The bell curve, and why it keeps showing up
The normal distribution is a symmetric bell defined by two numbers: its mean μ (the centre) and its standard deviation σ (the width). It appears everywhere for a structural reason: quantities that result from many small independent influences added together (measurement errors, heights, daily totals) pile up into this shape. That is the central limit theorem at work, and it is why the bell is the default model of variation.
Any normal question becomes a standard normal question through z: how many SDs from the mean is this value? Exam marks with μ = 70, σ = 10: a mark of 85 is z = 1.5, and Φ(1.5) = 0.9332 of marks fall below it. The probability of scoring between 60 and 85: Φ(1.5) − Φ(−1) = 0.9332 − 0.1587 = 0.7745. Above 90 (z = 2): 1 − 0.9772 = 0.0228, about 2.3% of the class.
The empirical rule
Within one SD of the mean: 68.3% of the distribution. Within two: 95.4%. Within three: 99.7%. These three numbers are the fastest sanity check in statistics: data claiming normality should roughly obey them (our commute sample: 65% and 95%), and a z of 1.96 earns its fame by bounding the middle 95% exactly, which is why 95% confidence intervals are built from it.
Assuming normality because it is convenient. Income, city sizes, reaction times and count data are not normal; they are skewed or discrete, and normal calculations on them misprice the tails badly (the interesting region). Check the shape first: a histogram, the 68-95 percentages, or the normal graph tool linked below.
Normal does not mean common, correct or good
The name is a historical accident (it was once thought to be the law of all variation). Plenty of healthy data are nowhere near normal, and plenty of normal-looking data describe things nobody wants. The distribution is a model: useful where it fits, replaceable where it does not.
Frequently asked questions
What are the parameters of a normal distribution?
Its mean mu, which centres it, and its standard deviation sigma, which sets its width. Two normals with the same mu and sigma are the same distribution; change either and the bell moves or stretches.
Why is the area under the curve exactly 1?
The curve is a probability density: area is probability, and total probability is 1 by definition. Heights on the curve are densities, not probabilities; only areas (ranges) carry probability.
What is the difference between normal and standard normal?
The standard normal is the member with mu = 0 and sigma = 1. Every normal converts to it via z = (x - mu) / sigma, which is why one table serves all normal calculations.
Are real data ever exactly normal?
Essentially never. Normality is a model approximation: the question is whether the approximation is close enough for the calculation at hand, checked with plots and the empirical rule, not whether it is perfect.