Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

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2 Pearson Education Limited Edinburgh Gate Harlow Essex CM2 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 214 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 1 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 1: ISBN 13: British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

3 The easiest person to deceive is one s own self. Edward Bulwer-Lytton Another issue can arise when graphs use nonlinear scales, meaning scales in which each increment does not always represent the same change in value. Consider Figure 31a, which shows how the speeds of the fastest computers have increased with time. At first glance, it appears that speeds have been increasing linearly. For example, it might look as if the speed increased by the same amount from 199 to 2 as it did from 195 to 196. However, if you look more closely, you ll see that each tick mark on the vertical scale represents a tenfold increase in speed. Now we see that computer speed grew from about 1 to 1 calculations per second from 195 to 196 and from about 1 million to 1 billion calculations per second between 199 and 2. This type of scale is called an exponential scale (or logarithmic scale) because it grows by powers of 1 and powers of 1 are exponents. (For example, 3 is the exponent in 1 3 = 1,.) It is always possible to convert an exponential scale back to an ordinary linear scale as shown in Figure 31b. However, comparing the two graphs should make clear why the exponential scale is so useful in this case: The exponential scale clearly shows the rapid gains in computer speeds, while the ordinary scale makes it impossible to see any detail in the early years shown on the graph. More generally, exponential scales are useful whenever data vary over a huge range of values. by THE WAy Computer Speed In 1965, Intel founder Gordon E. Moore predicted that advances in technology would allow computer chips to double in power roughly every two years. This principle is now called Moore s law, and it has held fairly true ever since Moore first stated it. Calculations per second 1,,, 1 1,, 1, 1 Billions of calculations per second Figure 31 Both graphs show the same data, but the graph on the left uses an exponential scale. Time UT To Think Based on Figure 31a, can you predict the speed of the fastest computers in 22? Could you make the same prediction with Figure 31b? Explain. ase study Asteroid Threat Asteroids and comets occasionally hit the Earth. Small ones tend to burn up in the atmosphere or create small craters on impact, but larger ones can cause substantial devastation. About 65 million years ago, an asteroid about 1 kilometers in diameter hit the Earth, leaving a 2-kilometer-wide crater on the coast of the Yucatan peninsula in Mexico. Scientists estimate that this impact caused the extinction of about three-quarters of all species living on Earth at the time, including all the dinosaurs. Clearly, a similar impact would be bad news for our civilization. We might therefore want to understand the likelihood of such an event. Figure 32 shows a graph relating the size of impacting asteroids and comets to the frequency with which such objects hit the Earth. Because of the wide range of sizes and time scales involved, both axes on this graph are exponential. The horizontal axis shows impactor (asteroid or comet) sizes, with each tick representing a power of 1. The vertical axis shows the frequency of impact; moving up on the vertical axis corresponds to more frequent events. With this double exponential graph, we can see trends clearly. For example, small objects of about 1 meter 116

1 hour by THE WAy Typical time between impacts 1 day 1 year 1 years 1, years 1 million years 1 million years Atmospheric explosion or small crater Tsunamis, widespread devastation, climate change

4 1 hour by THE WAy Typical time between impacts 1 day 1 year 1 years 1, years 1 million years 1 million years Atmospheric explosion or small crater Tsunamis, widespread devastation, climate change Mass extinction More than 15 impact craters have been identified around the Earth. One of the best preserved is Meteor Crater in Arizona (photo below). It formed about 5, years ago when an asteroid about 5 meters across crashed to Earth, releasing energy equivalent to that of a 2-megaton hydrogen bomb and leaving a crater more than a kilometer across and 2 meters deep. Once in Earth s history 1 m 1 m 1 m 1 km 1 km 1 km Impactor size (diameter) Figure 32 This graph shows how the frequency of impacts and the magnitude of their effects depends on the size of the impactor. Note that smaller impacts are much more frequent than larger ones. in size strike the Earth every day, but cause little damage. At the other extreme, objects large enough to cause a mass extinction hit only about once every hundred million years. The intermediate cases are probably the most worrisome. The graph indicates that objects that could cause widespread devastation such as wiping out the population of a large city can be expected as often as once every thousand years. This is often enough to warrant at least some preventive action. Currently, astronomers are trying to make more precise predictions about when an object might hit the Earth. If they discover an object that will hit the Earth, scientists will need to find a way to deflect it to prevent the impact. Percentage Change Graphs Is college getting more or less expensive? If you didn t look too carefully, Figure 33a might lead you to conclude that after peaking in the early 2s, the cost of public colleges fell during the rest of that decade. But look more closely and you ll see that the vertical axis on this graph represents the percentage change in costs. The drop-off therefore means only that costs rose by smaller amounts, not that they fell. Actual college costs are shown in Figure 33b, which makes it clear that they rose every year. Graphs that show percentage change are very common; you ll find them in the financial news almost every day. But as you can see, they can be very misleading if you don t realize that they are showing change rather than actual values. Changes in College Costs Actual College Costs 15% $3, Percentage change from previous academic year 12% 9% 6% 3% Private Public $25, $2, $15, $1, $5, Public Four-Year Private Four-Year (a) Figure 33 Trends in college costs: (a) annual percent change; (b) actual costs. Source: The College Board (b) Get your facts first, and then you can distort them as much as you please. Mark Twain 117

5 Pictographs Visual Displays of Data Pictographs are graphs embellished with additional artwork. The artwork may make the graph more appealing, but it can also distract or mislead. Figure 34 is a pictograph showing the rise in world population from 184 to 24 (numbers for future years are based on United Nations intermediate-case projections). The lengths of the bars correctly correspond to world population for the years listed. However, the artistic embellishments of this graph are deceptive in several ways. For example, your eye may be drawn to the figures of people lining the globe. Because this line of people rises from the left side of the pictograph to the center and then falls, it might give the impression that future world population will decline. In fact, the line of people is purely decorative and carries no information. The more serious problem with this pictograph is that it makes it appear that world population has been rising linearly. However, notice that the time intervals on the horizontal axis are not the same in each case. For example, the interval between the bars for 1 billion and 2 billion people is 123 years (from 184 to 1927), but the interval between the bars for 5 billion and 6 billion people is only 12 years (from 1987 to 1999). Pictographs are very common. As this example shows, however, you have to study them carefully to extract the essential information and not be distracted by the cosmetic effects. by THE WAy Demographers often characterize population growth by a doubling time the time it takes the population to double. During the late 2th century, the doubling time for the human population was about 4 years. If the population continued to double at this rate, world population would reach more than 3 billion by 21 and 19 billion by 22. By about 265, the human population would be so large that it would not fit on the Earth, even if everyone stood elbow to elbow everywhere. World Population (in billions of people) Figure 34 A pictograph of world population. The pictures add visual impact, but can also be misleading. Also notice that the horizontal scale (time) is not linear. Source: United Nations Population Division, future projections based on intermediate case assumptions. section 4 Exercises statistical literacy and Critical Thinking 1. Exaggerating a Difference. Under standard test conditions, the braking distance for a Honda Civic is 136 feet and the braking distance for a VW Jetta is 137 feet. Is that difference meaningful? How could a graph be constructed so that the difference is greatly exaggerated? 2. Graph of Populations. When constructing a graph showing the population of the United States and the population of Mexico, an illustrator draws two different people with heights proportional to the populations. Identify a way in which the graph might be misleading? What is the general name for such graphs that use drawings of people or objects? 3. Vertical scale. A line chart has a vertical scale with values of, 1, 1, 1, 1 and 1,. What is the name of such a scale? What is an advantage of such a scale? 4. sugar Cubes. To show how sugar production doubled from 199 to now, an illustrator draws two sugar cubes. The first cube is drawn with a length of 1 cm on each side and the second cube is drawn with a length of 2 cm on each side. What are the volumes of the two sugar cubes? Is the illustration misleading? If so, how? 118

Concepts and Applications 5. Car Mileage. Figure 35 shows the highway fuel consumption (mi/gal) for the Chevrolet Aveo and the Honda Civic. How is the graph misleading?

6 Concepts and Applications 5. Car Mileage. Figure 35 shows the highway fuel consumption (mi/gal) for the Chevrolet Aveo and the Honda Civic. How is the graph misleading? How could it be drawn so that it is not misleading? Highway fuel consumption (mi/gal) Figure 35 Car Mileage. Chevrolet Honda Civic 6. Comparing state Populations. Figure 36 depicts 21 populations for California, Texas, and New York (based on data from the 21 Census). How is this graph misleading? How could it be drawn so that it is not misleading? 9. Three-Dimensional Pies. The pie charts in Figure 38 give the percentage of Americans in three age categories in 199 and 25 (projected). a. Consider the 199 age distribution. The actual percentages for the three categories for 199 were 87.5% (others), 11.3% (6 84), and 1.2% (85+). Does the pie chart show these values accurately? Explain. b. Consider the 25 age distribution. The actual percentages for the three categories for 25 were 8.% (others), 15.4% (6 84), and 4.6% (8 5 +). Does the pie chart show these values accurately? Explain. c. Using the actual percentages given in parts a and b, draw flat (two-dimensional) pie charts to display these data. Explain why these pie charts give a more accurate picture than the three-dimensional pies. d. Comment on the general trends shown in the two pie charts. 199 Age Distribution 25 Age Distribution California Others Others Texas New York Figure 38 Source: U.S. Census Bureau. Figure 36 State Populations Population (millions) 7. Pictograph. Figure 37 depicts the amounts of daily oil consumption in the United States and Japan. Does the illustration accurately depict the data? Why or why not? Daily Oil Consumption (millions of barrels) 1. Moore s law. In 1965, Intel cofounder Gordon Moore initiated what has since become known as Moore s law: the number of transistors per square inch on integrated circuits will double approximately every 18 months. In the table below, the first row lists different years and the second row lists the number of transistors (in thousands) for different years. year Transistors year Transistors , , , , ,6, Figure 37 United States Japan 8. Pictograph. Refer to Figure 37 used in Exercise 7 and construct a bar chart to depict the same data in a way that is fair and objective. a. Construct a time-series graph of these data, using a uniform scale on both axes. b. Make an exponential graph of these data in which the subdivisions on the vertical axis are, 1, 1, 1, 1,, 1,, 1,, 1,,, and 1,,. c. Compare the graphs in parts a and b. 119

7 11. Percentage Change in the CPi. The graph in Figure 39 shows the percentage change in the CPI over recent years. In what year (of the years displayed) was the change in the CPI the greatest? What happened in 29? How do actual prices in 21 compare to those in 199? Based on this graph, what can you conclude about changes in prices during the period shown? Percent change from previous year Percent Change in CPI Year Figure 39 Source: U.S. Bureau of Labor Statistics. 12. seasonal Effects on schizophrenia? The graph in Figure 4 shows data regarding the relative risk of schizophrenia among people born in different months. purchasing power, which is adjusted for inflation with 1996 used as the reference year. The graph represents the years from 1955 to 211. Summarize what the graph shows. Minimum Wage and Its Purchasing Power $8 $7 Purchasing power $6 $5 $4 $3 $2 $1 Minimum wage Year Figure 41 Source: U.S. Department of Labor. 14. Double Horizontal scale. The graph in Figure 42 shows simultaneously the number of births in this country during two time periods: and When did the first baby boom peak? When did the second baby boom peak? Why do you think the designer of this display chose to superimpose the two time intervals, rather than use a single time scale from 1946 through 1994? Baby Boomers and Their Babies Relative risk January February March April May June July August Month of birth Figure 4 Source: New England Journal of Medicine. September October November December a. Note that the scale of the vertical axis does not include zero. Sketch the same risk curve using an axis that includes zero. Comment on the effect of this change. b. Each value of the relative risk is shown with a dot at its most likely value and with an error bar indicating the range in which the data value probably lies. The study concludes that the risk was also significantly associated with the season of birth. Given the size of the error bars, does this claim appear justified? (Is it possible to draw a flat line that passes through all of the error bars?) 13. Constant Dollars. The graph in Figure 41 shows the minimum wage in the United States, together with its 12 The baby-boom generation, born between 1946 and 1964, produced their own smaller boom between 1977 and Number of U.S. Births, 1946 through 1964 and 1977 through 1994 (in millions) First Baby Boom Second Baby Boom Year Figure 42 Source: Based on data from the National Center for Health Statistics. PRojECTs FoR THE internet & beyond 15. usa snapshot. USA Today offers a daily pictograph for its Snapshot. Find a snapshot from a recent issue of USA Today. Briefly discuss its purpose and effectiveness. 16. image search. Choose some topic that interests you for which you think that good statistical graphs should be available. Do an image search for the topic in Google, Bing, or other search engine. Does the search give you what you were looking for? Briefly discuss the value of the search results in terms of what you hoped to find.

Introductory Statistics Neil A. Weiss Ninth Edition

Introductory Statistics Neil A. Weiss Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: