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.
Where Inverse Z Score Is Used
Confidence intervals
Finding the critical z value for 90, 95 or 99 % confidence.
Hypothesis tests
Finding rejection-region cut-offs for a chosen α.
Percentiles
Finding the score that marks the 90th percentile.
Coursework
Checking reverse z-table lookups.
Common Inverse Z Score Mistakes to Avoid
- Check which tail your area refers to before entering it.
- Use the middle option for confidence levels.
- Round z to at least 2 decimals when working by hand.
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?
Apply the inverse standard normal function to the left-tail area: z = Φ⁻¹(p).
What is the z-score for 95% confidence?
1.96 for the middle 95%, which leaves 2.5% in each tail.
What is the inverse z score of 0.975?
1.96, because 97.5% of the standard normal area lies to the left of it.
How do I find a percentile value?
Find z for the percentile as a left-tail area, then compute x = μ + zσ.