Normal Distribution
The bell curve describes many natural phenomena, from heights to test scores. Its simplicity and ubiquity make it fundamental to statistics.
y = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
확률
Probability Visualization & Education
Understanding the mathematics of randomness
The bell curve describes many natural phenomena, from heights to test scores. Its simplicity and ubiquity make it fundamental to statistics.
y = (1/(σ√(2π))) × e^(-(x-μ)²/(2σ²))
Every outcome has equal probability. Like a fair die roll or a random selection from a range, uniformity represents perfect impartiality.
P(X) = 1/(b-a) for a ≤ x ≤ b
Models the time between rare events—waiting for a bus, radioactive decay, or customer arrivals. It captures the essence of aging.
f(x) = λe^(-λx) for x ≥ 0
Experiment with probability in real time
Roll virtual dice thousands of times and watch the distribution converge to uniformity.
Flip a virtual coin and observe how probability stabilizes with repetition (Law of Large Numbers).
Adjust mean (μ) and standard deviation (σ) to see how the curve breathes and transforms.
Core concepts explained with clarity and precision
Probability measures the likelihood of an event occurring. It ranges from 0 (impossible) to 1 (certain). It's the language we use to describe uncertainty in a quantifiable way.
Events are independent when the outcome of one doesn't affect the other. Most real-world events are interdependent, creating rich causal structures worth exploring.
The average outcome if an experiment is repeated infinitely. It guides decision-making in gambling, insurance, and economics—the rational choice maximizes expected value.
No matter the distribution of individual samples, their average approaches a normal distribution. This explains why so many phenomena follow the bell curve.
Where mathematics meets art
Probability isn't about chaos—it's about pattern. The same mathematical principles that govern a coin flip govern stellar distributions across galaxies. Randomness, properly understood, is deeply structured and fundamentally beautiful.
The normal distribution is nature's signature, appearing in measurements across countless domains. It emerges not through design, but through the mathematics of aggregation—a testament to the deep orderliness of the universe.
On hwaglyul.com, we celebrate this beauty. Every distribution is rendered as a visual poem, every simulation a dance of numbers, every calculation a step toward understanding the fundamental laws that govern reality.