The Normal Distribution

These functions give you access to the normal distribution. For more information about Fathom's distribution functions, click here.

For all of these functions, mu is the mean and sigma the standard deviation.

The defaults are mu = 0 and sigma = 1.

normalDensity(x, mu, sigma)

The probability density at x for the normal distribution with mean mu and standard deviation sigma.

If you plot the function normalDensity(x), you see the familiar bell curve.

normalCumulative(x, mu, sigma)

The cumulative probability function: the probability that a value from a normal distribution mean mu and standard deviation sigma will be less than or equal to x, the independent variable.

For example, normalCumulative(1.96) = 0.975. That is, 97.5% of the distribution is lower than +1.96.

Another example: normalCumulative(0) = 0.5.

normalQuantile(c, mu, sigma)

The quantile function: it gives the value for which the cumulative probability is c for the normal distribution with mean mu and standard deviation sigma.

So normalQuantile(0.025) = 1.96. That is, 1.96 is the x-value that captures 2.5% of the distribution.

Another example: normalQuantile (0.5) = 0.

Here are the three normal functions plotted together. Notice that the quantile function is the inverse of the cumulative function.

Notice also that the density function is the derivative of the cumulative function; similarly, the cumulative function, by design, is the area under the density function--its integral.

See also statistical functions (for mean( ) and its cousins).