Work backwards from a probability to the value: enter an area and say where it lies (left, right, center or both tails), add the mean and standard deviation, and get the matching x value and z-score, the same as invNorm on a TI-84 or NORM.INV in Excel.
Last updated:
A decimal between 0 and 1, e.g. 0.95, or a percent like 95%.
Where is the area?
Use 0 and 1 for the standard normal (z).
Enter your values to see the result.
Advertisement
Advertisement
How to use the Inverse Normal Distribution Calculator
Enter the area as a decimal (0.95) or a percent (95).
Say where the area is: to the left of x, to the right of x, in the center, or split over both tails.
Enter the mean and standard deviation, or keep 0 and 1 for z-scores.
Press Calculate to get the x value, the z-score and the working.
Check the shaded curve and the percentile table, or copy the results.
How to turn an area into a value
Every question an inverse normal calculator answers reduces to one step: find the left-tail area p, then compute z = Φ⁻¹(p) and x = μ + z·σ.
Question
Left-tail area to use
TI-84
Excel
Value below which a fraction A falls (percentile)
A
invNorm(A, μ, σ)
NORM.INV(A, μ, σ)
Cut-off for the top fraction A
1 − A
invNorm(1 − A, μ, σ)
NORM.INV(1 − A, μ, σ)
Interval holding the middle fraction C
(1 − C)/2 and (1 + C)/2
invNorm((1 ± C)/2, μ, σ)
NORM.INV((1 ± C)/2, μ, σ)
Two-tailed critical values for α
α/2 and 1 − α/2
invNorm(1 − α/2)
NORM.S.INV(1 − α/2)
Common z values
Left-tail area
z
Typical use
0.50
0
Median
0.75
0.6745
Third quartile
0.80
0.8416
80% power (one tail)
0.90
1.2816
Top 10% cut-off
0.95
1.6449
One-tailed α = 0.05; 90% confidence
0.975
1.9600
95% confidence; two-tailed α = 0.05
0.99
2.3263
One-tailed α = 0.01; 98% confidence
0.995
2.5758
99% confidence
0.999
3.0902
One-tailed α = 0.001
Worked example
Exam scores are normal with μ = 72 and σ = 9. To earn an A a student must beat 90% of the class. The left-tail area is 0.90, so z = Φ⁻¹(0.90) ≈ 1.2816 and x = 72 + 1.2816 × 9 ≈ 83.53. A score of about 83.5 or higher is needed.
Tip: if your data are clearly skewed (incomes, waiting times), normal percentiles can be misleading; use the actual data with a percentile calculator instead.
Frequently asked questions
What does an inverse normal calculator do?
A normal distribution calculator turns a value into a probability. The inverse goes the other way: given a probability (an area under the bell curve), it finds the value x that cuts off that area. For example, in a standard normal distribution the value with 95% of the area to its left is z ≈ 1.6449.
What is the formula for the inverse normal?
First find the standard normal quantile z = Φ⁻¹(p), where p is the area to the left. Then convert to your scale with x = μ + z·σ. There is no closed-form expression for Φ⁻¹, so it is computed numerically; this tool uses a rational approximation refined with Newton-type steps to near machine precision.
How do I use invNorm for a right-tail area?
invNorm on a TI-83/84 always uses the area to the left. For a right-tail area A, use invNorm(1 − A, μ, σ). For the top 10% of IQ scores (μ = 100, σ = 15), that is invNorm(0.90, 100, 15) ≈ 119.22. This calculator does the conversion for you when you choose 'Right of x'.
How do I find the z-scores for a confidence level?
Choose 'Center' and enter the confidence level. For 95% the area in each tail is 0.025, so z = Φ⁻¹(0.975) ≈ 1.95996, usually rounded to ±1.96. For 90% the value is ±1.6449 and for 99% it is ±2.5758.
What is the z-score for the 25th and 75th percentile?
The 25th percentile of the standard normal is z ≈ −0.6745 and the 75th percentile is z ≈ +0.6745. That is why the interquartile range of a normal distribution is about 1.349 standard deviations.
Can I enter a percentage?
Yes. Values from 1 to 100 are treated as percentages, so 95 means 0.95. Values between 0 and 1 are read as decimals. The area must be strictly between 0 and 1 because the tails of the normal curve never reach zero.
Is NORM.INV in Excel the same as invNorm?
Yes. Excel's NORM.INV(p, mean, sd) and NORM.S.INV(p) for the standard normal both take the area to the left, like invNorm on TI calculators. Google Sheets uses the same function names.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.