This article was downloaded by: [46.3.194.167] On: 18 November 2017, At: 06:07 Publisher: Institute for Operations Research and the Management Sciences (INFORMS) INFORMS is located in Maryland, USA INFORMS Transactions on Education Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org Blitzograms Interactive Histograms Sam Savage, To cite this article: Sam Savage, (2001) Blitzograms Interactive Histograms. INFORMS Transactions on Education 1(2):77-87. https:// doi.org/10.1287/ited.1.2.77 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval, unless otherwise noted. For more information, contact permissions@informs.org. The Publisher does not warrant or guarantee the article s accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright 2001, INFORMS Please scroll down for article it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org
Savage Sam Savage Department of Management Science and Engineering Stanford University Stanford, CA 94305-4026, USA savage@stanford.edu I Hear, I Forget I See, I Remember I Do, I Understand Chinese Proverb From Bicycles to Blitzograms You probably know how to ride a bicycle, but would you even recognize the equations of bicycle motion? The equations are clearly not necessary for riding a bike, but are they sufficient? Suppose, that upon seeing your first bicycle, you had derived the equations of motion from scratch, starting with F=ma. Would you shout Eureka then jump on and start riding? No, the equations of motion are irrelevant to riding a bicycle. Designing a bicycle maybe, but riding one, no way. We learn to ride through interactions with bicycles using our hands and the seats of our pants. Michael Polanyi (1983) has referred to this as tacit learning as opposed to the formal learning required to understand equations. I will demonstrate below, a path toward gaining a tacit understanding of the concept of probability distributions. Motivation Several years ago I came to the conclusion that many graduates of basic Statistics courses don t understand the Central Limit Theorem. As described below, I was wrong, they don t even grasp the concept of a distribution. I came to my initial conclusion by showing students (all of whom had taken statistics) the histogram of a uniform random variable generated by a spinner as shown in (and also available as SPINNER.xls (Savage, 1998a). After displaying the uniform histogram associated with the spinner, I would ask the students to draw the histogram of the average of two spins. Consistently about half would incorrectly draw another flat histogram, while the other half would correctly draw a centralized distribution, either triangular (technically correct), or bell shaped (close enough for me). From this experiment, which I repeated with consistent results, I incorrectly concluded that half of the students did not understand the Central Limit Theorem. After living under this misconception for about a year, I almost absent-mindedly asked a class, one day, to draw the histogram of a single spin before I had shown them the uniform results. I was shocked to find the same two drawings (flat and centralized) in roughly the same proportions for one spin that I had been getting for the estimate of the average of two spins. Those who draw a centralized distribution clearly do not grasp the concept of probability distributions. I have now repeated this experiment on over a thousand subjects, including Stanford graduate students, executives and military officers, all of whom have had at least one course in statistics. The roughly one half who draw a centralized distribution for a single spin have, in the words of Mark Twain, had their schooling interfere with their education. These results are discussed in (Savage, 1998b). Virtually all MBA and master s students in Operations Research and Management Science are expected to take a course in statistics, and many have had courses in stochastic modeling. I think this is a reasonable requirement, but if a large proportion of them can t accurately model a game board spinner, exactly what stochastic modeling skills do they have? Some Related Literature Further experiments with students perceptions of spinners performed by Armann Ingolfsson (1999), David Zalkind (1999), and others confirm that students have a poor intuitive grasp of the meaning of probability distributions at a very basic level. More generally, there is a growing literature on our poor ability to deal with uncertainty. Kahneman, Slovic, & Tversky in Judgement under 77
Uncertainty: Heuristics and Biases (1982) present a wide variety of well-documented and consistent errors in judgment, even among experts who have had substantial statistical training. Improving Statistical Reasoning - Theoretical Models and Practical Implications by Peter Sedlmeier (1999) reports extensively on errors in statistical reasoning, and also develops a formal model of statistical learning. He strongly suggests that students are able to obtain a better grasp of statistical concepts when they are presented in frequency vs. probability format. That is, people seem more likely to correctly answer questions of the form: How many items have property A vs. property B than they are questions of the form: Which is most likely, A or B". There are many layers of semantics and deduction involved in such reasoning, and a few sentences here, cannot do books like these justice. But as Sedlmeier says, in confirmation of the Chinese proverb above, As good teachers know, students learn better when they are actively involved in the learning process. He goes on to list numerous other sources of evidence that support learning by doing. Mindles (pronounced mine dells) Exhibit 1. Central Limit Theorem (CLT.gif) But how do you interact with a distribution? To interact with physical things we grasp them with our hands, usually by appropriately designed handles that make them easie r to use. Things to be grasped with our minds should similarly have appropriately designed mindles. Blitzograms The histogram has always been a pretty good mindle for distributions. Histograms are associated with counting things, which is aligned with Sedlmeier s observation that the frequency format is intuitive. In the old days, histograms had two problems. First, a histogram couldn t represent the distribution of a continuous random variable unless the bins became infinitesimal. This required the introduction of calculus, which further hampered a tacit understanding of the subject by most students. But that was before computer scientists discovered that for all practical purposes the smallest positive number is 2-64. Today, for all but the most exotic situations, we can forget about continuous random variables. Finite bins are all you need. The second problem with histograms was that traditionally they took a long time to compute and graph. The blitzogram is an instantaneous or, lightning fast, histogram (hence the name) that 78
Exhibit 2. Spinner.gif displays the frequency distribution of a dynamic data set. Hook the user to this data set through numeric inputs, function keys, or slide bars, and you get direct user interaction with a stochastic systems. Three Loose Classes of Blitzograms Three loose classes of blitzograms will be presented. Most examples are done in Excel spreadsheets, which may be downloaded or viewed as animated GIF files in your browser. One example was programmed in Java, and one was generated using the animation feature of XLSim, a simulation package that I am developing at AnalyCorp. 1. Sample Blitzograms These display histograms of samples of a given size, of a random variable. 2. Convergence Blitzograms These display the convergence of a simulation or some other process. The blitzogram demonstrating the central limit theorem on the title page is of this variety.. 3. Convolving Blitzograms These display the results of managerial actions resulting in the convolution of multiple distributions.if X capacity is fully consumed by the assignment in the previous step, then assign Y capacity to make the remaining number of units of A demanded in that period. 1. Sample Blitzograms Sample blitzograms display histograms of samples of a given size, of a random variable. In the examples here, the Data Table spreadsheet command will be used to nearly instantly generate a column of random varieties for which a new live histogram will be displayed each time the calculate key is pressed. The Data Table A little appreciated, but powerful command of the spreadsheet is known as the data table. This feature has been present since early versions of Lotus 1-2-3, yet it is regrettably not widely used. It allows a single spreadsheet formula to be parameterized on up to two parameters in Excel (and up to 3 in 1-2-3). For example, the file 3DGRAPH.xls, taken from exercise 1 (Savage, 1998a), contains a function of two variables, parameterized on x and y and then graphed. Changing the function in the driver formula of the table results in immediate changes to the graph as shown in 3DGRAPH.gif. If you want to try functions of your own in cell C10, you will need to use the Excel version described below. If you have Microsoft Excel you can download 3DGRAPH.xls (note if calculation is slow, you may need to close any workbooks containing references to user defined VBA functions). I present this example to my class and then ask the students to dream up the most aesthetic function they can. This exercise assists in a tacit understanding of functions of two variables, which I also find lacking in most students. But back to probability distributions. Monte Carlo Simulation in Data Tables In the mid 1980 s it was discovered that if the function in the table formula cell consisted of a random number generator of some sort, the resulting data table would create a Monte Carlo 79
simulation of the formula cell. I first heard about this from Charles Bonini of Stanford, in 1990, who attributed it to William Sharpe. Early versions of Lotus 1-2-3 had a command for creating frequency distributions of a given data set across a given set of bins, but the result was not dynamic. That is, if you changed the data or the bins, you needed to re-execute the command. Excel, however, had an array formula =FREQUENCY (Data_array,Bins_array) that yielded a dynamic frequency distribution. In about 1990, I tried using =FREQUENCY on the results of a Data Table simulation to create a live histogram on my 25 mhz 386. Although I coined the term blitzogram at the time, molassesgram would have been more appropriate. It was not until I reached 120 mhz around 1995, that the concept really worked. The file BLITZ1.xls, modeled after exercise 2.6 (Savage, 1998a) contains a blitzogram of 50 samples of a uniform random variable. Pressing the calculate key <F9> causes a new sample of 50 to be drawn as shown in BLITZ1.gif. Note: at the suggestion of a reviewer, the labels on the x axis, and scale of the y axis have been frozen in the gif version for a clearer picture of what is going on. This is only possible after one has determined the support of the distribution being Exhibit 3. 3Dgraph.xls modeled. Therefore the labels and scale in the xls version have been left dynamic so that arbitrary distributions may be tested. The most striking lesson of BLITZ1 is that a sample of 50 uniforms is not very uniform. BLITZ2.gif shows the results of 500 samples of RAND() and (RAND()+RAND()/2. The effects of the central limit theorem are clearly visible in the peaked distribution on the right. What is the shape of the distribution of the maximum of two uniform random variables? Plug =MAX(RAND(),RAND())) into BLITZ1.xls and find out, if you don t know already. As a point of reference, I assign this problem as a homework exercise to my students. Later I give an exam on which I ask them to draw the distribution of the minimum of two random variables. Nearly 100% of my students answer this correctly. I like to think that the time they have spent with blitzograms contributes to their success with this question. David Ashley of the University Missouri has developed some interesting variants of the data table blitzogram, (click here to download), which he uses with his own students. It should be noted that Professor John P. Gould of the University of Chicago, as well as others, no doubt, have arrived at this type of blitzogram independently. 80
2. Convergence Blitzograms Convergence Blitzograms display the convergence of a simulation or some other process. The earliest of these dates from the 19th century, as discussed Exhibit 4. Blitz1.xls Exhibit 5. Blitz2.xls below. As a modern example, the Crystal Ball simulation package has used this device for displaying convergence for some time. 81
The Spinner Again What would happen if you spun the spinner a really large number of times? Click here to find out. The Central Limit Theorem The blitzogram on the title page, Exhibit 1 CLT.gif, demonstrates the central limit theorem, by showing how averages of increasing numbers of independent random variates converge on the normal distribution. This was generated by running 100,000 Monte Carlo trials in XLSim of ten independent bi-modal distributions generated as follows: =IF(RAND()<0.3,gen_normal(0.3,0.05), gen_normal(0.6,0.05)) The averages of the first n variates were also calculated, for n=2..10. The first output cell of the simulation recorded one such random variate. The remaining nine output cells recorded the various averages. XLSim offers the option of scrolling through the histograms of all output cells to create a simple animation. It is interesting to note that the average of two of the bi-modal distributions was tri-modal. Quincunx - The Mother of All Blitzograms Sir Francis Galton, the father of Statistics also developed what must have been the first blitzogram. It consisted of a glass-encased array Exhibit 6. Quincunx of pins protruding from a board. The pins were arranged in such a pattern, that when ball bearings were dropped into the device, they were randomly reflected right or left at multiple stages, and finally dropped into bins. This resulted in a bell shaped pile of ball bearings at the bottom. It was with the help of the quincunx, as he called this device, that Galton discovered regression to the mean. The odd name derives from the pattern of dots on the 5 side of a die, which was used for the arrangement of the pins. I was convinced that a quincunx could be created in a spreadsheet, and about three minutes into the project had a sense of déjà vu. The structure was very similar to a binomial lattice option-pricing model. Where Galton thought of his random deflections as biological measures such as changes in height between generation, I decided to represent changes in stock prices as shown in exhibit 6. The Poisson Distribution It is sometimes instructive to see how theoretical distributions change with the underlying parameters. For example I introduce the Poisson distribution with the following story, some of which I believe is true. Poisson, an early 19th century French mathematican was studying data concerning the number of soldiers killed per month by horse kick in the Prussian army. Suppose that 82
there were three Horse Kick Deaths (HKD s) per month on average. Then one might model the situation as follows. The probability of one HKD on any given day is 10%. Multiply this by 30 days and you get 3 per month. Thus HKD s per month are binomially distributed with n=30 and p=0.10, right? Close, but no cigar you say, because we have ignored the chance that two soldiers get killed on the same day. No problem I say, we will divide the month into 60 half-days. Now the probability of an HKD on a single half-day is 5%, so HKD s per month are still binomially distributed, but now with n=60 and p=0.05. But two soldiers could receive fatal kicks on the same half-day you say. No problem I say, we will model quarter days by upping n to 120 and reducing p to 0.025. This could go on until we both dropped from exhaustion. When I ask students what this sort of process was called in their calculus course, few recognize it as a limit. POISSON.GIF shows a sequence of binomial distributions with increasing n approaching the Poisson. Notice how quickly the binomial achieves the limiting distribution. This was Exhibit 7. Poisson.gif generated from POISSON.xls, in which a slide bar is moved by the user to control the number n. I believe that movable controls play an important role in tacit learning with slide bars playing the role of handle bars on a bicycle. This example not only provides a convenient mindle for the Poisson distribution, but for the concept of a limit as well, which also seems to evade a tacit understanding by most students. 3. Convolving Blitzograms For the past decade, I have worked with Ben C. Ball of MIT on the problem of selecting optimal portfolios of petroleum exploration projects. This work is summarized in (Ball, 1999), see also Notes on Exploration and Production Portfolio Optimization. Early on we discovered that many managers lack a tacit understanding of the effects of diversification on portfolios. We developed the following example as a mindle for this often counter-intuitive subject. 83
The Safe and Risky Investment Suppose you are responsible for allocating $10 million between two exploration projects. One is relatively safe", the other relatively risky". Either project would require the full $10 million for a 100% interest. The chances of success are initially assumed to be independent. The probabilities and payoffs of the projects are reflected in table 1. Safe Risky Outcome NPV $Millions Independent Probability Dry Hole -10 40% Success 50 60% Dry Hole -10 60% Success 80 40% Table 1 This example has been intentionally constructed so that the expected returns of the two projects are the same as shown below: Safe: 60%*$50 + 40%*(-$10) = $26 Mil Risky: 40%*$80 + 60%*(-$10) = $26 Mil To attach a concrete risk to the outcomes, imagine that if you lose money you will also lose your job. Any allocation of your $10 million across these two projects will have the same expected value, but how do you minimize the probability of getting fired? To simplify the problem, just consider the investment strategies shown below. Safe Risky a. 100% 0% b. 90% 10% c. 20% 80% d. None of the Above Table 2 Allocation a obviously results in a 40% chance of getting fired, but do b or c improve on this? Most managers seem reluctant to transfer money from a safe investment to a risky one that has the same expected return. Of course, some allocation not shown above might result in a lower probability of getting fired than a, b or c, in which case you would choose d. You may wish to spend a minute trying to work this out intuitively before proceeding. Here s a hint. I once had the opportunity to pose this problem to William Sharpe (of CAPM fame) whose immediate response was to start writing equations. If Sharpe needs to write equations, most of us don t stand a chance of solving this on intuition alone. To determine the result of any particular allocation of investments it is necessary to look at the four possible joint outcomes of the two projects. Table 3 depicts this detail for strategy c above. This shows that strategy c results in only a 24% chance of losing your job, which turns out to be the best you can do (strategy b is 40%). A blitzogram adds further insight to this example. In this case, a weighted convolution of the safe and risky distributions is generated in real time by dragging a slide bar from left (100% safe) to right (100% risky). Click on the arrows at the ends of the slidebar in the graphic to change the percentage in the portfolio. Note the changes in probability of loss on the right of the screen. Click here for the graphic (require Internet Explorer to view). You may also download the Excel version in Blitz3.xls. Statistical Dependence Up to now we have assumed statistical independence between the two projects. We can also use this blitzogram to investigate the effects of positive and negative correlation between the two projects while maintaining the original Safe Risky Probability Return in $Million Result Success Dry Hole 60% x 60% 20% x $50 + 80% x (-$10) Keep Job = 36% = $2 Dry Hole Success 40% x 40% 20% x ($10) + 80% x $80 Keep Job = 16% = $62 Dry Hole Dry Hole 40% x 60% = 24% 20% x (-$10) + 80% x (-$10) = (-$10) Lose Job Table 3 84
Savage probabilities of success and failure. Note: given the probabilities in table 1, a correlation of +1 is not possible whereas one of -1 is possible. Observe the results of changing the correlation with a portfolio consisting of 60% safe and 40% risky". Again note the change in the probability of loss. The construction of Blitz3.xls Correlation Blitz3.xls starts with a column of 60 50"s and 40-10"s named safe and a parallel column of 60-10"s and 40 80"s named rdata". To control the correlation, a new column (risky) contains rdata shifted relative to the safe data. Note, when the data is shifted so that each success ("50") in the safe column lines up with a failure ("-10") in the risky column, a correlation of exactly -1 is achieved. As the risky column rotates more and more successes into an overlap with the successes in the safe column, the correlation increases to a maximum of 66.6% when all 40 of the risky column s successes line up with successes in the safe column. Weighted Portfolio A new column named Outcomes is now created with formulas of the type w 1 *S + w 2 *R. Where w 1 and w 2 are the percentages of the portfolio in safe and risky respectively, and S and R, are the corresponding cells in the safe and risky columns. The blitzogram is then created by using ={Frequency(Outcomes,Bins)} formula in cells J1:J12. An Alternate Formulation Blitz3.xls, may be generalized to model larger numbers of arbitrary distributions and to perform stochastic linear programming for optimal portfolios as will be shown in the next section. However, for modeling two investments with binary outcomes, Blitz3.xls represents a brute force approach. One of the reviewers of this article has created a more elegant formulation which is available for download as portfolio.xls. Blitzograms in Practice A number of us working in the petroleum industry have discovered, independently, that managers are not willing to accept the notion of an optimized portfolio until they have gained a tacit understanding of the underlying model. This is where blitzograms come in. EPPO.xls contains the prototype of a general single period Exploration and Production Portfolio Optimizer. This is a scenario optimization model described generally in (Ball, 1999) and in more detail in Notes on Exploration and Production Portfolio Optimization. Monte Carlo scenarios of 5 projects, generated to preserve statistical dependence, are created in columns (Table 4). Table 4 The model then creates a blitzogram based on a tentative budget allocation across the projects. By viewing the distributions associated with their investment decisions, managers are better able to understand the risk implications of their potential actions. Up to 100% of each project may be taken, subject to a budget constraint. Note, that by typing 100% into any particular project while leaving the others at zero, the blitzogram displays the distribution of that project alone. Typical financial portfolio models constrain the percentages in each instrument to sum to 100%. It is assumed that once the optimal mix is determined that it can be scaled up or down for any budget. In this model, however, the percentages represent the proportion of each project taken. Thus, with a large enough budget one could take 100% of each project. 85
Two views of EPPO.xls are shown. The first displays an investment in 50% of the third project, 75% of the 4th project, and 100% of the fifth project. The pie chart displays the proportion of the budget in each of the five projects. The final portfolio has been optimized using the Excel Solver to minimize the mean loss (the expected value of the portfolio given that it is negative) in cell C6, for a budget of $600,000 and required expected NPV of $1,900,000. Note this sort of scenario model may be easily modified to support other useful piecewise linear objective Exhibit 8. EPPO.xls functions. One these that seems especially promising is Conditional Value at Risk, see http://www.ise.ufl.edu/uryasev/finnews.pdf. Conclusion As computers become ever faster, more and more procedures that were once viewed as iterative will continue to become instantaneous. The blitzogram is the application of this trend to histograms, which I hope will lead to a better tacit understanding of probability distributions among both students and managers. 86
And this is not just an academic exercise. Commercial Monte Carlo simulation packages like @RISK and Crystal Ball, and my smaller INSIGHT.xla are widely available. I hope that managers of the future, familiar with tools like these, will stop demanding: A number for my report and start demanding A distribution for my simulation. Creating the Animations Most of the animations were created in Excel using Windows 95 and 98. Screen shots of various frames of the animation were taken using the <Alt><PrtSc> key combination, and pasted one by one into the Microsoft GIF Animator. This application may be downloaded at no charge from Microsoft. The results are rather primitive, but may be easily accomplished by any Excel user with a few minutes practice. My friend, Randy Schultz, used Java to program the Convergence Blitzogram with the spinner in the background. This sort of thing is a lot of work, even for a good programmer, but leads to the most control. An interesting alternative, which lies somewhere in between these two approaches, was developed for the animation of Blitz3.xls by Adam Brower of the Hermes Group. Here, Java script (much simpler than a Java program) is used to cycle through a sequence of GIF images at the user s command. The individual GIF files have been given the names 1 through n for this example. Use the View Source command in your browser to see this Java script. It may be easily modified to create animations of your own. No doubt many other software tools available on the market now and in the future, will make animations like these even easier to produce. References Ball, B.C. and S.L. Savage (1999), Holistic vs. Hole-istic Exploration and Production Strategies, Journal of Petroleum Technology. Ingolfsson, A. (1999), Obvious Abstractions: The Spinner Experiment, Interfaces, Vol. 29, No. 6, pp. 112-122. Kahneman, D., P. Slovic and A. Tversky (1982), Judgment under Uncertainty: Heuristics and biases, Cambridge University Press, Cambridge. Polanyi, M. (1983), Tacit Dimension, Peter Smith Pub, ISBN: 0844659991 Savage, S.L. (1998a), INSIGHT.xla Business Analysis Software for Microsoft Excel, - Text and Software, Exercise 1, Duxbury Press, Belmont, CA. Savage, S.L. (1998b), Statistical Analysis For The Masses", Statistics in Public Policy, Ed.: Bruce Spencer, Oxford University Press, http://www.stanford.edu/~savage/stat.pdf Sedlmeier, P. (1999), Improving Statistical Reasoning - Theoretical Models and Practical Implications, Lawrence Erlbaum Associates, Mahwah, NJ. Zalkind, D. (1999), Another Take on the Spinner Experiment, Interfaces, Vol. 29, No. 6, pp 122-126. 87