What Is Inverse Z Score?
An inverse z score, also called the inverse normal or the normal quantile, answers the opposite question to a z-table. Instead of asking what area lies below a z-score, it asks which z-score has a given area below it: z = Φ⁻¹(p).
Choose whether your probability is a left-tail area, a right-tail area, the middle area (a confidence level) or the combined area in both tails. The calculator returns the z cut-off, the matching x value, the area at the cut-off and a table of common critical values such as 1.645 for 90 % and 1.96 for 95 %.
The sections below cover the formula, the variables, a worked example for a 95 % interval and a percentile, and answers to common questions.
How to Use the Inverse Z Score Calculator
- Enter the probability or confidence level as a decimal.
- Say whether it is a left-tail area, right-tail area, middle area or combined tail area.
- Optionally enter a mean and standard deviation to convert z to x.
- Read the cut-off and the shaded area.
Inverse Z Score Formula
Inverse normal
Back to raw scale
Middle area to z
Inverse Z Score Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| p | Probability or area | strictly between 0 and 1 |
| z | Z-score cut-off | Φ⁻¹(p) |
| μ, σ | Mean and standard deviation for converting to raw units | |
| x | Raw value | μ + zσ |
Inverse Z Score Worked Example
For a 95% confidence interval the middle area is C = 0.95, so the left-tail area at the upper cut-off is (1 + 0.95) ÷ 2 = 0.975. Φ⁻¹(0.975) = 1.95996, usually rounded to 1.96. For test scores with μ = 100 and σ = 15, the 90th percentile is x = 100 + 1.28155 × 15 = 119.2.
With the example values loaded in the calculator above, the steps are:
- Find the z whose cumulative probability matches: z = Φ⁻¹(0.975) = 1.959964.
- Convert to the original scale: x = μ + z × σ = 0 + (1.95996) × 1 = 1.959964.
Result: z cut-offs enclosing the middle 95 % = ±1.959964; Equivalent x value(s) = -1.959964 and 1.959964.
How to Read an Inverse z-Score
The inverse z-score works backwards: you give an area and get the z value that cuts it off. For a 95% middle area, the upper tail cut-off has a left-tail area of (1 + 0.95) ÷ 2 = 0.975, and Φ⁻¹(0.975) = 1.96.
Enter a mean and standard deviation to convert the z value to a raw score, x = μ + zσ. Use the "area refers to" option to say whether your probability is the left tail, the right tail or the middle.
Where Inverse Z Score Is Used
Confidence intervals
Find z* for 90%, 95% or 99% confidence.
Percentiles
Find the value below which a given share of a normal distribution falls.
Cut-off scores
Convert a target percentile into a raw score.
Hypothesis tests
Find the critical value for a chosen α.
Common Inverse Z Score Mistakes to Avoid
- State whether the area is left tail, right tail or middle, because each gives a different z.
- A two-sided 95% interval uses the 0.975 point, not 0.95.
- For a raw score, multiply z by σ and add μ.
- The tails of the normal curve go on forever, so areas of exactly 0 or 1 have no finite z.
Inverse Z Score Terms and Definitions
- Critical value
- The z cut-off that separates the rejection region from the rest.
- Percentile
- A value below which a given percentage of observations fall.
Inverse Z Score FAQs
How do I find the z-score from a probability?
Find the z whose left-tail area equals the probability. For 0.975 it is 1.96. The calculator does this with the inverse of the normal CDF.
What is the z-score for 95% confidence?
1.96 for a two-sided interval, where each tail holds 2.5%. For a one-sided 95%, it is 1.645.
What is the inverse z score of 0.975?
It is 1.96, because 97.5% of the standard normal lies below it.
How do I find a percentile value?
Find the z for that percentile, then compute x = μ + zσ. The 90th percentile has z ≈ 1.2816.