About this calculator
The inverse option works the other way: give a probability and get the cut-off value, for example the score needed for the top 5%.
How to use it
- Enter the mean and standard deviation.
- Choose what to find.
- Enter x, the range or the probability.
The formula
z = (x − μ) ÷ σ; P(X < x) = Φ(z)
- μ
- mean
- σ
- standard deviation
- Φ
- standard normal cumulative distribution
Worked example
IQ scores with mean 100 and σ 15
- P(X < 120): z = 1.333, so P = 0.9088.
- The top 5% start at 100 + 1.645 × 15 = 124.67.
What the result means
About 68% of values lie within one standard deviation of the mean, 95% within two and 99.7% within three.
Assumptions
- The data are normally distributed.
Limitations
- Real data are often only roughly normal, especially in the tails.
Frequently asked questions
What is the standard normal distribution?
The normal distribution with mean 0 and standard deviation 1. Any normal value converts to it with z = (x − μ) ÷ σ.
How do I find the top 10% cut-off?
Choose the inverse option and enter 0.9 as the probability to the left.
Is P(X < x) the same as P(X ≤ x)?
Yes, for a continuous distribution.
Last reviewed on 6 October 2026. Found a mistake? Tell us.