Enter a probability (area to the left), along with the mean and standard deviation, to find the corresponding x-value or z-score.
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).
Using our calculator is very simple!
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.
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:
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.
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.
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%.
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:
Step 01: Find z score using invNorm (P)
We want invNorm (0.05) = z
To calculate this:
In our case, the nearest number is 0.04945, and it is in row -1.7, as picked from z table (as shown):
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:
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.
Yes, our calculator provides results for right-tail, left-tail, and two-tailed p values.
That means the value lies below the mean in the distribution.
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