The phrase “inverse normal distribution” refers to the technique of finding the x value of the normal random variable given the cumulative probability.
How to use a normal probability to find the value of a normal random variable X?
Suppose you are given that the random variable X follows normal distribution with mean 5 and standard deviation 2. You want to find the value of ‘c’ for which the cumulative probability is 0.71, that is, to find ‘c’ such that P(X≤c)=0.71.
The first thing to do is to convert the distribution to standard normal distribution. This can be done by subtracting the mean and then dividing by the standard deviation. The resulting standard normal variate shall be denoted as Z.
We get P(X-5/2≤c-5/2)=0.71 that is,
P(Z≤c-5/2)=0.71 where Z is the standard normal variate.
Now we look at the standard Z table:
We see that the required Z value lies somewhere between 0.55 and 0.56. Let us take the value approximately as the average of the two values, that is, 0.555.
Now set c-5/2=0.555 and solve to get c-5=1.11 that is,
c=6.11 and we hence get the required x value of the normal variate.