InvNorm Calculator

Use this InvNorm Calculator to find the X value or Z score for any normal distribution probability. Select a left tail, right tail, middle area, or two tail calculation and get instant results with a graph and complete steps.

What Is InvNorm? Meaning and Function Explained

InvNorm is short for inverse normal. The InvNorm function calculates the reverse of the normal cumulative distribution function (CDF). While the normal CDF tells you the probability of observing a value less than or equal to some number, InvNorm does the opposite: you enter a probability, and it returns the corresponding Z score or X value.

In formal notation, if Φ(z) = p, then Φ^-1(p) = z. Here, Φ represents the standard normal CDF, and Φ^-1 represents its inverse. This is the foundation behind finding Z scores from probabilities, calculating percentile cutoffs, and determining critical values for hypothesis testing.

The InvNorm function appears on graphing calculators like the TI-84 (as invNorm), TI-83, TI-89, and Casio models. It is also available in Excel (NORM.INV), Google Sheets (NORMINV), R (qnorm), and Python (norm.ppf). This online InvNorm calculator provides the same result without needing a graphing calculator.

Inverse Normal Distribution Formula

For a standard normal distribution (μ = 0, σ = 1):

z = Φ^-1(p)

For a custom normal distribution with mean μ and standard deviation σ:

x = μ + z × σ

The function Φ^-1 does not have a simple closed-form expression. Calculators and software use numerical approximations, such as rational function approximations refined with iterative methods, to achieve high accuracy.

InvNorm Calculator Modes: Tail, Center, and Two Tails

This calculator supports four probability modes. Choosing the correct mode depends on what region of the normal curve your probability describes:

• Left Tail: Use this when your probability represents the area to the left of the cutoff. This is the default on TI-84 and most software. If you need a percentile cutoff, left tail is the right choice.
• Right Tail: Use this when your probability represents the area to the right. The calculator automatically converts it to a left cumulative probability before computing. Common for questions like “what value is exceeded by only 5%?”
• Center (Middle Area): Use this when your probability is the area in the center of the distribution, symmetric around the mean. This is the standard approach for confidence intervals. Enter 0.95 to find the 95% confidence interval boundaries.
• Two Tails: Use this when your probability is the combined area in both tails. This is typical for two-tailed hypothesis tests. Enter the significance level α (such as 0.05) to find both critical values.

When to Use InvNorm on a Calculator

Use InvNorm any time you know a probability and need to find the corresponding value. Common situations include:

• Finding a percentile: “What score is at the 90th percentile?” Use left tail with p = 0.90.
• Finding a Z score from probability: “What Z score has 97.5% of the area to its left?” Use left tail with p = 0.975.
• Confidence intervals: “Find the 95% confidence interval critical values.” Use center mode with p = 0.95.
• Hypothesis testing: “Find the critical Z values for a two-tailed test at α = 0.05.” Use two tails mode with p = 0.05.
• Quality control: “What measurement is exceeded by only 1% of items?” Use right tail with p = 0.01.

If you have a value and need the probability instead, use the Normal CDF calculator — that is the reverse operation.

Left Tail InvNorm Example

Suppose test scores follow a normal distribution with mean 500 and standard deviation 100. What score marks the 90th percentile?

Here, p = 0.90 (left tail). Find z = Φ^-1(0.90) ≈ 1.28155. Then x = 500 + 1.28155 × 100 = 628.155. A student needs a score of approximately 628.16 to be in the 90th percentile. This is a classic use of InvNorm as a percentile calculator.

Try this example in the calculator.

Right Tail InvNorm Example

A factory produces bolts with diameters that follow a normal distribution (mean 10 mm, standard deviation 0.2 mm). What diameter is exceeded by only 5% of bolts?

Using right tail mode with p = 0.05: convert to left cumulative probability 0.95, find z ≈ 1.64485, then x = 10 + 1.64485 × 0.2 = 10.32897 mm.

Try this example in the calculator.

Center (Middle Area) and Confidence Interval Example

To find the 95% confidence interval boundaries for a standard normal distribution, use center mode with p = 0.95. The calculator finds lower z ≈ −1.95996 and upper z ≈ 1.95996. These are the familiar ±1.96 critical values.

For a custom distribution with mean 100 and standard deviation 15, the middle 95% of values fall between approximately 70.60 and 129.40.

Try this example in the calculator.

Two Tail InvNorm Example

In a two-tailed hypothesis test at the 0.05 significance level, you need to find the critical Z values. Enter p = 0.05 in two tails mode. The total tail probability splits equally: 0.025 in each tail. The critical values are z ≈ −1.95996 and z ≈ 1.95996.

Try this example in the calculator.

How to Find InvNorm on a TI-84 and TI-83 Calculator

On a TI-84 or TI-83 graphing calculator, press 2nd → VARS to open the DISTR menu, then select 3:invNorm(. The syntax is:

invNorm(area, μ, σ)

The area parameter is always the left cumulative probability. If you have a right tail probability of 0.05, enter 0.95 as the area. For the standard normal, you can omit μ and σ or use invNorm(area, 0, 1).

On newer TI-84 models (OS 2.55+), you may see a wizard that asks for area, μ, and σ separately. On a TI-89, the function is found under APPS → Stat/List Editor → F5 (Distr) → Inverse Normal.

Read the full guide: How to Use InvNorm on TI-84, TI-83, and TI-89.

How to Use InvNorm on a Casio Calculator

On Casio graphing calculators (such as the fx-9860 or fx-CG50), the inverse normal function is found under the Statistics menu. Navigate to STAT → DIST → NORM → InvN. Enter the tail setting and area, along with σ and μ. Note that menu labels may differ between Casio models.

Some Casio models allow you to select left, right, or central tail directly. Enter the cumulative probability values matching the cutoff you need.

Read the full guide: InvNorm on Casio Calculators.

How to Calculate InvNorm Without a Graphing Calculator

You do not need a physical graphing calculator to use InvNorm. Several options exist:

• This online calculator: This free InvNorm calculator works in any web browser on desktop or mobile. Enter your probability and get the Z score or X value instantly with steps.
• Z table (reverse lookup): Find your target probability in the body of a standard normal distribution table, then read the corresponding Z score from the row and column headers. This is less precise than a calculator.
• Excel or Google Sheets: Use =NORM.S.INV(p) for the standard normal or =NORM.INV(p, mean, sd) for a custom distribution.
• R or Python: Use qnorm(p) in R or from scipy.stats import norm; norm.ppf(p) in Python.

InvNorm in Excel, Google Sheets, R and Python

In all these tools, the probability p is the left cumulative probability. Read the full guide: InvNorm in Excel and Google Sheets.

Common Z Critical Values

InvNorm Calculator FAQs

Related Statistics Calculators

Calculation Method and References

This calculator uses a rational approximation algorithm for the inverse normal CDF, refined with Halley's method for higher accuracy. The implementation achieves precision to approximately 10 significant digits. For full details on the mathematical methodology, see the methodology page.

All calculations run entirely in your browser. No data is sent to any server. For more information, read our privacy policy.