Normal Distribution: How Things Go Most Of The Time

Normal distribution, often referred to as Gaussian distribution, is one concept that, once you understand, you start to observe everywhere. It’s a bell-shaped curve, fundamental to statistics and probability theory. The curve describes how data values distribute themselves around the mean in many natural and human-made processes. The shape is ad litteram bell-like, with a fat lump in the middle and two tails that stretch in either direction. See it as two exponential curves inversed on one another.

Gaussian distributions show up everywhere. Human height, intelligence, error-rate, are approximately normal. This is because normal distributions take random elements and average them out. They are the curves of averages, of the way things “normally” tend to “distribute” themselves. Simply put, if you fall into or near the middle of the curve, you are “normal” or “average”. If you are on either of the thin tails, you are an outlier. Somebody in the middle might be able to bench press 165 pounds, whilst an outlier could move 405. Sometimes it does not take much to exceed the normal distribution, and this is what makes top 1 percent performance doable without top 1 percent effort.

You are bound to run into bell curves now that you know what they look like. Length, height, and weight distribute normally. Hair, claws, nails, and teeth of biological specimen do too. Standardized testing scores in school tend to follow a normal distribution. Thus, generally speaking, it’s good to apply the normal distribution lens if the phenomenon you’re evaluating consists of a lot of different, small, independent forces. The value you get—that is, where the peak of the curve is—gives you a great idea of what to expect when taking up an endeavor or deciding on something. This is because, well, average things happen most of the time, so normal distribution calculations tend to be accurate approximations.

The key principles are rather self-explanatory. The curve is symmetric around the mean, the left and right sides being thin ends. In a perfect normal distribution, the mean, median, and mode are equal. Generally, 68 percent of data will fall within 1 standard deviation, 95 percent within 2, and 99.7 percent within 3—in other words, most of the values will average out and form the “bell”. It is ubiquitous in nature and a foundation of many other statistical methods, such as hypothesis testing, confidence intervals, and regression analyses. They have also found their use in machine learning, finance, and various sciences to predict outcomes.

What you must understand is that normal distribution is useful for approximation, not precise calculation. There is still going to be variability—or outliers. Extreme or “Black Swan” events happen, although rarely. You have to take these probabilities into account, especially if you use it to model risk or predict outcomes based on past trends. Else it can bite you. Averages happen most of the time. Until they do not. An example: evaluate your employees using it. Most will fall near the average. A few will overperform—the top performers, recognize them for promotions, while a few will underperform—the low performers, identify them as needing improvement.

Real life implications of normal distribution:

• Education: standardized tests like the SAT and GRE are often designed taking normal distribution into account, thus you can evaluate your score relative to the mean using z-scores—deviations from the mean;

• Finance: stock price changes, portfolio returns, and economic indicators often distribute normally, so you can use the bell curve to assess risk and model potential investment outcomes;

• Quality: in manufacturing, the product dimensions, weight, and defect rates often follow normal distributions, so you may monitor your production processes to ensure quality is within acceptable limits;

• Medicine: health factors like blood pressure, cholesterol levels, and other biological markers are modeled using normal distributions, thus, as a physician, you can infer lab results relative to population norms to diagnose conditions with relative accuracy;

• Machine learning: algorithms like Gaussian Naive Bayes and data standardization rely on normal distributions, so to ensure model accuracy you can ensure the data follows a bell curve or use transformations when it does not;

• Weather: even something as uncertain as the weather errs toward a normal distribution, thus you can predict the upcoming weather and identify anomalies or trends—climate science in a nutshell.

How to apply normal distribution as a thinking tool: (1) collect data, gathering a sample of observations to parse into your calculation; (2) visualize the distribution, plotting the data to observe whether it follows a bell-like shape; (3) check whether it is normally distributed; (4) apply probability to make predictions using your gathered data. Though be careful. Biological systems tend to follow normal distributions because they show how processes were optimized by virtue of evolution. Real-world data, however, can deviate from perfect normality. Address these possibilities. Extreme values can ruin your calculations.

Thought-provoking insights. “Nature prefers balance, but the extremes shape the narrative.” highlights how most events center around the mean but outliers get the most of our attention—the winners and losers. “In randomness lies order.” reflects how seemingly random variations can converge into predictable patterns. “The bell curve humbles us all.” suggests how predictable human behavior is and how little variability there actually is. The most successful men and women on the planet are just like you. You can do it too.

Questions to reflect on:

1. How does understanding normal distribution help you interpret your field?

2. What are some real-life phenomena I run into that follow a normal distribution?

3. How can outliers affect the normal distribution of a given data set?

4. What predictions and/or decisions could normal distribution help me with?

5. What are the limitations of normal distribution in statistical analyses?

Quotes to stretch your neurons:

1. "Statistics are ubiquitous. The only thing which varies is the amount of understanding that accompanies it." - W. Edwards Deming, American engineer and statistician.

2. "Statistical thinking will one day be as necessary a qualification for efficient citizenship as the ability to read and write." - H.G. Wells, English writer.

3. "All models are wrong, but some are useful." - George E.P. Box, British statistician.

Example use cases:

1. Quality control: companies use normal distribution to analyze manufacturing processes, seeing that products meet quality standards by monitoring variations and identifying deviations.

2. Finance: analysts use normal distribution to model stock returns and assess investment risks, helping them make informed decisions based on the probability of outcomes in lieu of guesswork.

3. Medicine: medical researchers utilize normal distribution to analyze patient data, such as blood pressure or cholesterol levels, to determine what constitutes a "normal" and/or "optimal" range to identify potential health risks.

4. Sociology: researchers in psychology and social sciences use normal distribution to examine human behavior and traits, like intelligence-quotient scores or height, which follow a normal distribution pattern.