Why the Normal Distribution Is Called "Normal"
The normal distribution (Gaussian distribution) is a probability distribution that appears across countless natural and social phenomena. Height, weight, IQ test scores, and measurement errors all tend toward normality when many independent factors combine additively. The Central Limit Theorem provides the mathematical guarantee for this convergence.
The name "normal" does not imply that this distribution is normative or standard. It earned the label historically because it was the earliest studied and most widely applied distribution. In reality, many important phenomena - income distributions, city population sizes, earthquake magnitudes - do not follow normal distributions at all.
The 68-95-99.7 Rule
The most practically useful property of the normal distribution is the fixed relationship between distance from the mean and probability. Exactly 68.3% of observations fall within one standard deviation of the mean, 95.4% within two standard deviations, and 99.7% within three. This is known as the "68-95-99.7 rule."
MyRank's height ranking exploits this property. Given that Japanese adult males have a mean height of 171.0 cm with a standard deviation of 5.5 cm, a person standing 182 cm tall is approximately two standard deviations above the mean, placing them at the 97.7th percentile (top 2.3%). When data follows a normal distribution, only two parameters - mean and standard deviation - suffice to calculate any percentile precisely.
When Normality Fails
Income distribution is the canonical example of non-normality. Income has a lower bound of zero (it cannot be negative) and no upper bound, producing a long right tail. Such distributions are modeled using log-normal or Pareto distributions. Assuming normality when calculating income percentiles underestimates top earners and overestimates the position of lower earners.
This is precisely why MyRank's income ranking uses empirical cumulative distribution functions from actual data rather than assuming normality. Selecting the appropriate statistical model based on the actual shape of the data distribution is a prerequisite for accurate ranking calculations.
Intuitive Understanding of Standard Deviation
Standard deviation measures "how spread out the data is," but many find it difficult to grasp intuitively. The simplest interpretation is: "how far a typical individual deviates from the average."
A standard deviation of 5.5 cm for height means that a randomly selected person's height will fall within about 5.5 cm of the mean with 68% probability - this is "normal." Being more than 11 cm (two standard deviations) from the mean is "quite unusual" (under 5%), and more than 16.5 cm (three standard deviations) is "extremely rare" (under 0.3%).
The Practical Relationship Between Rankings and Normal Distribution
In normally distributed data, percentile "density" is highest near the center. Small value changes near the 50th percentile produce large rank shifts, while large value changes near the 1st or 99th percentile barely move one's rank at all.
This matches intuition. The difference between 170 cm and 172 cm significantly affects rank position, but the difference between 190 cm and 192 cm has negligible impact. Understanding that the "weight of one rank position" varies with distributional shape helps avoid overreacting to small fluctuations in one's position.