Equations · Probability and statistics
Normal distribution
The density of the bell curve, centred on the mean μ with spread σ.
f(x) = (1 / (σ√(2π))) e^(−(x − μ)² / (2σ²))
Aethel says
f of x equals the quantity 1 over the quantity sigma the square root of 2 pi, times e to the power of negative the quantity x minus mu, squared over the quantity 2 sigma squared
What it says
The curve peaks at the mean and falls away symmetrically. About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
The factor in front makes the total area one. Sums of many small independent effects tend towards this shape, which is why it appears so often: the central limit theorem.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| μ | mu | the mean, where the curve peaks |
| σ | sigma | the standard deviation, the width of the curve |
| e | e | Euler's number, about 2.71828 |
| π | pi | about 3.14159 |
Where you meet it
Statistics, measurement error, finance and machine learning.
Questions
How do you say Normal distribution out loud?
f(x) = (1 / (σ√(2π))) e^(−(x − μ)² / (2σ²)) is read: “f of x equals the quantity 1 over the quantity sigma the square root of 2 pi, times e to the power of negative the quantity x minus mu, squared over the quantity 2 sigma squared.” That is exactly what Aethel says when it meets the formula in a book.
What does μ mean in Normal distribution?
μ (said “mu”) is the mean, where the curve peaks.
What does σ mean in Normal distribution?
σ (said “sigma”) is the standard deviation, the width of the curve.
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