About the Inverse Normal Distribution Calculator
This inverse normal distribution calculator works backwards from a probability to a value. Give it an area under the normal curve, the mean μ and the standard deviation σ, and it returns the cut-off x and the matching z-score — exactly what the invNorm function on a TI-84 or NORM.INV in Excel does. Choose whether the area sits in the left tail, the right tail, the middle of the curve, or split across both tails.
Students use it to find percentiles ("what score puts you in the top 10%?"), critical values for confidence intervals and hypothesis tests (1.96 for 95%), and cut-offs for grading on a curve. Analysts use it to set control limits, service levels and safety stock: the stock level that covers demand 95% of the time is the 95th percentile of the demand distribution.
Enter the area as a decimal between 0 and 1 (0.95, not 95). For central and two-tailed areas the calculator returns both bounds, symmetric around the mean. Use a mean of 0 and a standard deviation of 1 to get z-scores from the standard normal distribution.
With the default inputs, the x value is 124.6728. Change any value above to recalculate instantly.
How to use the inverse normal distribution calculator
- 1Enter the area (probability) as a decimal between 0 and 1.
- 2Enter the mean and standard deviation (use 0 and 1 for z-scores).
- 3Choose whether the area is in the left tail, right tail, centre or both tails.
- 4Read the x value and z-score; for central or two-tailed areas read both bounds.
Formula and method
The normal cumulative distribution function Φ(z) gives the area to the left of a z-score. The inverse normal function runs it backwards: given an area, it returns the z-score with that much probability to its left. The calculator first converts your request into a left-tail area (for example, a right-tail area of 0.05 becomes 0.95 on the left), inverts Φ numerically to high precision, and then converts the z-score to your scale with x = μ + zσ.
For a central area the remaining probability is split evenly between the two tails, so the bounds are μ ± zσ with z = Φ⁻¹((1 + p)/2). For a two-tailed area p, each tail holds p/2. These are the same critical values used for confidence intervals and two-sided tests.
- p
- The area (probability) you entered
- Φ⁻¹
- Inverse of the standard normal cumulative distribution function
- z
- Standard score (number of standard deviations from the mean)
- μ, σ
- Mean and standard deviation of the distribution
- x
- Cut-off value on the original scale
Worked examples
95th percentile of IQ scores (μ = 100, σ = 15)
The left-tail area is 0.95, so z = Φ⁻¹(0.95) ≈ 1.6449. Converting to the IQ scale: x = 100 + 1.6449 × 15 ≈ 124.67, so 95% of people score below about 124.7.
Critical value for a 95% confidence interval
A central area of 0.95 leaves 0.025 in each tail, so z = Φ⁻¹(0.975) ≈ 1.96. That is why 95% confidence intervals use estimate ± 1.96 standard errors.
Top 10% cut-off for exam scores (μ = 70, σ = 8)
A right-tail area of 0.10 means 0.90 lies to the left, so z = Φ⁻¹(0.90) ≈ 1.2816 and x = 70 + 1.2816 × 8 ≈ 80.25. Students scoring above about 80.3 are in the top 10%.
Frequently asked questions
What does invNorm do?+
invNorm(area, μ, σ) on TI calculators returns the value x such that the area to its left under a normal curve with mean μ and standard deviation σ equals the given area. It is the inverse of normalcdf and matches NORM.INV in Excel and qnorm in R.
How do I find a z-score from a percentile?+
Convert the percentile to a decimal left-tail area and invert the standard normal CDF. The 90th percentile is z = Φ⁻¹(0.90) ≈ 1.2816, the 97.5th percentile is 1.96, and the 50th percentile is 0.
How do I use the inverse normal for a right-tail area?+
Subtract the right-tail area from 1 to get the left-tail area, then invert. For the top 5%, use 0.95 on the left, which gives z ≈ 1.6449. This calculator does the conversion for you when you select Right tail.
What are the common critical z values?+
Two-sided: 1.645 for 90%, 1.960 for 95% and 2.576 for 99% confidence. One-sided: 1.282 for 90%, 1.645 for 95% and 2.326 for 99%.
Why can the area not be exactly 0 or 1?+
The normal distribution extends to infinity in both directions, so no finite value has exactly 0% or 100% of the area to its left. Areas very close to 0 or 1 give very large negative or positive z-scores.