InvNorm Calculator

What Is the Inverse Normal Distribution?

The inverse normal distribution function – commonly called invnorm – works backwards from the standard normal CDF. Instead of asking “what’s the probability of being at or below a value?”, invnorm asks:

Given a probability, what value corresponds to that cumulative area under the normal curve?

This reversal is essential for statistical analysis. When you know the probability you need (like 95% confidence) but need to find the actual cutoff value, the inverse normal function delivers that answer. The calculator above computes Φ⁻¹(p), the inverse of the standard normal cumulative distribution function, then transforms it to your specific distribution using the mean and standard deviation you provide.

Mathematically, if you enter a probability p, the invnorm calculator finds the unique value x where P(X ≤ x) = p. For the standard normal distribution (μ=0, σ=1), this returns the z-score. For any other normal distribution, the calculator applies the transformation x = μ + z × σ to give you the actual value in your distribution’s units.

Tips/Instructions for Using this InvNorm Calc

Enter your desired cumulative probability as a decimal (0.95) or percentage (95%). Then specify your distribution’s mean (μ) and standard deviation (σ). If you’re working with the standard normal distribution, leave these at 0 and 1 respectively.

Select the appropriate probability type based on your problem:

• Left Tail: Use when your probability represents the area to the left of the unknown value. This is the default for most invnorm calculations and answers “below what value does p% of the data fall?”
• Right Tail: Use when your probability represents the area to the right. This answers “above what value does p% of the data fall?”
• Center: Use for confidence intervals where your probability represents the symmetric middle area. Entering 0.95 returns the z-score that captures 95% of the distribution between -z and +z.
• Two Tails: Use when your probability represents the combined area in both tails. Common for hypothesis testing where you need critical values for a two-tailed test.

The calculator instantly returns both the z-score and the x-value for your distribution, along with a visualization showing exactly where your result falls on the normal curve.

InvNorm on TI-84 and Other Calculators

The invnorm function appears on most statistical calculators under slightly different names.

On the TI-84 and TI-83 series, you’ll find it as invNorm( under the DISTR menu (2nd → VARS). The syntax is invNorm(area, μ, σ) where area is your cumulative probability from the left.

On Casio calculators, the function is typically labeled InvNormCD or accessed through the Statistics mode. TI-Nspire users can access it through Menu → Statistics → Distributions → Inverse Normal.

Our online invnorm calculator provides the same results as these handheld calculators but adds visual feedback and handles all tail types automatically. It’s particularly useful when you need to verify calculator results or don’t have your device handy.

Common InvNorm Applications

1) Finding Critical Z-Values for Confidence Intervals

The most frequent use of invnorm is determining critical values for confidence intervals.

For a 95% confidence interval, you need the z-score that captures 95% of the distribution in the center. Using the center tail option with probability 0.95, the invnorm calculator returns z* = 1.96. This means your confidence interval extends 1.96 standard errors in each direction from the sample mean.

Common critical z-values every statistics student should know: z* = 1.645 for 90% confidence, z* = 1.96 for 95% confidence, and z* = 2.576 for 99% confidence.

The invnorm calculator derives these instantly for any confidence level you need.

2) Determining Percentile Cutoffs

Invnorm directly answers percentile questions. If exam scores follow a normal distribution with mean 500 and standard deviation 100, what score marks the 90th percentile? Enter probability 0.90, μ=500, σ=100 using left tail. The calculator returns x = 628.2, meaning a score of 628 or higher places a student in the top 10%.

This application extends to any normally distributed measurement: height percentiles for growth charts, income percentiles for economic analysis, or performance percentiles for employee evaluations.

3) Setting Thresholds in Quality Control

Manufacturing processes use invnorm to establish specification limits. If a filling machine targets 500mL with standard deviation 5mL, and you want only 1% of bottles underfilled, what’s your minimum acceptable fill? With probability 0.01 (left tail), μ=500, σ=5, invnorm returns 488.4mL. Any bottle below this threshold falls in the bottom 1% and should trigger quality review.

Similarly, you can set upper specification limits using the right tail option to control overfilling or identify unusually high readings that warrant investigation.

Making Sense of Your InvNorm Results:

The inverse normal function returns values that map directly to positions on the bell curve.

Here’s how to interpret common results:

• Z-score of 0: Corresponds to probability 0.50 (50th percentile). The value equals the mean—exactly at the center of the distribution.
• Positive z-scores: Values above the mean. A z-score of 1.0 corresponds to probability 0.8413, meaning 84.13% of values fall below this point.
• Negative z-scores: Values below the mean. A z-score of -1.0 corresponds to probability 0.1587, meaning only 15.87% of values fall below this point.
• Extreme z-scores (beyond ±3): Correspond to probabilities very close to 0 or 1. These represent rare values in the extreme tails occurring less than 0.3% of the time.

The relationship between probability input and z-score output follows a predictable pattern. Small probabilities (close to 0) produce large negative z-scores. Large probabilities (close to 1) produce large positive z-scores. The transformation is nonlinear—equal changes in probability don’t produce equal changes in z-score, especially in the tails.

InvNorm vs Normal CDF: The Difference

The normal CDF and invnorm are mathematical inverses of each other. The normal CDF takes a value and returns a probability. Invnorm takes a probability and returns a value. If normCDF(1.96) = 0.975, then invNorm(0.975) = 1.96.

Use the normal CDF when you have a specific value and want to know what percentage of the distribution falls at or below it. Use invnorm when you have a target probability or percentile and need to find the corresponding value.

Think of it this way: normal CDF answers “how unusual is this value?” while invnorm answers “what value marks this level of unusualness?”

The Math Behind This Calculation

The inverse normal distribution has no closed-form solution—you cannot write a simple algebraic formula that directly computes it. Instead, invnorm relies on numerical approximation algorithms that iteratively approach the correct answer.

This calculator uses the rational approximation method developed by Abramowitz and Stegun, which provides accuracy to at least 6 decimal places across the entire probability range. The algorithm divides the probability space into regions and applies optimized polynomial approximations for each region.

For the standard normal distribution, the invnorm function satisfies the relationship: if z = Φ⁻¹(p), then Φ(z) = p, where Φ is the standard normal CDF. The general formula for any normal distribution is:

This transformation shifts the result by the mean (μ) and scales it by the standard deviation (σ), converting the standardized z-score into the actual value within your distribution.

Quick FAQs

What probability should I enter for a 95% confidence interval?
For a two-sided 95% confidence interval, use the center tail option with probability 0.95. This returns z* = 1.96. Alternatively, using left tail with probability 0.975 gives the same z-score, since 95% in the center means 2.5% in each tail.

Why does invnorm return an error for probability 0 or 1?
The inverse normal function is undefined at exactly 0 and 1. A probability of 0 would require negative infinity, and probability 1 would require positive infinity. Enter values very close to these bounds (like 0.0001 or 0.9999) for extreme percentiles.

How do I find the z-score for a specific percentile?
Enter the percentile as a decimal probability using left tail. For the 85th percentile, enter 0.85. The returned z-score tells you how many standard deviations above the mean this percentile falls.

What’s the difference between invnorm and norminv?
These are different names for the same function. TI calculators use “invNorm,” Excel uses “NORM.INV” (or “NORMINV” in older versions), and statistical software like R uses “qnorm.” They all compute the inverse of the normal cumulative distribution function.

Can I use invnorm for non-normal distributions?
No. The inverse normal function specifically applies to normal (Gaussian) distributions. For other distributions, you need their specific inverse CDF functions—such as inverse t-distribution for small samples or inverse chi-square for variance analysis.