Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Question

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... Sigmas away from the mean?

Answer 1

The probability distribution function (pdf) of a normal distribution with mean $\mu$ and standard deviation $\sigma$ is defined to be:

$$f(x) = \dfrac{1}{\sigma \sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

For $x= \mu\pm\sigma$ this simplifies to $f(\mu\pm\sigma) = \dfrac{1}{\sigma\sqrt{2\pi e}}$

For $x= \mu\pm 2\sigma$ this simplifies to $f(\mu\pm 2\sigma) = \dfrac{1}{\sigma e^2\sqrt{2\pi}}$

For $x= \mu\pm 3\sigma$ this simplifies to $f(\mu\pm 3\sigma) = \dfrac{1}{\sigma e^4\sqrt{2\pi e}}$

As you can see, it doesn't simplify completely and will depend on what $\sigma$ is (unlike the cdf such as in the "rule of 3").

For the standard normal distribution where $\mu = 0$ and $\sigma = 1$ it can simplify a bit further.

$N(0) = \dfrac{1}{\sqrt{2\pi}}\approx 0.399$

$N(\pm 1) = \dfrac{1}{\sqrt{2\pi e}} \approx 0.24$

$N(\pm 2) = \dfrac{1}{e^2\sqrt{2\pi}} \approx 0.054$

$N(\pm 3) = \dfrac{1}{e^4\sqrt{2\pi e}} \approx 0.0044$

Personally, I don't find these numbers helpful or useful to memorize, so I'm not sure what you would want to do with them except use them to draw a more accurate normal curve. Any math using normal distributions generally uses calculus and integrals to find area under the curve or tables which already computed those areas for us. Specific values of the function are less relevant.