Enter a probability (area), choose your tail direction, and specify the mean (μ) and standard deviation (σ) to find the corresponding x-value on the normal distribution. The Inverse Normal Distribution Calculator also returns the Z-score for your result — useful for hypothesis testing, quality control, and standardized scoring.
x-value
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Z-score
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Area to the Left of x
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Area to the Right of x
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The inverse normal distribution works backward from the cumulative normal distribution. Instead of finding the probability from a known x-value, it finds the x-value that corresponds to a given probability (area under the curve). It is often called the quantile function or percent-point function of the normal distribution.
Invnorm (inverse normal) finds the x-value such that the area under the normal curve up to that point equals the specified probability. For example, invnorm(0.95, 0, 1) returns the z-score (≈1.6449) below which 95% of values fall in a standard normal distribution.
Use the normal distribution when you have an x-value and want to find the probability. Use the inverse normal distribution when you already know the probability (or percentile) and need to find the corresponding x-value. A classic use case is determining cutoff scores for the top 5% of test takers.
The Z-score is calculated as Z = (x − μ) / σ, where μ is the mean and σ is the standard deviation. This standardizes the x-value to the standard normal distribution (mean 0, SD 1). Our calculator returns the Z-score automatically alongside the x-value.
Set the probability to 0.30, the tail to 'Left tail', and use your distribution's mean and standard deviation. For a standard normal distribution (μ=0, σ=1), this gives x ≈ −0.5244, meaning 30% of values fall below that point.
The tails are the extreme left and right ends of the normal distribution curve. The left tail represents low values, and the right tail represents high values. Two-tailed calculations split the probability between both extremes, which is common in hypothesis testing.
Yes. Simply enter your distribution's mean (μ) and standard deviation (σ). The calculator converts the result from the standard normal distribution to your specified distribution using the formula x = μ + Z × σ.
IQ scores follow a normal distribution with μ = 100 and σ = 15. To find the minimum IQ score for the top 2% of the population, set probability = 0.98, tail = left, mean = 100, SD = 15. The result is approximately x = 130.8, confirming the common 'genius' threshold.