Distribution Fundamentals: Some Things Are Far From Average

Although normal distributions are most well-known in statistics, many processes follow abnormal or non-normal distributions. Non-bell-shaped curves, that is. In many cases, curves are shaped entirely differently: beta, exponential, gamma, log normal, uniform, skewed, symmetric, and countless other distributions exist. For example, bacteria growth follows an exponential curve while the lifetime of products adheres to a Weibull distribution. Unlike normal distribution — which is symmetric, unimodal, and characterized by low tails — abnormal distributions are skewed, with multiple peaks, or even flat shapes.

Left-skewed distributions have a fat left tail. Right-skewed distributions are their right counterpart, where most values cluster to the left. For example, income tends to be right-skewed — most people earn moderate incomes while a few earn disproportionate amounts. Heavy tails has a higher likelihood of extreme values. Financial returns often follow these, meaning rare but extreme events like market crashes occur more frequently that a bell curve would predict. Multimodal distributions have multiple peaks, indicating the presence of distinct subpopulations. Test scores from a diverse student body fall under these, since there can be a significant difference in students’ educational backgrounds.

Flat distributions are where outcomes are all equally likely, like rolling a fair dice. Each outcome from 1 to 6 has the same probability. Lognormal distributions arise when a variable is the product of many independent factors. The size of cities, stock prices, and biological measures follow lognormal curves. A pareto distribution characterizes phenomena with the Pareto proportion, where a small number of occurrences account for the majority of the effect. Wealth distribution, in economies where a small fraction of the population holds the majority of the money, fall under this umbrella.

The point is that real-world phenomena cannot adequately be described by any one model. Real-world processes involve multiple underlying factors and multiplicative processes that result in non-bell-curves. If a dataset combines groups with different characteristics, the distribution suddenly has more than one peak value. In fields like finance, natural disasters, and social phenomena, the presence of rare but significant events forms heavy tails. Think length of stay data: most patients leave the hospital after a short time, but a handful have complications and stay many times longer. Using statistics in day-to-day decision-making is great. Just don’t let over-simplification lead you to false conclusions.

Real-world examples of abnormal distributions:

• Income: in many economies, most people earn moderate incomes while a tiny percentage earn exceptionally high amounts. The right-skewed distribution means the income is pulled upward by high earners.

• Wealth: wealth is even more concentrated than income. Often, a small fraction of the population has a large share of total wealth, a classic example of the Pareto curve.

• City Size: the distribution of city population tends to follow Zipf’s law. A few megacities house enormous populations, while most cities are exponentially smaller. This heavy-tailed curve shows that the largest cities are so much larger than the average that they contain the majority of the people.

• Asset Returns: stock and other financial asset returns often exhibit heavy tails. Extreme fluctuations — both gains and losses — occur much more frequently than people think. This non-normal behavior is critical to manage risk, as it means rare events like market crashes are more common than a bell curve would suggest.

• Earthquake Magnitude: the frequency and magnitude of earthquakes follows the Gutenberg-Richter curve. Small earthquakes are common. Large ones are rare. But their impacts are disproportionate, skewing the graph.

• Viral Content: the number of likes, views, and shares on online platforms follows a heavy-tailed distribution. Most content receives modest attention, but some posts go viral and attract millions of views. This explains why averages are misleading in digital media analytics.

How you might use abnormal distributions as a mental model: (1) close your eyes — always begin by visualizing the distribution of data to better understand it; (2) assume abnormality — choose models that welcome skewness and heavy tails, and graph it out to really see the data in action; (3) manage your eggs — in fields like finance, consider the implications of heavy tails and plan for the possibility of extreme events that deviate significantly from the average with stress tests and scenario analyses; (4) make informed decisions — be cautious about using the mean as the sole measure, acknowledging that a few extreme values can distort averages and mislead your choices; (5) segment data where applicable — if tracking for marketing, recognize that the average customer doesn’t always encompass the majority experience, and categorize them into groups (e.g. high spenders, average spenders, old, young, tech-savvy, etc.) to better understand customer actions.