A power law distribution is a statistical distribution that follows the form of y = ax^k. They’re seen everywhere, I can almost guarantee that you have seen one. Next time you are looking for a tool to analyze a situation, see if a power log distribution can help.

The top 5 key properties (according to me) of a power law distribution are:  

• Heavy Tail - Power law distributions have a “fat” or “heavy” tail, meaning that extremely large values have a higher probability of occurring compared to a normal distribution.
• Scale Invariance - The distribution retains its general shape even when scaled up or down. There is no characteristic scale. Zooming in on the head or tail looks similar to the whole distribution.
• Straight Line on Log-Log Plot - Taking the logarithms of both the x and y axis values produces a linear relationship, evident due to the power relationship between x and y.
• Source of Self-Organizing Behavior - Power laws often arise from systems with feedback loops, networks, recursion, or optimization taking place as they grow and evolve. The processes lead to self-organization following a power law pattern.
• Rare Events Have Large Impacts - Due to the heavy tail, the largest events in a power law distribution are substantial outliers that have meaningful impact on the total system behavior. These extremely rare events carry significance despite their infrequency.

Here are 5 compelling places where you have most likely seen power log distributions:  

• Sales of Consumer Products - The volume of sales across consumer products follows a power law distribution, with a small number of products selling in extremely high volumes (blockbusters) and increasingly more niche products selling fewer units.
• Population of Cities - The populations of cities within a given country or geographical area roughly follow a power law distribution, with a few megacities and many smaller cities.
• Frequency of Words - Power laws have long been observed in the relative word frequencies across languages where most words occur rarely, and some extremely frequently.
• Website Traffic - The distribution of traffic across websites follows a power law distribution. A very small number of websites (like Google, Facebook, Amazon) drive massive amounts of internet traffic, while increasingly more websites have progressively fewer visitors.
• Citations of Scientific Papers - The accumulated citations received by published scientific papers follows a power law distribution where most papers get few if any citations, while a select set of papers attract a disproportionately high number of citations over time.

I noticed an interesting real-world manifestation of a power law distribution in my daughter’s early development. Around 2 years old, she rapidly progressed in her language abilities. Reflecting back, major milestones seem to follow a doubling pattern - she started walking confidently around 1 year old, and demonstrated greater mobility and happiness around 6 months. Extrapolating this pattern, age 4 marks a transition into social schooling, age 8 potentially coincides with biological changes from the onset of puberty, 16 encompasses newly acquired driving privileges and economic participation, and while skipping age 32, for being too close to me, ahead to age 64, retirement age shapes lifestyle changes for many at that stage of life. Although imprecise, the major inflection points in human development from birth through late adulthood appear to loosely conform to a power law distribution, with capacity, freedom, and responsibilities doubling over set intervals. While more research would be needed to formalize this as a principle, the anecdotal pattern correlates to observable real-world behavior.