# Inverse Cumulative Normal Probability Calculator

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Instructions:
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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 \(\Pr(X \le x) = p\).

## More about this Inverse Cumulative Normal Probability Calculator

This
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Inverse Cumulative Normal Probability Calculator
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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:
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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

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 \( X = 11.545\).