1, 2, 3, 4, 5, 6. Into L2 , enter 11, 10, 16, 6, 5, 2. Press WINDOW. Set Xmin = .5 , Xmax = 6.5 , Xscl = (6.5 – .5)/6 , Ymin = –1 , Ymax = 20 , Yscl = 1 , Xres = 1 . Press 2 nd Y=. Start by pressing 4:Plotsoff ENTER. Press 2 nd Y=. Press 1:Plot1 . Press ENTER. Arrow down to TYPE. Arrow to the 3 rd picture (histogram). Press ENTER. Arrow down to Xlist: Enter L1 (2 nd 1). Arrow down to Freq. Enter L2 (2 nd 2). Press GRAPH. Use the TRACE key and the arrow keys to examine the histogram. Try It The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted. 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2; 2 3; 3; 3; 3; 3; 3; 3; 3 20 student athletes play one sport. 22 student athletes play two sports. Eight student athletes play three sports. Fill in the blanks for the following sentence. Since the data consist of the numbers 1, 2, 3, and the starting point is 0.5, a width of one places the 1 in the middle of the interval 0.5 to _____, the 2 in the middle of the interval from _____ to _____, and the 3 in the middle of the interval from _____ to _____. Using this data set, construct a histogram. Number of Hours My Classmates Spent Playing Video Games on Weekends 9.95 10 2.25 16.75 0 19.5 22.5 7.5 15 12.75 5.5 11 10 20.75 17.5 23 21.9 24 23.75 18 20 15 22.9 18.8 20.5 Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram. Try It The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram. 22 35 15 26 40 28 18 20 25 34 39 42 24 22 19 27 22 34 40 20 38 and 28 Use 10–19 as the first interval. Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals. Frequency Polygons Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the x -axis and y -axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them. A frequency polygon was constructed from the frequency table below. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 5 5 59.5 69.5 10 15 69.5 79.5 30 45 79.5 89.5 40 85 89.5 99.5 15 100 The first label on the x -axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the x -axis. The point labeled 54.5 represents the next interval, or the first “real” interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the x -axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side. Try It Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in . Age at Inauguration Frequency 41.5–46.5 4 46.5–51.5 11 51.5–56.5 14 56.5–61.5 9 61.5–66.5 4 66.5–71.5 2 Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets. We will construct an overlay frequency polygon comparing the scores from with the students’ final numeric grade. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 5 5 59.5 69.5 10 15 69.5 79.5 30 45 79.5 89.5 40 85 89.5 99.5 15 100 Frequency Distribution for Calculus Final Grades Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 10 10 59.5 69.5 10 20 69.5 79.5 30 50 79.5 89.5 45 95 89.5 99.5 5 100 Try It We will construct an overlay frequency polygon comparing the scores from with the students’ final test scores in algebra. Frequency Distribution for Algebra Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 10 10 59.5 69.5 5 15 69.5 79.5 40 55 79.5 89.5 35 90 89.5 99.5 10 100 Calculus Test Scores v Algebra Test Scores Image Insert Constructing a Time Series Graph Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected. One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don‘t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph. To construct a time series graph, we must look at both pieces of our paired data set . We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur. The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only. Year Jan Feb Mar Apr May Jun Jul 1 181.7 183.1 184.2 183.8 183.5 183.7 183.9 2 185.2 186.2 187.4 188.0 189.1 189.7 189.4 3 190.7 191.8 193.3 194.6 194.4 194.5 195.4 4 198.3 198.7 199.8 201.5 202.5 202.9 203.5 5 202.416 203.499 205.352 206.686 207.949 208.352 208.299 6 211.080 211.693 213.528 214.823 216.632 218.815 219.964 7 211.143 212.193 212.709 213.240 213.856 215.693 215.351 8 216.687 216.741 217.631 218.009 218.178 217.965 218.011 9 220.223 221.309 223.467 224.906 225.964 225.722 225.922 10 226.665 227.663 229.392 230.085 229.815 229.478 229.104 Year Aug Sep Oct Nov Dec Annual 1 184.6 185.2 185.0 184.5 184.3 184.0 2 189.5 189.9 190.9 191.0 190.3 188.9 3 196.4 198.8 199.2 197.6 196.8 195.3 4 203.9 202.9 201.8 201.5 201.8 201.6 5 207.917 208.490 208.936 210.177 210.036 207.342 6 219.086 218.783 216.573 212.425 210.228 215.303 7 215.834 215.969 216.177 216.330 215.949 214.537 8 218.312 218.439 218.711 218.803 219.179 218.056 9 226.545 226.889 226.421 226.230 225.672 224.939 10 230.379 231.407 231.317 230.221 229.601 229.594 Try It The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO 2 emissions for the United States. CO2 Emissions Year Ukraine United Kingdom United States 1 352,259 540,640 5,681,664 2 343,121 540,409 5,790,761 3 339,029 541,990 5,826,394 4 327,797 542,045 5,737,615 5 328,357 528,631 5,828,697 6 323,657 522,247 5,656,839 7 272,176 474,579 5,299,563 Uses of a Time Series Graph Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot. How NOT to Lie with Statistics It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a "sense" of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book How to Lie with Statistics . It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently. Again, the goal is to enlighten with visuals that tell the story of the data. Pie charts have a number of common problems when used to convey the message of the data. Too many pieces of the pie overwhelm the reader. More than perhaps five or six categories ought to give an idea of the relative importance of each piece. This is after all the goal of a pie chart, what subset matters most relative to the others. If there are more components than this then perhaps an alternative approach would be better or perhaps some can be consolidated into an "other" category. Pie charts cannot show changes over time, although we see this attempted all too often. In federal, state, and city finance documents pie charts are often presented to show the components of revenue available to the governing body for appropriation: income tax, sales tax motor vehicle taxes and so on. In and of itself this is interesting information and can be nicely done with a pie chart. The error occurs when two years are set side-by-side. Because the total revenues change year to year, but the size of the pie is fixed, no real information is provided and the relative size of each piece of the pie cannot be meaningfully compared. Histograms can be very helpful in understanding the data. Properly presented, they can be a quick visual way to present probabilities of different categories by the simple visual of comparing relative areas in each category. Here the error, purposeful or not, is to vary the width of the categories. This of course makes comparison to the other categories impossible. It does embellish the importance of the category with the expanded width because it has a greater area, inappropriately, and thus visually "says" that that category has a higher probability of occurrence. Time series graphs perhaps are the most abused. A plot of some variable across time should never be presented on axes that change part way across the page either in the vertical or horizontal dimension. Perhaps the time frame is changed from years to months. Perhaps this is to save space or because monthly data was not available for early years. In either case this confounds the presentation and destroys any value of the graph. If this is not done to purposefully confuse the reader, then it certainly is either lazy or sloppy work. Changing the units of measurement of the axis can smooth out a drop or accentuate one. If you want to show large changes, then measure the variable in small units, penny rather than thousands of dollars. And of course to continue the fraud, be sure that the axis does not begin at zero, zero. If it begins at zero, zero, then it becomes apparent that the axis has been manipulated. Perhaps you have a client that is concerned with the volatility of the portfolio you manage. An easy way to present the data is to use long time periods on the time series graph. Use months or better, quarters rather than daily or weekly data. If that doesn't get the volatility down then spread the time axis relative to the rate of return or portfolio valuation axis. If you want to show "quick" dramatic growth, then shrink the time axis. Any positive growth will show visually "high" growth rates. Do note that if the growth is negative then this trick will show the portfolio is collapsing at a dramatic rate. Again, the goal of descriptive statistics is to convey meaningful visuals that tell the story of the data. Purposeful manipulation is fraud and unethical at the worst, but even at its best, making these type of errors will lead to confusion on the part of the analysis. References Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker “Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013). “Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013). “Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013). “Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013). “CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013). “Births Time Series Data.” General Register Office For Scotland, 2013. Available online at http://www.gro-scotland.gov.uk/statistics/theme/vital-events/births/time-series.html (accessed April 3, 2013). “Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013). Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach . CRC Press: 1980. “Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013). Chapter Review A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually goes on y -axis with the frequency being graphed on the x -axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time. Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table. Data Value (# cars) Frequency Relative Frequency Cumulative Relative Frequency What does the frequency column in sum to? Why? 65 What does the relative frequency column in sum to? Why? What is the difference between relative frequency and frequency for each data value in ? The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value. What is the difference between cumulative relative frequency and relative frequency for each data value? To construct the histogram for the data in , determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling. Answers will vary. One possible histogram is shown: Construct a frequency polygon for the following: Pulse Rates for Females Frequency 60–69 12 70–79 14 80–89 11 90–99 1 100–109 1 110–119 0 120–129 1 Actual Speed in a 30 MPH Zone Frequency 42–45 25 46–49 14 50–53 7 54–57 3 58–61 1 Tar (mg) in Nonfiltered Cigarettes Frequency 10–13 1 14–17 0 18–21 15 22–25 7 26–29 2 Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger. Depth of Hunger Frequency 230–259 21 260–289 13 290–319 5 320–349 7 350–379 1 380–409 1 410–439 1 Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values. Use the two frequency tables to compare the life expectancy of males and females from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of females compared to males? Life Expectancy at Birth – Females Frequency 49–55 3 56–62 3 63–69 1 70–76 3 77–83 8 84–90 2 Life Expectancy at Birth – Males Frequency 49–55 3 56–62 3 63–69 1 70–76 1 77–83 7 84–90 5 Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births. Sex/Year 1855 1856 1857 1858 1859 1860 1861 Female 45,545 49,582 50,257 50,324 51,915 51,220 52,403 Male 47,804 52,239 53,158 53,694 54,628 54,409 54,606 Total 93,349 101,821 103,415 104,018 106,543 105,629 107,009 Sex/Year 1862 1863 1864 1865 1866 1867 1868 1869 Female 51,812 53,115 54,959 54,850 55,307 55,527 56,292 55,033 Male 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321 Total 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354 Sex/Year 1870 1871 1872 1873 1874 1875 Female 56,431 56,099 57,472 58,233 60,109 60,146 Male 58,959 60,029 61,293 61,467 63,602 63,432 Total 115,390 116,128 118,765 119,700 123,711 123,578 The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for a city during the period from 1961 to 1973. Year 1961 1962 1963 1964 1965 1966 1967 Police 260.35 269.8 272.04 272.96 272.51 261.34 268.89 Homicides 8.6 8.9 8.52 8.89 13.07 14.57 21.36 Year 1968 1969 1970 1971 1972 1973 Police 295.99 319.87 341.43 356.59 376.69 390.19 Homicides 28.03 31.49 37.39 46.26 47.24 52.33 Construct a double time series graph using a common x -axis for both sets of data. Which variable increased the fastest? Explain. Did the city's increase in police officers have an impact on the murder rate? Explain. Homework Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows: Publisher A # of books Freq. Rel. Freq. 0 10 1 12 2 16 3 12 4 8 5 6 6 2 8 2 Publisher B # of books Freq. Rel. Freq. 0 18 1 24 2 24 3 22 4 15 5 10 7 5 9 1 Publisher C # of books Freq. Rel. Freq. 0–1 20 2–3 35 4–5 12 6–7 2 8–9 1 Find the relative frequencies for each survey. Write them in the charts. Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not? Make new histograms for Publisher A and Publisher B. This time, make bar widths of two. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer. Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group. Singles Amount($) Frequency Rel. Frequency 51–100 5 101–150 10 151–200 15 201–250 15 251–300 10 301–350 5 Couples Amount($) Frequency Rel. Frequency 100–150 5 201–250 5 251–300 5 301–350 5 351–400 10 401–450 10 451–500 10 501–550 10 551–600 5 601–650 5 Fill in the relative frequency for each group. Construct a histogram for the singles group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis. Construct a histogram for the couples group. Scale the x -axis by $50 widths. Use relative frequency on the y -axis. Compare the two graphs: List two similarities between the graphs. List two differences between the graphs. Overall, are the graphs more similar or different? Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x -axis by $50, scale it by $100. Use relative frequency on the y -axis. Compare the graph for the singles with the new graph for the couples: List two similarities between the graphs. Overall, are the graphs more similar or different? How did scaling the couples graph differently change the way you compared it to the singles graph? Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences. Singles Amount($) Frequency Relative Frequency 51–100 5 0.08 101–150 10 0.17 151–200 15 0.25 201–250 15 0.25 251–300 10 0.17 301–350 5 0.08 Couples Amount($) Frequency Relative Frequency 100–150 5 0.07 201–250 5 0.07 251–300 5 0.07 301–350 5 0.07 351–400 10 0.14 401–450 10 0.14 451–500 10 0.14 501–550 10 0.14 551–600 5 0.07 601–650 5 0.07 See and . In the following histogram data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where both boundary values are included). In the following histogram, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where values on both boundaries are included). Compare the two graphs: Answers may vary. Possible answers include: Both graphs have a single peak. Both graphs use class intervals with width equal to $50. Answers may vary. Possible answers include: The couples graph has a class interval with no values. It takes almost twice as many class intervals to display the data for couples. Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same. Answers may vary. Compare the graph for the Singles with the new graph for the Couples: Both graphs have a single peak. Both graphs display 6 class intervals. Both graphs show the same general pattern. Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different. Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison. Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows. # of movies Frequency Relative Frequency Cumulative Relative Frequency 0 5 1 9 2 6 3 4 4 1 Construct a histogram of the data. Complete the columns of the chart. Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each. The percentage of people who own at most three T-shirts costing more than $19 each is approximately: 21 59 41 Cannot be determined c If the data were collected by asking the first 111 people who entered the store, then the type of sampling is: cluster simple random stratified convenience Following are the rates of unmarried adults for the 50 U.S. states and Washington, DC. State Percent (%) State Percent (%) State Percent (%) Alabama 32.2 Kentucky 31.3 North Dakota 27.2 Alaska 24.5 Louisiana 31.0 Ohio 29.2 Arizona 24.3 Maine 26.8 Oklahoma 30.4 Arkansas 30.1 Maryland 27.1 Oregon 26.8 California 24.0 Massachusetts 23.0 Pennsylvania 28.6 Colorado 21.0 Michigan 30.9 Rhode Island 25.5 Connecticut 22.5 Minnesota 24.8 South Carolina 31.5 Delaware 28.0 Mississippi 34.0 South Dakota 27.3 Washington, DC 22.2 Missouri 30.5 Tennessee 30.8 Florida 26.6 Montana 23.0 Texas 31.0 Georgia 29.6 Nebraska 26.9 Utah 22.5 Hawaii 22.7 Nevada 22.4 Vermont 23.2 Idaho 26.5 New Hampshire 25.0 Virginia 26.0 Illinois 28.2 New Jersey 23.8 Washington 25.5 Indiana 29.6 New Mexico 25.1 West Virginia 32.5 Iowa 28.4 New York 23.9 Wisconsin 26.3 Kansas 29.4 North Carolina 27.8 Wyoming 25.1 Construct a bar graph of the unmarried adult rates of your state and the four states closest to your state. Hint: Label the x -axis with the states. Answers will vary. Frequency the number of times a value of the data occurs Histogram a graphical representation in x - y form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency. The graph consists of contiguous rectangles. Relative Frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes

Math