Probability · Interactive · September 13, 2026
Every Gaussian bell shape comes from the same formula. Only two numbers, μ and σ, decide where it sits and how wide it spreads. Drag the sliders below and watch each piece of the formula light up as it moves the curve.
The normal distribution shows up everywhere in probability: as the limit in the central limit theorem, as the marginal law of Brownian motion at a fixed time, and as the default noise model in most of statistics and machine learning. Its density is usually memorized as a single expression, which hides the fact that each part of it has one clear job.
The goal of this post is to separate those jobs. The mean controls location, the standard deviation controls spread, and the constant in front exists only to make the total probability equal to one.
Slides the whole curve along the x-axis. The peak always sits directly above μ, and nothing about the curve's shape changes.
A small σ makes a tall, narrow spike (values cluster tightly around μ). A large σ makes a short, wide curve (values scatter further out).
The 68–95–99.7 rule: how much probability sits within one, two and three standard deviations of the mean, for any normal random variable, regardless of μ and σ.
≈ 68.3% of the probability lies within one standard deviation of the mean. This is the darkest band on the chart.
≈ 95.4% lies within two standard deviations, the range most "typical" outcomes fall into.
≈ 99.7% lies within three. Anything beyond this is a genuine tail event.
Shape versus location, and why linear combinations of independent normals stay normal.
Changing μ only translates the curve: every point on it moves by the same amount. Two normals with the same σ but different μ are exact copies of each other, just shifted. For example, N(10, 1) has exactly the shape of N(0, 1), centered 10 units to the right.
Changing σ rescales the width while the total area under the curve stays exactly 1, so a narrower curve must get taller to compensate. Compare N(0, 1/4), which is tall and narrow, with N(0, 4), which is short and wide. The peak height is 1/(σ√(2π)).
For independent normals X and Y, αX + βY is again normal, with mean αμX + βμY and variance α²σX² + β²σY². The bell shape is preserved under linear operations; only its two numbers change.