Calculator Inputs
Enter a probability, choose a tail interpretation, and set the normal distribution parameters. The calculator returns z-scores and x cutoffs from the inverse cumulative distribution.
Plotly Graph
The graph shows the normal curve for your mean and standard deviation. The highlighted region matches the selected probability mode.
Example Data Table
| Scenario | Probability | Mean | Standard Deviation | Approximate z Result | Approximate x Result |
|---|---|---|---|---|---|
| Left tail cutoff | 0.9500 | 0 | 1 | 1.6449 | 1.6449 |
| Right tail cutoff | 0.0500 | 100 | 15 | 1.6449 | 124.6735 |
| Central area bounds | 0.9500 | 50 | 10 | ±1.9600 | 30.4000 to 69.6000 |
| Outside area bounds | 0.1000 | 0 | 1 | ±1.6449 | -1.6449 to 1.6449 |
Formula Used
For a normal variable
X ~ N(μ, σ²), convert between x and z with:z = (x - μ) / σ and x = μ + σz
If
P(X ≤ x) = p, then:z = Φ⁻¹(p)x = μ + σz
If
P(X ≥ x) = p, then the equivalent left probability is 1 - p:z = Φ⁻¹(1 - p)x = μ + σz
If the central probability between symmetric bounds is
p, each tail is:(1 - p) / 2Upper bound:
zᵤ = Φ⁻¹((1 + p) / 2)Lower bound:
zₗ = -zᵤ
If the combined outside probability is
p, each tail is p / 2:z = Φ⁻¹(1 - p/2)Bounds are
-z and +z.This page uses a rational approximation for
Φ⁻¹, which is accurate for practical calculator work.
How to Use This Calculator
- Enter the probability as a decimal between 0 and 1, such as
0.95. - Choose the correct tail mode. Left tail, right tail, central area, and outside area each solve a different inverse probability question.
- Enter the mean and standard deviation for your normal distribution. Use
0and1for the standard normal case. - Set the number of decimal places and graph width to match your reporting needs.
- Press Calculate Inverse Normal. The result appears above the form, directly below the header.
- Review the returned z-score, x cutoff, bounds, and tail areas in the result cards.
- Use the CSV or PDF buttons to export the current results for reporting, review, or documentation.
FAQs
1) What does the inverse normal CDF calculator return?
It returns the z-score or x-value that matches a specified probability under a normal distribution. Depending on the selected mode, it can also return lower and upper symmetric bounds.
2) When should I use left tail versus right tail?
Use left tail when you know the cumulative probability below a cutoff. Use right tail when the probability is above the cutoff. The two modes solve different directions of probability.
3) What is the difference between central area and outside area?
Central area finds symmetric bounds around the mean containing a chosen probability. Outside area finds symmetric cutoffs so the combined probability beyond both bounds equals the entered value.
4) Can I use values other than mean 0 and standard deviation 1?
Yes. The calculator works for any normal distribution with a positive standard deviation. It converts the inverse z result back to x using the distribution’s mean and spread.
5) Why must the probability stay between 0 and 1?
Probabilities at 0 or 1 would imply infinite cutoffs for a normal distribution. The inverse normal function is therefore only defined for probabilities strictly inside that open interval.
6) What does the plotted shaded region mean?
The shaded part of the curve represents the probability you entered under the selected mode. It visually confirms whether you solved a left tail, right tail, middle area, or outside area problem.
7) Are the results suitable for confidence intervals and critical values?
Yes. Central and outside modes are especially useful for common statistical tasks such as confidence intervals, significance thresholds, and symmetric critical value lookups.
8) What export options are included here?
You can export the current result set as CSV for spreadsheet work or as PDF for sharing, printing, or attaching to reports and study notes.