The Monte Carlo Method vs. the Normal Distribution: Approximating Uncertainty in the Absence of “Headwinds”

Disclaimer: This post is rather wonkish with statistical and metallurgical discussions.

In a previous post I outlined the use of Bill James (founder of Sabermetrics / Moneyball influencer) similarity index to a metallurgical engineering project (Link). In the example, a statistical model was developed for projecting the strength of a particular alloy with respect to changes in a processing parameter. A Monte Carlo simulation was used to evaluate the distribution in the projected strength as a result of 1,000’s of changes in the model inputs. The result of the simulation gives you the approximate probability of the various outcomes.

The Monte Carlo Method

The Monte Carlo method, as you may have guessed, derives its name from the Monte Carlo casino in Monaco. The approach was invented by a Polish mathematician named Stanislaw Ulam as part of the Manhattan Project (Source). The inspiration for Ulam came from playing solitaire and wondering an easy way to calculate the probability of winning the game, eventually leading him to apply this logic to neutron diffusion (Source). The method involves taking a model and feeding in distributions of the various inputs and recording the outputs over hundreds or thousands of iterations. An example of a distribution generated from a Monte Carlo simulation results for work performed on developing an improved age practice for 7068 aluminum is provided below. The simulation was performed by changes in the time and temperature adjustments to the two step age practice performed following solution heat treatment.


The Monte Carlo method as a tool for generating a distribution of probable outcomes, differs from the classical example taught in Stats 101 courses. Classical empirical modeling is typically introduced using linear regression (think Excel and the linear trend line). Linear regression models are developed such that a line is drawn through the average or expected outcome for an input variable or set of input variables. The residuals or difference between the actual and projected (expected) values at a given point are assumed to be normally distributed and any residual outside of 2 standard deviations from the projected value is considered an “outlier” (Source). This “lazy” approach to modeling the distribution of outcomes can be effective; however, comes with the risk of greatly underestimating the probability of “unlikely” outcomes.

When the Normal Distribution Fails

The defining characteristic of the normal distribution is its central tendency or in layman’s terms, the majority of the data is clustered around the mean. The image below outlines this concept by highlighting the percentage of the data in each area with respect to the number of standard deviations (σ) from the mean (μ). From the image you can see as only 0.1% of the data lies beyond the 3σ point. This feature makes the normal distribution easy to illustrate and drives its use as the basis for tools such as control charts.


Image source

Nassim Nicholas Taleb (NNT), in his book The Black Swan, exhausts the fact that using the Normal (Gaussian) Distribution is dangerous for approximating the likelihood of seemingly low probability outcomes. NNT states that things that are normally distributed face “headwinds” which make probabilities drop faster and faster as you move away from the mean (e.g. height, IQ, etc.). If the “headwinds” are removed the resulting outcomes will become significantly asymmetrical (think 80/20 Pareto principle). NNT illustrates this point by contrasting wealth distribution in Europe and contrasting it with what the distribution would look like if it were normally distributed.

Wealth Distribution in Europe:

  • People with wealth greater than €1 million: 1 in 63
  • Higher than  €2 million: 1 in 125
  • Higher than  €4 million: 1 in 250
  • Higher than  €8 million: 1 in 500
  • Higher than  €16 million: 1 in 1,000
  • Higher than  €32 million: 1 in 2,000
  • Higher than €320 million: 1 in 20,000
  • Higher than €640 million: 1 in 40,000

Normal Wealth Distribution:

  • People with wealth greater than €1 million: 1 in 63
  • Higher than  €2 million: 1 in 127,000
  • Higher than  €4 million: 1 in 886,000,000,000,000,000
  • Higher than  €8 million: 1 in 16,000,000,000,000,000,000,000,000,000,000,000


The above example demonstrates if wealth was  normally distributed, the likelihood of a Bill Gates or Warren Buffett is incomputable and provides a simple lesson in the fragility of the normal distribution when it comes approximating the probability of unlikely outcomes.

Wrapping it Up

“All models are wrong, some models are useful”

-George Box, Industrial Statistician

In process engineering, “head winds” to borrow the term from NNT are made up of the controls imposed on the process inputs. These controls form the basis for the Y = f(x) philosophy touted by Six Sigma books to demonstrate that if the inputs to a process are “in control” the resulting outputs will be as well. The problem with this logic is that it implies that the organization attempting to control the process has identified all the necessary input variables and deployed adequate controls (i.e. “headwinds).

Recently, I fell victim to this oversimplification after resurrecting a model which used the “lazy” approach to modeling uncertainty discussed above and applying it to a process where the “headwinds” (i.e. controls on raw material) had been removed. The result was a drastic underestimation of the probability of an undesirable outcome (production of material outside of the specification limits). Using the “lazy” approach (+/- 3σ) to modeling the probable outcomes the likelihood of nonconformity ended up being an order of magnitude higher than originally projected, DOH!.

Lesson Learned: Avoid the “lazy” approach and embrace the Monte Carlo!




Remembering Hans Rosling, Co-founder of the GapMinder Foundation


I had the privilege to see Dr. Hans Rosling speak at the 2012 ARPA-E Energy Innovation Summit. An event where Elon Musk announced Tesla would be repaying their government loans early and Secretary Steven Chu kicked off his talk with a dirty joke. The punchline, “Yes, but my previous husband was an entrepreneur. Every night he would stand at the end of the bed and tell me how great it was going to be”. With all the great technology being exhibited and guest speakers like Elon Musk, Dr. Chu, T. Boone Pickens, and several congressmen & women, Dr. Rosling blew them all out of the water! Dr. Rosling put an amazing show using his revolutionary data visualization tools as he painted a picture of global health and economics which he used to promote his battle against preconceived notions. Cheers to you Dr. Rosling!

If you enjoyed the video check out Dr. Rosling’s organization or his book  Factfulness.

“I keep saying the sexy job in the next ten years will be statisticians. People think I’m joking, but who would’ve guessed that computer engineers would’ve been the sexy job of the 1990s?”

-Hal Varian, Chief Economist at Google

Attributes to Look for When you Must Select an “Expert”

If you enjoyed our last post where we explored the effectiveness of financial titans, an example of how medical professionals can be fallible, and how incentives lead realtors to to work against your best interest (Link), but find yourself still in need an “Expert” seek out these traits adapted from The Signal and the Noise by Nate Silver.


Attributes to Look For:

  • Multidisciplinary: Demonstrates the ability to incorporate ideas from other fields or disciplines
  • Adaptable: Finds new approaches or willing to pursue multiple approaches at the same time
  • Self-Critical: Willingness to take ownership for mistakes or failed predictions
  • Tolerant of Complexity: Understanding that the universe is complex and recognizes that some things are unpredictable (e.g. Black Swans)
  • Cautious: Express projected outcomes in probabilistic terms
  • Empirical:Rely more on observations and data than theory

Attributes to be Cautious Of:

  • Specialized: Career dedicated to working on one or two great problems. Skeptical of the opinion of “outsiders”
  • Stalwart: Singular approach to problem solving. New data is used to refine the original model
  • Stubborn: Blames others or “bad luck” for mistakes and errors
  • Order-seeking: Expects things to abide by relatively simple governing relationships
  • Confident: Speak in terms of certainty
  • Ideological: Expects solutions to be a manifestation of some grander theory


The two lists outlined above signify that the “best” experts will be reluctant to make bold statements regarding  projected outcomes. This reluctance may be perceived as a “weakness”, as it violates laws of persuasion. Always be on your guard for Gurus preaching certainty as they are of course trying to sell you something!

Persuasion Tip #9: Display confidence [either real or faked] to improve your persuasiveness. You have to believe yourself, or at least appear as if you do, in order to get anyone else to believe

Win Bigly by Scott Adams

Forget Gurus! Why you should do your own homework.

“How could I have been so mistaken as to have trusted the experts?”

-John F. Kennedy after Bay of Pigs

Turning to experts for solutions to problems can be appeasing for a number of reasons the most common of which is the comfort in outsourcing the decision making process. In my own experience I’ve found organizations love to delegate decisions, especially technical decisions, to an ordained group of “Experts”. These shaman will fly in on Monday afternoons and confer in conference rooms until Thursday at which point they will collectively provide their OPINION on the matter at hand. This OPINION will taken as fact via a combination of confirmation bias and office politics. Decisions will be made based on the new found “facts” and opportunities will be lost.

This post; however, is not about the use of Guru’s in the corporate world, but how we turn to these anointed professionals  for help making decisions in our everyday life. In this post we will explore the effectiveness of investment professionals, political pundits, doctors and realtors with the goal of providing you with a little extra motivation to do your own research.

Investment Professionals:

Turn on Fox Business, MSNBC, or pop open the Wall Street journal and you can witness the worship of investment “greats” providing you a myriad of explanations of where the market is going or what the hot stock is. Though Wall Street analysts make average salaries measured in the million$, are these analysts anymore effective than dart throwing chimps? The answer is no!

Need further evidence?! A 1995 study of the highest paid Wall Street analysts invited by Barron’s to its annual roundtable to make recommendations revealed investments made based on the projections of these oracles merely matched the average market return (Source: Mlodinow). Furthermore, studies from 1987 and 1997 found that the recommendations from the television show  Wall $treet Week significantly lagged the overall market performance, while a Harvard study of 153 investment newsletters indicated “no evidence of stock pickling ability” (Source: Mlodinow).

You may now be asking yourself how these stock market gurus who apparently have at best the insight of a dart throwing chimp be touted as “beating the market” and be paid outrageous amounts of money? The answer is the illusion of patterns. Humans have evolved to be excellent at pattern recognition, allowing us to perform feats such as circumnavigating the globe with the stars. Excellent abilities to recognize patterns is a double edged sword, as we can also identify patterns where none exist. The example below provides a distribution of the top 300 fund managers and the number of consecutive years they beat the market (S&P 500) and the success of students correctly guessing a series of coin flips. The two distributions are almost identical providing insight that the ability of fund managers to beat the market equivalent to the probability of correctly calling “heads” or “tails”. As the population of coin flipping students or fund managers increases the probability of a small number of “extraordinary performers” increases to the point of certainty.

bell curve

Top 300 Fund Managers vs. 300 Coin Flipping Students (Source: Mlodinow)

Political Pundits:

After recognizing the failure of political scientists to accurately predict the fall of the Soviet Union, Phil Tetlock (then a professor at UC Berkeley) undertook a 15 year study to evaluate  the accuracy of political predictions. Tetlock’s findings were published in his book, Expert Political Judgement.  Tetlock’s findings concluded that experts were barely better than random chance (think dart throwing chimps) at predicting events. In fact, events Tetlock’s “experts” predicted had a zero percent chance of occurring, actually occurred about 15% of the time, while absolute certain events (per the “experts”) did not occur about 25% of the time.  Tetlock has noted that the more interviews an expert participated in the worse his or her prediction accuracy (Source: N. Silver)!

In his own review of political pundits, Nate Silver evaluated predictions made by panelist on the television program The McLaughlin Group. Silver analyzed almost 1000 predictions and found that the pundits were about as accurate as a coin flip. Furthermore, Silver noted that “experts” on The McLaughlin Group predictions were influenced by their political affiliations (Source: N. Silver).

Persuasion Tip #1: When you identify as part of a group, your opinions tend to be biased toward the group consensus.

Win Bigly by Scott Adams

Medical Doctors:

As it may be easy to “write off” investment analysts and political pundits as modern day snake oil salesmen, lets take a brief look at Medical Doctors. Studies have shown that radiologists fail to identify lung disease in about 30 percent of the X-ray results they read, despite the clear presence of the disease on the film (Source: Malkiel). Other experiments have shown that professional psychiatrists were unable to distinguish between the sane & insane (Source: Malkiel).

A classic example is a study by the American Child Health Association performed in the 1920’s were 1,000 children from the New York City public schools were examined by physicians to determine the need for a tonsillectomy (Source: Malkiel). Of the original 1,000 students, 611 (61.1%) were identified as needing their tonsils removed. The remaining students were then evaluated by another group of physicians who then selected 174 (44.7%) as requiring a tonsillectomy. The remaining group of 215 students were evaluated by a third set of physicians who concluded that 99 (46%) of those students were in need of the operation. The final 116 children were examined a final time and the next group of physicians recommended that 51 (43.9%) students needed a tonsillectomy. The results of this study indicated that parents taking their children to a New York City physician in the 1920’s for tonsil issues were effectively paying doctors to flip a coin!


Realtors, or as I refer to them as the most morally bankrupt reptiles on the plane (I’ve recently had some bad experience with realtors), are another example of anointed professionals who often fail to deliver the insight and results their clients expect.  Realtors, as the video from the Freakonomics Movie, demonstrates how real estate agents are incentivized to work against their clients to quickly ascertain a sale.

If you have found yourself complaining about the time on the market of your home or lack of open houses and your realtors only advise is “lower the price” the video below is essential. The previous examples of investment professionals and medical doctors hinted at the influence of random luck, the example of real estate professionals provides insight into how incentive systems lead anointed professionals to provide you with misinformation. Leverage websites such as and data from your local realtor association to generate your own insight.

Wrapping it Up:

Gurus and professionals are not infallible as the examples provided indicate. In some cases as we explored with investment professionals and tonsillectomy diagnosis, professionals may about as effective as dart throwing chimps or coin flipping. Professionals will be influenced by the consensus of the groups they are in, as was the case with political pundits. Lastly, they may be incentivized to act against your best interests (think real estate agents). Thus, it is essential to do your own homework! Filter the signal from the noise in the data you collect and mindful of the bias of Guru’s as a result of incentives and affiliations.

“All professions are conspiracies against the laity.”

-George Bernard Shaw, Major Barbara

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