Normal Distribution Quantile Calculator
Find normal-distribution cutoffs from a lower-tail, upper-tail or central probability with your entered mean and standard deviation.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Numerically invert the standard-normal cumulative probability and transform the result with the entered mean and standard deviation.
Cutoff=μ+σ×Φ⁻¹(p); central upper z=Φ⁻¹[(1+p)/2]
Worked example
For mean 100 and standard deviation 15, the cutoff with 95% probability below it is about 124.672804. A central 95% interval uses approximately ±1.959964 standard deviations.
Assumptions and input details
Probabilities from 0.0001% through 99.9999% avoid infinite endpoints. Positive standard deviation is required. Numerical bisection inverts the normal CDF. This is an assumed population-distribution cutoff, not an empirical percentile rank, prediction guarantee or confidence interval for a sample mean.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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