InvNorm Calculator

InvNorm Calculator

What Is InvNorm?

InvNorm (inverse normal) is a statistical function that finds the value on a normal distribution curve that corresponds to a given cumulative probability. In other words, if you know the probability (area under the curve to the left of a point), InvNorm tells you what that point is.

This is the reverse of the normal CDF (cumulative distribution function). While the CDF tells you the probability of getting a value less than x, InvNorm tells you what x is for a given probability.

The InvNorm Formula

InvNorm(p, μ, σ)
p = cumulative probability (0 to 1)
μ = mean of the distribution
σ = standard deviation

For standard normal (μ=0, σ=1):
InvNorm(0.975) = 1.96 (95% confidence)

The calculation uses an approximation algorithm since there is no closed-form solution for the inverse of the normal CDF. Common approximations include the rational approximation method and the Beasley-Springer-Moro algorithm.

Common InvNorm Values

Several InvNorm values appear frequently in statistics:

InvNorm(0.90) = 1.282 — 80% confidence interval
InvNorm(0.95) = 1.645 — 90% confidence interval
InvNorm(0.975) = 1.960 — 95% confidence interval
InvNorm(0.995) = 2.576 — 99% confidence interval

Applications

InvNorm is used in hypothesis testing, confidence interval construction, quality control, finance (Value at Risk), and any field requiring statistical inference from normally distributed data.

How to Use This Calculator

Enter the probability (between 0 and 1), along with the mean and standard deviation. The calculator instantly shows the corresponding x-value and z-score. For the standard normal distribution, use mean=0 and standard deviation=1.

Understanding Your Results

The z-score tells you how many standard deviations from the mean the result is. The x-value is the actual point on your distribution. A probability of 0.5 always returns the mean, since half the distribution lies below the mean.

Frequently Asked Questions

InvNorm finds the x-value (or z-score) that corresponds to a given cumulative probability on a normal distribution curve.

InvNorm(0.975) = 1.96 for the standard normal distribution. This is the z-score used for 95% confidence intervals.

Enter the cumulative probability (area to the left) as a decimal between 0 and 1. For example, 0.95 for the 95th percentile.

The standard normal distribution has mean μ=0 and standard deviation σ=1. Any normal distribution can be standardized to this form.

InvNorm does the reverse of a z-table lookup. Instead of finding probability from z-score, it finds z-score from probability.

Use InvNorm for constructing confidence intervals, finding critical values in hypothesis testing, and calculating percentiles of normal distributions.