How Normal is the Normal Distribution

Todd Moses
Mar 13, 2019 · 7 min read

A young man in Naval Uniform sat at a table with beads of sweat forming around his short cropped hair. His left wrist was bound with the index finger laid flat on the wooden surface. Above his finger rested the chef’s cleaver with the front point pointed downward, piercing the wax film of the table. The rear of the sharp blade ready to dissect with the slightest push by the firm worn hand of his captor.

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“The rules are simple,” spoke the older man holding the cleaver. “Strike your lighter ten times,” he gleefully explained with grip tightening, “If any of the ten strikes fail, I take the finger.” “But I get the car if each strike works,” the Sailor interjected with a hint of trepidation. “Yes, yes” assured the man holding the knife with slight annoyance.

The Navy man’s right hand held a metallic lighter with his finger on the wheel. “One,” spoke the captor. With slight pause the seaman flipped the wheel of the lighter. A spark and flame appeared. “Two,” came the voice with increased anticipation. The lighter flicked again with spark and flame. “Three,” burst the man with a gleeful shout.

Like much of probability, the foundations for the Normal Distribution comes from gambling. In 1733, Abraham de Moivre discovered that the random act of flipping a coin (Binomial Distribution) approached a very smooth curve as the number of iterations increased. Since that time, it was found that the Normal Distribution appears in many natural phenomenon. Making it a basis for Statistical Inference.

It seems that most, if not all, measures dealing with humans fit the Normal Distribution or Bell Curve as it is commonly called. Everything from employee salaries, Intelligence Quotient (IQ), Grade Point Average (GPA), likelihood for chronic illness, and more fit into this model. At first, one may assume this is due to such measures being contrived to fit a simple model like the assumption in High School Physics that gravity is constant. However, looking deeper into the subject brings up some interesting discoveries.

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With the sailor’s right thumb indented onto the wheel, a strike occurs and flame appears. “Four,” came the voice in a more relaxed tone. With fingers damp from fear, the strike came and as in slow motion a flame appeared.

“Five,” said the man holding the knife with a slight smile forming. The sailor’s face wet with perspiration, blew liquid from his upper lip onto the table. Nervously moving his right thumb in the familiar motion, the lighter produced flame as he loudly exhaled. “Six,” the man continued without emotion.

While many distributions have a bell shape, the Normal Distribution has several unique properties. For one, it is symmetric around its mean and is the only distribution where the mean and variance calculated from a set of independent samples are independent of each other. In other words, when taking random samples, one would expect the greatest number of samples to appear in the middle section and for that section to have equal distance from the left and right extremes. To be precise, a Normal Distribution needs 68% of the area within one standard deviation of the mean.

In Major League Baseball, players abilities follow the right tail of the Normal Distribution. This is due to the fact that to become a professional baseball player in the Major League, one must be among the best. The same is true for other professions where only the very best can participate. For example, the top ranked professional schools (Business, Law, and Medical) only accept those at the far right of the standardized test distribution.

While the average IQ is 100, those far outside of this range in either direction often suffer from some form of mental illness. In addition, many on the extremes have difficulty in careers, marriages, and life. In fact, most of the truly revered professions rarely have persons with IQ scores above 130 and the median hovers around 115. This includes College Professors, Medical Doctors, and Engineers.

In measuring employee performance, many managers find that it is those in the middle part of the distribution that are the most effective at their jobs. The reason why appears to be that the least competent employees take time from those just towards the right tail with needs for assistance. Those at the highest levels, farthest to the right, are often preoccupied with solving hard problems that may or may not need solving and become board with most of the needed tasks.

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The sailor in near panic felt the perceived sharpness of the blade slightly above his skin. He attempted to control the fear by imagining the nearly new Cadillac just outside of the room. The one he would possibly drive home after this ordeal was finished. Thinking of its metallic blue finish sparkling in the sunlight, he felt his heart rate slow. Imagining the smell of the leather upholstery from behind the wheel he felt tension escape. With newly gained calm, the seaman allowed his thumb to roll back the striker and watched as a flame arose.

“Seven,” announced the older man as his watch slightly moved from the anticipation of making the cutting stroke. “That’s a nice watch,” the sailor proclaimed staring at the gold Rolex on the wrist of the man waiting to harm him. “Seven,” the man announced in response. “If I make this one, can I have your watch,” the young seaman interrupted. Pausing, the man holding the knife looked at his watch, examining it closely. “Seven,” he said firmly.

With a flick of his thumb, the lighter clicked. A moment of terror engulfed the sailor as he waited for the flame to appear. Watching intently the tip of the lighter, a flicker of yellow appeared then a complete flame dancing with orange. “Eight,” demanded the older man appearing disappointed by the result.

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Most of Statistical Inference is based upon the idea that data fits the pattern of the Normal Distribution. In coin flips, this means as the number of iterations approaches infinity, the number of heads and tails will grow closer to a 50% distribution of each. However, that is over many flips. In the short term, the result is random, given a fair coin.

So it seems that the more samples one has of random events, the less random they become. A notion that keeps insurance companies profitable. The problem is that this is not the same as prediction. Consider that out of a billion coin flips, the mean of the distribution will trend toward 50% while the path to get their will be random. That is, no one can know the result of the next flip.

As the sailor looked at the lighter in his right hand, he felt the moisture from his skin, becoming irritated by it. Letting out a breath he was unaware of holding, his thumb almost on command began the movement of striking the flint. The sound seemed repulsive now as he watched the spark barely appear. In near defeat, he equally waited for the pain of amputation and for sight of flame. After an intense microsecond, it was the flame that occurred.

“Nine,” came the voice of the captor as his grip slightly moved on the knife. “Wait,” pleaded the young sailor, “I need a break.” “Nine,” said the man again, ignoring the request. “No,” shouted the sailor in exasperation. “Please,” he requested with a slight whimper. “Nine,” replied the voice.

With watery eyes and nose dripping, the sailor looked at the knife positioned just above his finger. The blade was almost a mirror finish. A slight smudge was present just past the handle where the man’s thumb must have touched while adjusting his grip. The sailor’s mind began to wander, thinking of his first pocket knife. Then quickly back to the present horror with the sound of “Nine.”

This time the sound seemed robotic, lifeless. Outside of conscious action, the sailor’s thumb began to move the wheel. Each increment of movement slipped along his skin from the moisture until his thumb passed completely over the wheel. Unlike before, the level of force onto the wheel was slight and no spark appeared. Anticipating the consequence, the seaman closed his eyes and waited for the combination of sharp and dull pain that awaited.

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Conclusion

The story was an adaptation of the short story, “Man from the South” by Roald Dahl. It first appeared in the magazine Collier’s in 1948. Since then it has been adapted into television, radio, and film productions. While a bit dramatic, It seems a good illustration of how the Normal Distribution is never guaranteed for small samplings.

What is probably more fascinating is the number of data sets that form a Normal Distribution. It seems that measuring anything in regards to rankings, it appears. Be it defects in code, effectiveness of developers, and salaries of management.

As always, thank you for reading. To learn more about the author, visit https://toddmoses.com.

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