# Inverse Cumulative Normal Probability Calculator

Instructions: Compute the inverse cumulative normal probability score for a given cumulative probability. Give a cumulative probability $$p$$ (a value on the interval [0, 1]), specify the mean ($$\mu$$) and standard deviation ($$\sigma$$) for the variable $$X$$, and the solver will find the value $$x$$ so that $$\Pr(X \le x) = p$$.

Pop. Mean ($$\mu$$)
Pop. St. Deviation ($$\sigma$$)
Cumulative Probability ($$p$$)

This Inverse Cumulative Normal Probability Calculator will compute for you a score $$x$$ so that the cumulative normal probability is equal to a certain given value $$p$$. Mathematically, we find $$x$$ so that $$\Pr(X \le x) = p$$.

Example: Assume that $$X$$ is a normally distributed variable, with mean $$\mu = 500$$ and population standard deviation $$\sigma = 100$$. Let us assume we want to compute the $$x$$ score so that the cumulative normal probability distribution is 0.89. First, the z-score associated to a cumulative probability of 0.89 is

$z_c = \Phi^{-1}(0.89) = 1.227$

This value of $$z_c = 1.227$$ can be found with Excel, or with a normal distribution table. Hence, the X score associated with the 0.89 cumulative probability is

$x = \mu + z_c \times \sigma = 500 + 1.227 \times 100 = 622.7$

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