The following diagram therefore describes the pdf of any normal distribution.
Observe how the tails of the distribution fade away.
- The distribution almost disappears at 3σ
from µ
- The probability (area) further than 2σ
from µ
is small — only about 1/20 of the total area.
To be more precise, for all normal distributions,
- \(P( X \text{ is within } \sigma \text{ of } \mu) = 0.6827\)
- \(P( X \text{ is within } 2\sigma \text{ of } \mu)
= 0.9545\)
- \(P( X \text{ is within } 3\sigma \text{ of } \mu)
= 0.9973\)
The next probabilities are particularly important.
- \(P( X \text{ is within } 1.645 \sigma \text{ of } \mu) = 0.90\)
- \(P( X \text{ is within } 1.960 \sigma \text{ of } \mu)
= 0.95\)
- \(P( X \text{ is within } 2.576 \sigma \text{ of } \mu)
= 0.99\)
In particular, there is a 95% probability that a normally distributed random variable \(X\) is within 1.96 standard deviations of the distribution's mean.