The invNorm( Command

Command Summary

Calculates the inverse of the cumulative normal distribution function.

Command Syntax

invNorm(probability[,μ, σ])

Menu Location


  1. 2ND DISTR to access the distribution menu
  2. 3 to select invNorm(, or use arrows.

Calculator Compatibility


Token Size

2 bytes

invNorm( is the inverse of the cumulative normal distribution function: given a probability, it will give you a z-score with that tail probability. The probability argument of invNorm( is between 0 and 1; 0 will give -1E99 instead of negative infinity, and 1 will give 1E99 instead of positive infinity

There are two ways to use invNorm(. With three arguments, the inverse of the cumulative normal distribution for a probability with specified mean and standard deviation is calculated. With one argument, the standard normal distribution is assumed (zero mean and unit standard deviation). For example:

for the standard normal distribution

for the normal distribution with mean 10 and std. dev. 2.5


This is the only inverse of a probability distribution function available (at least on the TI-83/84/+/SE calculators), so it makes sense to use it as an approximation for other distributions. Since the normal distribution is a good approximation for a binomial distribution with many trials, we can use invNorm( as an approximation for the nonexistent "invBinom(". The following code gives the number of trials out of N that will succeed with probability X if the probability of any trial succeeding is P (rounded to the nearest whole number):


You can also use invNorm() to approximate the inverse of a t-distribution. Since a normal distribution is a t-distribution with infinite degrees of freedom, this will be an overestimate for probabilities below 1/2, and an underestimate for probabilities above 1/2.


Unlike the normalpdf( and normalcdf( commands, the invNorm( command does not have a closed-form formula. It can however be expressed in terms of the inverse error function:

\begin{align} \operatorname{invNorm}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1) \end{align}

For the arbitrary normal distribution with mean μ and standard deviation σ:

\begin{align} \operatorname{invNorm}(p,\mu,\sigma)=\mu+\sigma\operatorname{invNorm}(p) \end{align}

Related Commands


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