# 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 $$p$$.

### How does this invnorm calculator?

In simple words, this calculator finds a z-score associated to a given probability value. The invnorm calculator z-score that is found is then converted to the required X score

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

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

### Can you use invNorm without calculator?

The short answer is NO. Normal probability calculations are complex mathematical operations for which you need either a normal table, or a statistical software, or a hand calculator.

### Inverse normal distribution

The normal distribution and its inverse are broadly used in statistics, and it is worth it having a deep understanding of it.

The normal distribution computes probabilities associated to scores, where the inverse normal distribution computes scores that are associated to given probabilities.

### Should I use a normal table or a calculator?

By far, a calculator will be better, with some caveats:

• Using a inverse normal calculator will provide far more precise results than a table
• Tables are usually accurate up to 4 or 5 digits, and calculators are typically accurate up to 15 digits most typically
• With a calculator you can type in any value, whereas with a table you may need to interpolate for values that are not actually in the table
• Sometimes instructors asks specifically to use table values, so in that case you will need to learn how to use them

### The Standard Normal Distribution

If you are dealing specifically with the standard normal distribution, you could check this Inverse Cumulative Standard Normal Probability Calculator .

Other graph creators that you could use are our normal probability plot , normal distribution grapher or our Pareto chart marker .

### Example: Inverse Cumulative Normal distribution

Question: Assume that X has a normal distribution with a mean of 12, and a population standard deviation of 3.5. Find the score of the distribution such that 45% of scores of the population are to its left.

Solution:

The following are the population mean $$(\mu)$$, population standard deviation $$(\sigma)$$ and cumulative probability provided:

 Population Mean $$(\mu)$$ = $$12$$ Population Standard Deviation $$(\sigma)$$ = $$3.5$$ Cumulative Probability = $$0.45$$

We need to find a score $$x$$ so that the corresponding cumulative normal probability is equal to $$0.45$$. Mathematically, $$x$$ is such that:

$\Pr(X \le x) = 0.45$

where $$X$$ is normally distributed with a population mean $$\mu = 12$$ and a population standard deviation $$\sigma = 3.5$$

First, we corresponding $$z$$ score so that the cumulative standard normal probability distribution is equal to $$0.45$$. So we are looking for $$z_c$$ such that

$\Pr(Z \le z_c) = 0.45$

By using a normal distribution table (or some statistical calculator like Excel can be used too), we find that $$z_c = -0.13$$. Hence, the X score associated with a cumulative probability of $$0.45$$ is

In other words, we have found a z-score $$z_c = -0.13$$ with the property that

$\Pr(Z \le -0.13) = 0.45$

Hence, the X score associated with a cumulative probability of $$0.45$$ is

$\begin{array}{ccl} X & = & \mu + z_c \times \sigma \\\\ \\\\ & = & 12 + (-0.13) \times 3.5 \\\\ \\\\ & = & 11.545 \end{array}$

Therefore, the $$X$$ for which the cumulative normal probability is $$0.45$$ is $$X = 11.545$$.