Tech Tips: Normal Distributions

Calculating Heights of a Normal Curve

Recall that a normal distribution with mean $\mu$ and standard deviation $\sigma$ is one characterized by the function: $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}$$ To find this value (i.e., the height of $f$ at $x$),


Calculating Areas Under a Normal Curve

To find the probability that a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$ results in a value less than $x$ (i.e. the area under the normal curve to the left of $x$.),


Inverse Normal Functions

Suppose one wishes to find the $x$ value for which a normally distributed random variable with mean $\mu$ and standard deviation $\sigma$ will produce an outcome less than $x$ with some given probability of $p$. Equivalently, one seeks the $x$ value where there is an area of $p$ to the left of $x$ and under the related normal curve.


Simulating Random Variables following a Normal Distributions

To generate random values following a normal distribution with mean $\mu$ and standard deviation $\sigma$,