The normal distribution (the bell curve)
A random variable has a distribution — and distributions come in canonical shapes. One shows up so often it earned a name: the normal distribution, or "the bell". Let's see how it works and why it appears everywhere.
PDF — density: the height itself is not a probability; the probability of an interval = the area under the curve over it. Total area = 1.
A symmetric "bell": most values crowd near the center, thinning toward the edges. μ sets where the center stands, σ how widely values fan out (σ stretches the bell wider and lower, μ slides it along the axis). Appears wherever a quantity is the sum of many small independent random influences — heights, measurement errors, part deviations. Press "pour" — the real sample\'s histogram approaches the theoretical density.
Before you is the normal distribution curve. Horizontal axis — the value; vertical — how often it occurs. Most values crowd near the middle, and they thin out toward the edges. Hence the bell shape.
What it means
The main gift of the bell is describing a whole mountain of data with just two numbers. Normally, to say anything about a quantity you need the data itself. But if the shape is normal, then μ (where the centre sits) and σ (how widely everything spreads) contain the rest: to answer "what share of people are taller than 190 cm", μ = 178 and σ = 7 is enough — no census required. Two numbers stand in for thousands of observations without losing anything that matters.
The second gift is a common language for things that cannot be compared. Height in centimetres, test scores in points, a part's deviation in microns do not line up directly. But the z-score puts any value on one scale — "how many σ it sits from the centre" — and a person 192 cm tall (z = 2) turns out to be exactly as unusual as a student scoring two sigmas above average. That is why z-scores are everywhere, from medical norms to credit scoring.
And the practical upshot: all of this holds only while the shape really is bell-like. So the first step is not computing σ and percentages but looking at the histogram and making sure you have a bell in front of you, not an asymmetric tail pretending to be one.
Where it shows up
Adult height fits the bell almost perfectly. Medical norms are built the same way: values near the middle count as "normal", and deviation is measured in σ.
Manufacturing too: a part beyond ±3σ is flagged as defective — the famous "three-sigma rule".
The bell became fashionable in the 19th century thanks to Adolphe Quetelet: he noticed that the heights and weights of thousands of soldiers fall onto the normal curve and invented the concept of the "average man" — and, along the way, the body-mass index we still use.
Definitions
When the method lies (assumptions)
The 68-95-99.7 rule works only if the data really is close to normal. Real quantities are often skewed (incomes, session times, receipts) — the shares inside σ-intervals differ there, and an "outlier beyond 3σ" may be an ordinary tail value.
Normality tends to appear in sample MEANS (that is how the CLT works), not in raw data. Before applying the rule to individual observations, check the shape on a histogram.