Inverse Normal (InvNorm) Calculator
Enter a cumulative probability to find the z-score, or add a mean and standard deviation for the raw value.
The inverse normal function answers the reverse of the normal distribution: given a cumulative probability, it returns the value below which that proportion falls. For the standard normal, a probability of 0.975 gives a z-score of 1.96. With a mean and standard deviation, it returns the matching raw value.
How to Use This Calculator
Enter the cumulative probability โ the area to the left under the normal curve โ as a number between 0 and 1. Leave the mean at 0 and standard deviation at 1 for a standard-normal z-score, or enter your own mean and standard deviation to get the corresponding value in your distribution.
This is the function labelled invNorm on graphing calculators, used constantly for confidence intervals and percentiles.
How the Calculation Works
The normal cumulative distribution turns a value into a probability; the inverse turns a probability back into a value. There is no simple closed form, so the calculator uses a well-known rational approximation accurate to many decimal places.
For a non-standard normal, the z-score is scaled and shifted: multiply by the standard deviation and add the mean. That converts the standard-normal answer into the value in your own distribution.
z = invNorm(p); value = mean + z ร SD
Worked Example
For a probability of 0.975 with the standard normal (mean 0, SD 1), invNorm gives z = 1.96 โ the familiar value behind a 95% confidence interval, which leaves 2.5% in each tail.
With a mean of 100 and standard deviation of 15, the same probability gives 100 + 1.96 ร 15 = 129.4, the 97.5th percentile of that distribution.
Assumptions and Limitations
- Uses the cumulative probability to the left (lower tail)
- Probability must be strictly between 0 and 1
- Uses an accurate rational approximation of the inverse normal
- Assumes a normal distribution
When to Use an Inverse Normal Calculator
Statistics students use this to find the z-score or value that corresponds to a given probability โ the inverse of the normal distribution. Run it to get critical values for confidence intervals and hypothesis tests, or to find a percentile cutoff. Enter the probability to get the matching z or value.
Related Calculators
Discriminant Calculator
Find bยฒ โ 4ac and what it tells you about a quadratic's roots.
Open calculator โLinear Regression Calculator
Least-squares line, slope, intercept and Rยฒ from data.
Open calculator โSample Variance Calculator
Sample and population variance and standard deviation from data.
Open calculator โPoisson Distribution Calculator
Probability of k events given an average rate ฮป.
Open calculator โFrequently Asked Questions
What does invNorm do?
It returns the value below which a given proportion of a normal distribution falls. Given a cumulative probability, it gives the z-score (or, with a mean and SD, the raw value) โ the reverse of the normal CDF.
What is invNorm(0.975)?
About 1.96 for the standard normal. It is the z-score with 97.5% of the area to its left, which is why 1.96 appears in 95% confidence intervals.
How do I get a value instead of a z-score?
Enter your distribution's mean and standard deviation. The calculator computes value = mean + z ร SD, converting the standard-normal z-score into your distribution's value.
What probability do I enter?
The cumulative probability to the left of the value โ the area under the curve up to that point. For a right-tail or two-tail problem, convert it to a left-tail probability first.
Is invNorm the same as the z-table in reverse?
Yes. A z-table converts z-scores to probabilities; invNorm converts probabilities back to z-scores, using a precise approximation instead of a printed table.
Looking for a different calculator?
CalculatorPlus has free tools across finance, construction, math, health and more โ each one showing the formula and a worked example.