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The Normal Distribution

The bell curve: centred at μ, spread by σ, with total area 1. Probabilities are areas, and every area comes from one function once x is converted to a z-score.

Step 1 of 12

Speed

The standard normal distribution: mean μ = 0, standard deviation σ = 1. The total area under the curve is 1, and the area over any stretch of the axis is the probability of landing there.

P(-1 \le X \le 1) = \Phi\!\left(\tfrac{1 - 0}{1}\right) - \Phi\!\left(\tfrac{-1 - 0}{1}\right) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = \mathbf{0.6827} \\[6pt] f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad \text{peak } f(\mu) = \frac{1}{1\sqrt{2\pi}} = 0.399

μ (mean)0σ (standard deviation)1a (lower bound)-1b (upper bound)1

The Normal Distribution

The bell curve: centred at μ, spread by σ, with total area 1. Probabilities are areas, and every area comes from one function once x is converted to a z-score.

Step 1 of 12

Speed

The standard normal distribution: mean μ = 0, standard deviation σ = 1. The total area under the curve is 1, and the area over any stretch of the axis is the probability of landing there.

P(-1 \le X \le 1) = \Phi\!\left(\tfrac{1 - 0}{1}\right) - \Phi\!\left(\tfrac{-1 - 0}{1}\right) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = \mathbf{0.6827} \\[6pt] f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad \text{peak } f(\mu) = \frac{1}{1\sqrt{2\pi}} = 0.399

μ (mean)0σ (standard deviation)1a (lower bound)-1b (upper bound)1

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The Normal Distribution

See also

The Normal Distribution

See also