InvNorm Calculator

This calculator helps to determine the x-value (or z-score) from a normal distribution based on the probability (P), mean (μ), and standard deviation (σ).

How to Use the Calculator?

Using our calculator is very simple!

Enter the cumulative probability, mean, and standard deviation
Click ‘Calculate’

Statistical Results Provided by Calculator:

Z-Score: Represents how much standard deviation a value is from the mean within a standard normal distribution.
X-Score: This is the actual value (x) for a given probability in a normal distribution.
Graph: A visual representation showing where the value lies under the normal curve for different probability cases.
Step Wise Calculations: Detailed calculations to help you understand how the calculator derived the results.

What Is Invnorm?

It is an inverse statistical function that applies backward on a cumulative distributive function (CDF) to find the x-value using the known probability value.

Inv Norm Formula:

There is no direct formula for the inverse normal function, but it uses numerical methods to find the z-score such that the area under the normal curve to the left of z equals the given probability.

Inverse Normal Calculator & Distribution Explained:

Our calculator uses the normal distribution formula to show the area under the bell curve, which corresponds to the given probability value. It helps you to calculate inv norm for the following graphs:

Left Tail (P(X < x) = p)

the graph represents the left tail invnorm

The tool uses the left tail when you want the z-score where the given area lies to the left of the value.

Example: To find the z-score for the lowest 5% of a dataset using invNorm(0.05).

For these values, our calculator will return a negative z-score, representing values in the lower tail.

Right Tail (P(X > x) = p)

The right tail represents the upper end of the distribution.

Since the invNorm function always calculates the area to the left, the calculator subtracts your desired right-tail probability from 1 to give you the final result.

Example: For the top 5%, the tool uses invNorm(1 - 0.05) → invNorm(0.95).

This returns a positive z-score.

Two Tails (P(|X − μ| > x) = p)

In some cases, such as confidence intervals or two-tailed hypothesis tests, the invNorm calculator might need to find values in both tails of the distribution.

Example: For a 95% confidence level, 5% is split between both tails, so each tail contains 2.5%.

Left cutoff: invNorm (0.025)
Right cutoff: invNorm (0.975)

Worked Example (Inverse Normal Distribution):

Suppose you want to calculate the x value (raw score) that separates the lowest 5% of data in a normal distribution with the following perspectives:

Cumulative Probability = 0.05
Mean = 100
Standard Deviation = 15

Calculations:

Step 01: Find z score using invNorm (P)

We want invNorm (0.05) = z

To calculate this:

Find out the nearest value to the provided p-value (0.05) in the z table, as shown:
Check the corresponding row
Add the numbers

In our case, the nearest number is 0.04945, and it is in row -1.7, as picked from z table (as shown):

**Standard Normal Z-Table (Negative Z-Scores)**
Z \\ Col	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
-1.7	0.0446	0.0454	0.0461	0.0468	0.0475	0.0495	0.0505	0.0512	0.0519	0.0526

By adding these, we get;

z = -1.7 + 0.04945

z = -1.650 (you can confirm it with our z score calculator as well)

Step 02: Convert z score to x score

By using the invnorm formula:

x = μ + z * σ

Substituting the values:

x = 100 + (-1.650 * 15)

x = 100 - 24.675

x = 75.325

Graph:

Faqs:

What is the difference between Invnorm and normalcdf?

Normal Cumulative Distribution Function (CDF):

Calculates the probability that a randomly selected value from a normal distribution will fall within that range.

Inverse Normal Distribution Function (InvNorm):

Calculates the x-value (or z-score) that corresponds to that cumulative probability.

Can I use the calculator for right tail probabilities?

Yes, our calculator provides results for right-tail, left-tail, and two-tailed p values.

What if my result is negative?

That means the value lies below the mean in the distribution.

InvNorm Calculator