Inverse Normal Distribution Calculator (InvNorm)
Enter a cumulative probability, choose your tail direction, and supply the mean and standard deviation. The calculator returns the x-value and z-score that correspond to that probability in the normal distribution. All four tail modes are supported: left-tailed, right-tailed, two-tailed outside, and two-tailed within. Results update instantly as you type.
Formula
Worked example
IQ scores follow a normal distribution with mean 100 and standard deviation 15. What IQ score separates the bottom 20% from the top 80%? Set p = 0.20 (left-tailed), mu = 100, sigma = 15. z = InvNorm(0.20) = -0.8416. x = 100 + (-0.8416)(15) = 100 - 12.62 = 87.38. So 20% of IQ scores fall below 87.38.
What is the inverse normal distribution?
The normal distribution has a bell-shaped probability density function. The cumulative distribution function (CDF) maps any real number x to the probability that a random draw from the distribution falls at or below x. The inverse of that operation, sometimes written InvNorm or the probit function, answers the reverse question: given a probability p, what value x produces that cumulative area? This calculator solves that inverse problem for any mean, standard deviation, and tail direction.
How to use this calculator
Choose a tail mode first. Left-tailed is the default: you supply P(X < x) and get x. Right-tailed gives you the x above which a fraction p of the distribution falls. The two two-tailed modes work with symmetric intervals around the mean: "outside" finds the cutoffs beyond which a fraction p of the distribution lies, while "within" finds the interval that contains exactly p of the probability (used for confidence intervals). Enter your probability, mean, and standard deviation and the result appears immediately. The z-score is always shown alongside the x-value so you can compare results across different distributions.
The formula and algorithm
The probit function has no closed form, so it is computed numerically. This calculator uses Peter Acklam's rational approximation, which divides the interval (0, 1) into three regions and applies a different rational polynomial in each. The maximum relative error is less than 1.15 x 10^-9, which is more than sufficient for any practical statistical work. Once the standard normal z-score is found, the x-value follows from the linear transformation x = mu + z * sigma, where mu is the mean and sigma is the standard deviation.
Left-tailed, right-tailed, and two-tailed modes
The tail mode determines which area of the curve p represents. In the left-tailed mode, p is the area to the left of x, so p = 0.95 gives the 95th percentile. In the right-tailed mode, p is the area to the right, so p = 0.05 gives the same 95th percentile. In the two-tailed outside mode, p is the combined area in both tails; this is used when you want to find the rejection region for a two-tailed hypothesis test at significance level alpha = p. In the two-tailed within mode, p is the area inside the symmetric interval; this is the confidence level, so p = 0.95 returns the bounds of a 95% confidence interval.
Practical applications
Inverse normal calculations appear in many real-world settings. In quality control, a manufacturer might ask what product dimension corresponds to the bottom 1% of output, to set a rejection threshold. In finance, value-at-risk models use the inverse normal to find the loss exceeded with a given probability. In standardized testing, a score at the 90th percentile is found by computing InvNorm(0.90) on the score distribution. In hypothesis testing, the critical z-value for a 5% significance level is InvNorm(0.975) = 1.96 for a two-tailed test or InvNorm(0.95) = 1.645 for a one-tailed test.
Common z-score critical values
| Confidence level | Significance (alpha) | z-score (each tail) | x1 (mu=0, sigma=1) | x2 (mu=0, sigma=1) |
|---|---|---|---|---|
| 80% | 0.20 | 1.2816 | -1.2816 | 1.2816 |
| 85% | 0.15 | 1.4395 | -1.4395 | 1.4395 |
| 90% | 0.10 | 1.6449 | -1.6449 | 1.6449 |
| 95% | 0.05 | 1.9600 | -1.9600 | 1.9600 |
| 99% | 0.01 | 2.5758 | -2.5758 | 2.5758 |
| 99.5% | 0.005 | 2.8070 | -2.8070 | 2.8070 |
| 99.9% | 0.001 | 3.2905 | -3.2905 | 3.2905 |
Standard cutoffs used in confidence intervals and hypothesis testing (two-tailed within mode, standard normal).
Frequently asked questions
What is the difference between the normal distribution calculator and this one?
The normal distribution calculator (forward direction) takes an x-value and returns the cumulative probability P(X < x). This calculator is the reverse: you supply the probability and it returns the x-value. They are mathematical inverses of each other.
What does InvNorm mean on a TI-84 calculator?
InvNorm is the name Texas Instruments uses for the inverse normal function on their graphing calculators. It takes three arguments: the area (left-tail probability), the mean, and the standard deviation. The result is the x-value such that the area to the left under the normal curve equals the specified probability. This calculator performs the same computation online.
What is the z-score for a 95% confidence interval?
For a 95% confidence interval, the two-tailed within probability is 0.95. This means each tail contains (1 - 0.95) / 2 = 0.025 of the probability. The z-score is InvNorm(1 - 0.025) = InvNorm(0.975) = 1.9600. The confidence interval runs from mu - 1.96 * sigma to mu + 1.96 * sigma. Select "Two-tailed within" mode and enter p = 0.95 to see this directly.
How do I find the 90th percentile of a normal distribution?
Use left-tailed mode, set p = 0.90, and enter your mean and standard deviation. The calculator returns the x-value such that 90% of the distribution lies below it. For the standard normal (mu = 0, sigma = 1) this is z = 1.2816. For IQ scores (mu = 100, sigma = 15) the 90th percentile is approximately 119.2.
What probability values are valid?
The probability p must be strictly between 0 and 1 (exclusive). A value of exactly 0 or 1 would correspond to negative or positive infinity, which is not a finite number. Values very close to 0 or 1 are handled correctly by the rational approximation used here, but values outside (0, 1) produce no result.
Can I use this for a non-standard normal distribution?
Yes. Enter any mean and any positive standard deviation. The calculator first finds the standard normal z-score and then applies the linear transformation x = mu + z * sigma to convert to your distribution. For example, with mu = 500 and sigma = 100 (a common SAT-style scale), the 95th-percentile score is 500 + 1.6449 * 100 = 664.5.
What is the probit function?
The probit function is another name for the inverse of the standard normal CDF. The word "probit" (probability unit) was coined by Chester Bliss in 1934 for use in bioassay and dose-response analysis. In statistics it is written as Phi^-1(p) or InvNorm(p). This calculator evaluates it using a high-accuracy rational polynomial approximation valid for any p in (0, 1).