InvNorm Calculator

Enter the cumulative probability, mean, and standard deviation to calculate the corresponding Z-score or X-value.

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InvNorm Calculator:

The online InvNorm calculator allows you to quickly compute the inverse normal distribution for given probabilities, mean, and standard deviation. It also generates a visual graph showing left-tail, right-tail, and two-tail areas based on the entered values.

What Is Inverse Normal Distribution?

The inverse normal distribution is used to find a specific value (X) corresponding to a cumulative probability in a normal distribution. Unlike the standard normal process that finds probabilities for a given value, this method finds the value for a specified probability.

Inverse Normal Formula:

The InvNorm calculation uses the following key parameters:

Cumulative Probability (Required): The probability that a variable is less than or equal to a given value.
Mean (μ) (Required): The average value of the distribution.
Standard Deviation (σ) (Required): Measures the spread of the data around the mean.

Mathematically, the inverse normal function is the reverse of the cumulative distribution function (CDF). The CDF of a normal distribution is expressed as:

f(x, μ, σ) = (1 / (σ√(2π))) ∫^x_-∞ e^{-((t - μ)² / (2σ²))} dt

Where:

μ = mean
σ = standard deviation
x = value of the variable

Since there’s no simple algebraic formula for the inverse normal distribution, calculators use numerical approximations and algorithms to determine X from a given probability.

How To Use The InvNorm Calculator?

Follow these steps to calculate the inverse normal value easily:

Input the cumulative probability, mean, and standard deviation.
Press the “Calculate” button.
The calculator provides left-tail, right-tail, and two-tail results, along with the confidence interval, and highlights the corresponding area on a graph.

Applications of Inverse Normal Distribution

The inverse normal distribution is widely used to find specific values linked to probabilities:

Determining critical values in hypothesis testing.
Estimating Value-at-Risk (VaR) in finance.
Setting quality control limits in manufacturing.
Supporting statistical inference by quantifying uncertainty for expected outcomes.

References:

Wikipedia: Inverse distribution.

Fiveable: Inverse Normal Distribution Guide.

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