The Numerate Leader E-Book

Thomas A. King

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 382

Veröffentlichungsjahr: 2021

Beschreibung

Learn how to make informed decisions through statistical reasoning! Using a qualitative approach to introduce statistical reasoning, The Numerate Leader: How to Pull Game-Changing Insights from Statistical Data is a cutting-edge book that helps the reader extract information from unfamiliar data sets. Combining introductory statistics with a few ideas from the philosophy of science, this work helps generalists find patterns that may be expected to recur in the future. Identifying one or two such relationships can be a game-changer for the reader and their employer or client. Thomas A. King's revelatory writing is easy to understand and conversational in tone. King makes the complex, tedious topics that you studied in the classroom--but likely didn't yet understand--easily comprehensible. Historical examples and humorous anecdotes illuminate technical concepts so that readers may pull insights from data sets and then explain conclusions reached through effective storytelling. What's more, the book is fun to read. A natural teacher, King emphasizes that complex software is unnecessary for success in this field. Readers, however, will find: * Real-life examples that help put statistical concepts into an understandable context * A glossary of important statistical terms and their use * An appendix detailing ten math facts numerate people should know Perfect for undergraduate and graduate students entering advanced data analytics courses, as well as data analysts and c-suite executives just starting out, The Numerate Leader is key in helping develop the skills to identify provisional relationships between disparate data sets and then assess the significance of conclusions reached.

Leseprobe

The Numerate Leader

How to Pull Game-Changing Insights from Statistical Data

Thomas A. King

Copyright © 2022 by Thomas A. King. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the Web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at www.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at [phone], outside the United States at [phone], or fax [phone].

Wiley publishes in a variety of print and electronic formats and by print-on-demand. Some material included with standard print versions of this book may not be included in e-books or in print-on-demand. If this book refers to media such as a CD or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com. For more information about Wiley products, visit www.wiley.com.

Library of Congress Cataloging-in-Publication Data

Names: King, Thomas A., 1960- author. | John Wiley & Sons, Inc., publisher.

Title: The numerate leader : how to pull game-changing insights from statistical data / Thomas A. King.

Description: Hoboken, New Jersey : Wiley, [2022] | Includes bibliographical references and index.

Identifiers: LCCN 2021027855 (print) | LCCN 2021027856 (ebook) | ISBN 9781119843283 (cloth) | ISBN 9781119843306 (adobe pdf) | ISBN 9781119843290 (epub)

Subjects: LCSH: Statistical literacy. | Commercial statistics.

Classification: LCC QA276 .K483 2021 (print) | LCC QA276 (ebook) | DDC 001.4/22—dc23

LC record available at https://lccn.loc.gov/2021027855

LC ebook record available at https://lccn.loc.gov/2021027856

Cover Image: Edmond Halley: ©GeorgiosArt/Getty Images

Comet: ©ZU_09/Getty Images

Postage Stamp: ©IkonStudio/Getty Images

Cover Design: Wiley

PREFACE

Since you've cracked open this book, permit me a few guesses. In your educational journey, you have taken at least one introductory statistics class. That course involved associating types of story problems with as many formulas. Success meant matching formulas to stories and then plugging and chugging to arrive at so-called answers that could be measured to decimal places.

You then prepared for a final exam by cramming facts into short-term memory. You arrived at the testing location at the appointed hour, attacked assigned questions, submitted the completed test, celebrated with friends, and then cleared your mind to gird for the next task on that semester's to-do list. Today, any effort to recall lessons learned yields little beyond hazy memories.

This story played out four times in my academic travels. I studied introductory statistics in high school, college, a master's program designed to teach accounting to liberal arts students, and then an MBA program. Receiving decent grades from each trip to the well, I viewed myself as quantitatively literate and accepted an insurance job with confidence. The Greeks had another word for this trait – hubris, outrageous arrogance, which brings misfortune to those so afflicted.

My assignment was simple: review historical car accident data to set future prices. If prices were too high, we wouldn't sell any insurance; if too low, resulting claims costs would swamp any premiums collected and bring financial loss. Should pricing actions bring profitable growth, I would be rewarded; failure, encouraged to pursue other career opportunities.

I stumbled through early assignments with a track record unblemished by success. Time spent in the classroom offered little benefit when faced with messy, real-world data. Sufficiently self-aware to recognize that I did not know what I was doing, I sought help from more informed coworkers.

The Beatles sang how they got by with a little help from their friends. Support offered by numerate colleagues salvaged my career. My mentors invested time to teach me how to think about the basic concepts discussed in this book.

I was then able to make a decent living applying these tools to reveal patterns buried in unfamiliar data sets, a skill rewarded by labor markets. Evidence of success was the ability to pay off our mortgage, send three kids to private colleges, and then secure my wife's permission to become an underpaid university professor at middle age.

More interestingly, Peter Lewis, Progressive's leader for much of my career there, set up a process to hire and train dozens of people like me. Analytic skills unleashed by this business model helped our company, one of many in a fragmented, mature industry, grow organically from obscurity into a Fortune 500 firm. In this process, Peter earned his way onto the Forbes 400, a compilation of the richest people in the world. Using this one data point, I infer that numerate organizations create wealth.

I now teach accounting to undergraduates, graduate students, and executives. Conversations with our alumni have convinced me that the power of statistical reasoning lies not in fancy tools or sophisticated software. Rather, informed decision-making results from the ability to remember and apply a few basic ideas. This book is a (mostly) qualitative introduction to these ideas.

My thesis is that anyone armed with a basic understanding of the ideas that follow may surface information from unfamiliar data sets. Turning data into information allows ordinary people to make material contributions to their employers and society.

This book is designed to raise the statistical game of a generalist who works on an organization's front line. The target reader is too busy with budget deadlines, screaming customers, supply chain problems, and human resource mishaps to learn exotic math.

To borrow a phrase from political adviser James Carville, It's the ideas, stupid. This book is not about which buttons to push when firing up a computer. All software discussions provided within these pages are limited to a few functions in Microsoft Excel. This is the book to read before enrolling in advanced data analytics classes.

Numerate employees who use these tools identify provisional relationships. We then invite statistical experts, whom I respectfully call propeller heads, to kick the tires and determine whether proposed ideas have merit. If experience is any guide, identifying just one or two insights that stand up to scrutiny will be a game-changer for the reader's career and their employer's financial prospects.

The core skill is finding patterns that may be expected to recur in the future. The recipe discussed in the following pages blends concepts you learned in introductory statistics laced with a few ideas from the philosophy of science. You have already learned – but likely don't yet understand – what's covered in the pages that follow. What is new is how the material is explained. My hope is that my storytelling skills will allow you to put previously learned material to much more productive use.

As a warning, I am not a math person. Statisticians reading this book will quickly tick off justifiable criticisms about incompleteness and lack of rigor. My defense is that nontechnical explanations are a necessary step to help generalists uncover interesting questions that permit propeller heads to work their magic.

Academics teach doctoral students to write in dry, serious prose. I have deliberately chosen to ignore my scholarly training and write in the first person with a conversational voice. Doing so, I hope, will make statistics less scary and spark conversations that will make the world a better place. Numerate people seek to be approximately right rather than precisely wrong.

Writing a book is more of a journey than a task. An African proverb says that if one wishes to travel quickly, then go alone; if one wishes to travel far, then go together. My journey followed the second approach, and I'm too embarrassed to share how much time went into the creation of this slim volume.

Any effort to thank all those who contributed to my journey would rival a bad Oscars' acceptance speech or, even worse, the drudgery of the Iliad's catalogue of ships. Instead, I simply express gratitude to the people of The Prairie School, Twin Disc, Harvard College, New York University, Arthur Andersen, Harvard Business School, Progressive Insurance, and Case Western Reserve University. Colleagues and mentors at these organizations invested countless hours in my intellectual development.

I do wish to acknowledge my paternal grandfather, who encouraged me to write and stressed the importance of an active voice, strong verbs, and the measured use of prepositions, and my father, who taught me that there is never a single, correct answer to any significant management problem.

A special shout-out goes to my academic mentor, Gary Previts, who gave me the opportunity to sample the academic life. My students and colleagues at the Weatherhead School of Management pushed me to refine my thinking with patience and good humor. Finally, whatever success I have achieved is due to the boundless support given by wife, best friend, and life partner, Yvonne.

Responsibility for errors and omissions in the following pages rests squarely on my shoulders.

Chagrin Falls, Ohio

May 2021

1NUMERACY

Our journey begins with the story of Edmond Halley (1656–1742). Son of a wealthy soap maker, he studied mathematics at Oxford and became a respected astronomer. The question of how gravitational force influences the shape of celestial orbits led him to travel to Trinity College, Cambridge, to seek help from the reclusive Isaac Newton.

Upon their meeting, Halley recognized the scope of Newton's astonishing genius. The young astronomer pressed his hero to publish thoughts in a book that came to be titled Mathematical Principles of Natural Philosophy, which has my vote for the most significant scientific book ever written. Among other things, Newton demonstrated that the sun holds a planet in an elliptical orbit.

Halley then used this principle to review past observations of comets. These bright, fleeting objects have attracted attention since biblical times. The Bayeux Tapestry shows a comet streaking through the sky before the Norman conquest of England in 1066. Halley reasoned that comets follow the same laws of motion as other celestial objects.

Armed with Newton's inverse-square law, Halley studied records of past comet observations and attempted to infer orbital periods. He concluded that a comet he observed in 1682 was the same object identified in the tapestry and then seen by Peter Apian in 1531 and Johannes Kepler in 1607. In 1705, Halley predicted that this object would return to view in 1758.

Sixteen years after Halley's death, stargazers turned to the skies to search for the returning comet. On Christmas Day in 1758, an astronomer in Germany spotted the object that would become known as Halley's Comet.

This anecdote illustrates numeracy in action. Halley identified a pattern buried within data and then used this pattern to make a bold prediction outside of the domain of the original data set. This was some trick because the gravitational pulls of Jupiter and Saturn give the comet an irregular orbital period. What's more, this object is visible for just a few weeks out of each orbit. Halley crafted a successful prediction from thin data.

Halley's story leads to the summary idea of this book:

Numeracy is the craft of statistical reasoning.

Permit me some ink to unpack the four key words in this short sentence.

Numeracy. I first came across the term numeracy as a college student, when I read a paper written by the statistician Andrew Ehrenberg (1977). He railed against colleagues at the prestigious Royal Statistical Society who amassed gobs of data and then did little with it. Educated math whizzes, surrounded by rich data sets, often show little skill at extracting meaningful information and then presenting it clearly. A useful statistician – for whom Ehrenberg bestows the label of numerate – identifies a cool “so what” and then presents the finding in a way that communicates it clearly.

I cannot begin to list all the horrifically bad presentations I have endured in my career. Time and again, well-trained people gushed waterfalls of data without providing any meaningful insight. Apparently, the presenters were saying, I worked so hard on this analysis, and I want you to see evidence of how much time was spent on the project. Ugh.

Occasionally, however, I watch a presentation or read a paper where the presenter distills hours of work into a simple summary that builds a bridge between evidence and conclusion. When I'm blessed to receive the work of a numerate person, I feel enormous gratitude for the gift of information delivered in a concise, pleasing manner.

Halley showed numeracy by distilling his work on comets into a breathtakingly simple conclusion: the celestial body in question would again be visible in 1758. Simplicity is a foundational idea of this book.

Craft. A typical U.S. university has a school of arts and sciences. Disciplines taught there, ranging from soft humanities to hard physical sciences, expose students to two broad categories of scholarship. The softer arts encourage expression of individual points of view while the sciences emphasize agreed-upon answers.

No two classics students will draw identical conclusions about the role of Achilles in the Iliad, but every chemistry student should agree on how to balance a reaction equation involving sodium and hydrochloric acid. Scientific statements may be disproven, while those in the humanities may be argued without end.

Falling between arts and sciences are crafts, pursuits requiring a type of thinking rarely taught in higher education. An example of a craft is metalworking. Grades of steel have chemical properties that lend themselves to scientific study. Other factors, however, influence a metalworker's ability to cut a certain piece of metal to meet required specifications.

On a particular day, the air has a certain temperature, humidity, and pressure. The grinding machine is at a certain stage in its maintenance cycle. The steel blanks have idiosyncratic properties associated with the production lot made at a point in time at a particular mill. I doubt that any scientific model will ever be able to incorporate how these and dozens of other variables influence the shaping of metal.

When I was in college, I worked in a factory that made transmission assemblies. An unshipped unit in final assembly needed roller bearings with dimensions specified to a couple ten-thousandths of an inch – a ridiculously tight tolerance in an age before the spread of numerically controlled machines. My job was to deliver steel blanks to a master machinist who was “voluntold” to cut the bearings immediately. A baseball analogy would be asking a player on his scheduled day off to step out of the dugout in the bottom of the ninth inning to hit a home run.

A crowd formed around the machinist, who examined the blanks, inspected the grinding machine, and took note of myriad factors that were lost on me. He set up his equipment and began cutting. A quality control professional used precise calipers to measure the dimensions of the output. The first few bearings failed quality control. The machinist made adjustments and then produced a series of bearings that each met the required size standards. The rest of us watched in awe.

In that moment, this craftsman garnered more respect among colleagues than any investment banker, management consultant, university professor, or business executive I've ever met. None of us present that day could come close to doing what he just did. I considered dropping out of college to become his apprentice.

This machinist – whom I remember as Yoda – worked a craft, a discipline that straddles the domains of art and science. The science of metallurgy informs us of processes used to transform steel. However, the scope of this science is not sufficiently developed to tell us how to handle every situation we may face. At that point, a craftsman blends individual judgment with formal training to accomplish a desired task.

This judgment is not easily codified or documented, hampering the ability of a master to pass along know-how to an apprentice. The master offers coaching, but the apprentice assumes responsibility for finding their own way while learning from the master.

Numeracy is a craft. There is some science embedded in the tools used to reveal patterns buried in data. However, this science is not sufficiently robust to instruct people what to do in all cases. Simply buying a computer loaded with statistical software gets one nowhere fast. Numerate people use their wits to sort through quirks embedded in unfamiliar data sets. An effective craftsman blends school-taught technique with hard-won experience to sort through the problem at hand.

The little science given in this book merely repeats what many other reference books on statistics share. Any decent teacher may explain the dozen or so concepts discussed here. The real magic comes from you using your judgment to apply them to circumstances associated with data sets in your life. The art of numeracy is learned but not taught, and I hope that the storytelling in this book serves as a catalyst to help you cultivate this skill faster than what would have been accomplished if you had not read these pages.

Halley's prediction required blending the science of Newton's inverse-square law with the art of estimating masses and distances of significant bodies within the solar system using incomplete astronomical data available at that time.

Statistics. Statistics, a subset of mathematics, studies how one may make uncertain inferences about a broader (and often unobservable) population from the study of properties of a small sample. A classic example is when your grandmother prepared homemade soup. After combining and heating the ingredients, she undoubtedly stirred the liquid and sipped a spoonful to assess the mixture. She was able to draw conclusions about the entire pot from a small taste.

Statistics may be viewed as the opposite of probability, the study of the likelihood of future events occurring based on known frequency distributions. We know that a fair, flipped coin has a 50% chance of landing as a head. Since fair coin flips are independent events, we may conclude that there is a one-in-four chance that this coin will land as heads in two consecutive trials.

The field of probability arose in the seventeenth century as gamblers sought to understand how much money should be wagered in games of chance. Use of probability theory requires that the objects studied – be they coins, cards, dice, or roulette wheels – have knowable probability distributions. This pursuit was popularized in the 2008 movie 21, where a group of card-counting MIT students used probability theory to make money at blackjack tables at Las Vegas casinos. Unfortunately, most phenomena in our lives are more complicated than games of chance.

Frank Knight, a little-remembered economist, made this distinction by contrasting risk and uncertainty (1921). Risk refers to circumstances where things we study have known probability distributions. As shown in the movie 21, decks of cards meet this standard. In a well-shuffled deck of 52 cards, there is a 1-in-13 chance that the next card drawn will be a queen. Peter Bernstein wrote the classic discussion of how leaders over the years have used tools of probability to bring risk under control (1996).

By contrast, uncertainty represents situations where outcomes have unknown (and perhaps unknowable) frequency distributions. Helping people make sense of uncertain situations is the contribution of this book. Whether a firm should expand into a new market is such a problem. The likelihood of outcomes may be expressed in approximate terms (Boss, there's a decent chance that our product will catch on with French consumers) but almost never as a precise number.

Knight argued that it is difficult to earn sizable profits from situations involving simple risks. Others may perform the same calculations and neutralize any advantage gained from applying probability tools to the problem at hand. However, the ability to tame uncertainty offers the prospect of substantial economic rewards.

Inferences arising from statistical analysis are always uncertain. Our sample may have been too small or not representative of the broader population. Users of statistical analysis must accept that conclusions reached may be completely wrong. The reward for trying, however, is that, with practice, numerate people are better positioned to reap economic profits described by Knight.

Halley demonstrated numeracy by offering a range of dates for his prediction. He did not express his prediction as a point estimate (i.e., a particular day) but instead offered a confidence interval (a year) in which his prediction was expected to be realized. To borrow a line from the movie Love Story, statistics means never having to say you're certain.

Reasoning. Reasoning, the final operative word in our definition, means connecting dots. A well-reasoned argument shows the bridge that links the evidence to the conclusion.

Halley connected the dots by combining deduction (applying the general principle of Newton's inverse-square law of gravitation to the particulars of comets orbiting the sun) and induction (generalizing from specific messy, incomplete observational data) to reach a justifiable conclusion. Halley went even further by expressing his conclusion in such a way that it could be disproven. The fact that the comet returned in the predicted year does not prove that Halley was right but making a prediction that came to pass gave the guy a lot of street cred.

Numeracy is thus the ability to turn raw data into information. Claude Shannon, a polymath working at AT&T's Bell Labs, wrote one of the most significant papers of the twentieth century (1948). Bearing the sterile title “A Mathematical Theory of Communication,” the article argues that information is something real that may be defined, measured, and managed.

Shannon's crucial point is that information is a surprise. If a message is predictable, then no one is surprised to receive it, and it thus contains little information. An example would be an old friend repeating an oft-told story, where you pretend to listen as the tale inches toward its predictable conclusion. In contrast, a message is informative if it is unexpected. A great murder mystery meets this standard because few readers connect the clues to solve the crime.

Imagine going to a college reunion, where former classmates cluster in groups to catch up on life events. The groups collectively produce a stable level of background noise. From time to time, this baseline is punctuated by shrieks as one group reacts to the sharing of an unexpected, juicy piece of gossip. Outbursts reveal elevated levels of information transfer. A second analogy for a surprise is the large stock price reaction to a release of significant but previously undisclosed information about a company's business prospects.

In Shannon's world, information reduces uncertainty. The purpose of studying numeracy is to extract information from raw data. To use an academic word, numeracy mediates the process of extracting information from available data.

Insight gained – uncertainty reduced – better positions us to cope with the messy world surrounding us. Using Knight's framework in a commercial setting, information reduces uncertainty and supports efforts to earn economic profits. My former employer and its CEO earned considerable monetary rewards from investments that boosted employee numeracy.

The remaining pages present the argument that unlocking the promise of numeracy comes from understanding a limited number of basic ideas, illustrated as letters, symbols, or brief formulas.

I warn you that time invested in this book is no guarantee of success. Learning a craft takes practice, but practice is not a sufficient condition for achieving desired goals. The essayist Malcolm Gladwell studied factors associated with extreme levels of performance across many realms of human achievement (2008). He found that raw talent is a poor predictor of success.

Figure 1.1 Numeracy Helps Extract Information from Data

Instead, Gladwell believed that highly successful people are willing to invest 10,000 hours to develop a skill. Regardless of discipline – chess, music, sports, computer programming, whatever – experts put in the hours to hone their craft. A 10,000-hour investment represents spending four hours per day, five days per week, over a decade to develop a skill. The machinist I admired had made such an investment to learn his craft.

My belief, however, is that a few hours spent reading this book will accelerate the learning process, much like a catalyst speeds a chemical reaction without direct participation. Time spent with this book allows the reader to realize a greater return from their 10,000-hour journey. If I've done my job well, reading this book will allow you to make more meaningful contributions to organizations that you serve.

The good news is that data analysis becomes easier and more enjoyable with practice. I have reached the stage where it is fun to explore new data. Really.

Before closing this chapter, let me discuss what numeracy is not. First, numeracy is not the answer to all problems of management. The craft of statistical reasoning is merely additive to any effort to predict or explain things.

My beloved Cleveland Browns, an American football team, have not done well in the (many) years associated with my writing this book. As I pen these words, there are 32 teams in the National Football League, and my Browns are one of just four to have never made a Super Bowl appearance. Player turnover has been so high that I cannot name all the starting quarterbacks since the team's 1999 reincarnation. During the two seasons ending in December 2017, the Browns won just one game and lost 31 (or a winning percentage of 3.125% [1-in-32]), a rate statistically worse than flipping a coin – evidence that something was broken.

On a trip to Kansas City, I had the opportunity to meet an executive with the Kansas City Chiefs, a rival team that flourished when my Browns languished. I asked him why my team has fared so poorly. Without hesitation, he replied that we had placed excessive reliance on “Harvard stats guys.” His criticism was that the Browns had given too much authority to highly educated people who put inordinate faith in statistics. Good leadership mixes art with science. Numbers are helpful but cannot provide all the answers.

Second, numeracy does not mean mathematical rigor. Any reader armed with a decent grounding in high school math is capable of understanding concepts discussed in this book. The role of the numerate person is to identify provisional patterns buried in raw data. Use of these patterns, should they stand up to scrutiny, may transform organizations. Progressive's experience in using patterns to predict insurance accidents is a notable example. Success does not require the reader to learn trendy data analytics tools, such as data mining, artificial intelligence, or machine learning, that have garnered lots of attention in the business press.

What is needed is the help of trained statisticians to kick the tires of provisional patterns surfaced by numerate generalists. After we do our work, we invite propeller heads to use skills I will never acquire. These experts, armed with advanced degrees in mathematics or computer science, articulate hypotheses, aggregate data from various sources, and then use a host of advanced techniques to rule out proposed ideas unsupported by data. Propeller heads extend the work of the numerate. Generalists and propeller heads are teammates, not rivals.

Propeller heads are trained professionals with individual identities, not unlike doctors. Physicians have received considerable education and training to offer specialized health care services. A patient with a vision problem would typically not reach out to a dermatologist for help. Numerate generalists involved in, say, refining an opinion poll would typically not ask an econometrician (one who works with large, structured data sets) to try to disprove a hypothesis.

To take the medical analogy further, a generalist knows that no two experts will agree on everything. A patient suffering from macular degeneration may receive conflicting diagnoses and prescribed treatments from two ophthalmologists. A numerate employee may similarly receive conflicting advice and conclusions from statisticians. Welcome to the game. Finding ways to work with propeller heads is part of the 10,000-hour journey to become numerate.

To sum up, this book seeks to cultivate numeracy. Whereas a literate person can read and write, a numerate person is able to count and compare. These skills better enable someone to make informed predictions or explanations that help organizations flourish in an uncertain world.

Prediction – a means of coping with danger – lies at the core of human survival. Whoever makes better predictions garners advantage over others. The ancient Greek philosopher Thales (circa 624–546 BCE) allegedly used powers of observation and reasoning to corner the market for olive presses based on weather forecasts that foretold a healthy crop yield. He also predicted a solar eclipse.

A second purpose of numeracy is to support efforts to explain why things have happened as they have. Dinosaurs are extinct. Prediction offers little help in assessing their prospects. However, an explanation for their extinction offers a useful story that better equips us to plan for environmental change.

Any student of management needs to improve his or her ability to predict or explain things. Numeracy lies at the heart of these skills. This is the book I wish someone had handed me when I began my career.

Recap of Chapter 1 (Numeracy)

Numeracy is the craft of finding patterns buried in data sets.

Numerate people convert raw data into uncertainty-reducing information.

The craft of numeracy permits more informed predictions or explanations.

The foundation of numeracy rests on a dozen or so statistical ideas presented in this book.

2ZERO [0]

Perhaps the most underrecognized idea in human thought is the number zero, an idea that emerged in many places in ancient times. The signifier, an oval-shaped symbol, is recognizable to schoolchildren around the world. The signified, the concept associated with the oval, is the absence of something to be measured. The ability to show the presence of an absence makes numeracy possible.

The number zero confers at least three benefits. First, it serves as a placeholder so that we may measure differences between numbers. Second, zeroes may sometimes be placed to the right of decimal places to create ever-more-precise measures for items of interest. But, most importantly, zero allows the establishment of a baseline so that we may calculate ratios. Numerate people need to be aware of when we may not be able to calculate differences, ever-more-precise measures, or ratios.

Placeholders. Let's start with the number 3,210. The figure has a value equal to the sum of three thousands, two hundreds, one ten, and no single units [(3 × 1,000) + (2 × 100) + (1 × 10) + (0 × 1)]. If we wanted to add or subtract a similar figure to this value, we would align counts for thousands, hundreds, tens, and ones for the two figures involved and then perform the arithmetic on counts within component categories.

Children in primary school learn to align two or more numbers and then borrow or carry from adjacent categories to perform long addition and subtraction. The use of zeroes in our Arabic numeral system permits first-graders to calculate differences between large numbers.

Now, imagine we use a number system with an absence of the number zero. An example is one used in Ancient Rome. Roman numerals assign symbols for certain quantities. The letters M, D, C, X, V, and I represent, respectively, the values for 1,000, 500, 100, 10, 5, and 1.

To cope with not having zeroes, we place smaller values to the left of a larger value to signify subtraction or on the right to signify addition. Thus, IX means that the reader should subtract 1 from 10 to arrive at 9, and XI means that the reader should add 1 to 10 to arrive at 11.

To demonstrate how horribly cumbersome this approach is, consider a stylized example presented in Table 2.1. Here we show the results of a toll-taker collecting a tax from pedestrians who cross a bridge in the Roman Empire. The civil servant simply tallies the number of people who cross each week. The figure grows over the week as the cumulative number of pedestrian crossings increases.

Table 2.1 Pedestrian Bridge Crossings over a Week

Cumulative Counts

Daily Counts

Day

Roman

Arabic

Roman

Arabic

Sunday

IX

   9

IX

  9

Monday

XLIX

   49

XL

 40

Tuesday

XCVIII

   98

XLIX

 49

Wednesday

CDXCIX

  499

CDI

401

Thursday

MIV

1,004

DV

505

Friday

MCXCVII

1,197

CXCIII

193

Saturday

MDVII

1,507

CCCX

310

The toll-taker reports to an administrator (the tax man) who wants to know how many people cross the bridge daily in order to make granular predictions of future tax revenue.

The records show that, through Wednesday, CDXCIX people (or 499, using Arabic numerals) had crossed the bridge, and this cumulative number had grown to MIV (1,004) through Thursday. Suppose the tax man asks how many people crossed on Thursday.

Here's my response to the boss's question:

The use of zeroes as placeholders, provided in the Arabic numbering system, makes this subtraction problem child's play, whereas the absence of zeroes in the Roman system makes the problem quite difficult. Zeroes let us measure differences easily.

Precision. When placed to the right of a decimal place, zeroes help us measure things with increasing granularity. Suppose an interior designer asks us to measure the width of the family room in our house. We offer three responses:

5 meters

5.0 meters

5.00 meters

All three measurements are equal to about 16 feet, 5 inches. Yet, each response conveys different information. A measurement made to the nearest hundredth of a meter permits more exact space planning than would be available for a measurement made to the nearest tenth. The first measure is useless if we need to buy new furniture or a rug.

However, we may not always be able to measure things with increasing precision. Let us now add three words to our vocabulary: dichotomous, discrete, and continuous. Whenever looking at a new data set, please classify observations into one of the following three buckets.

Dichotomous variables display just one of two possible values. A switch is on or off, an employee is present or absent, or a flipped coin lands as a head or tail. Yes, we may experience unusual circumstances (say, coins landing on edges), but these events are outliers – something considered in Chapter 7.

People tally dichotomous variables, such as the number of days an errant employee fails to show up for work at the factory. We need to be careful when selecting statistical tools to evaluate dichotomous phenomena, such as whether or not a firm declares bankruptcy within a given year.

Discrete variables may assume certain values. A rolled die displays a value of one through six. A dealt card from a standard deck must display one of four suits (heart, diamond, club, or spade), but the suit color is dichotomous (either red or black). In the United States, one may buy shoes in sizes 8, 8½, or 9 but not in, say, size 82/3. People count discrete variables, such as the number of completed transmission assemblies shipped by the factory in a month.

An issue with discrete variables is determining the thing to be counted. There's the joke about a cashier working in the express lane of a grocery store in Cambridge, Massachusetts. A shopper approaches with a cart packed full of groceries. The employee nods to the overhead sign warning “Twelve or fewer items” and asks the shopper whether he went to Harvard and can't count or MIT and can't read.