How to use this calculator
- Enter or choose mean (μ).
- Enter or choose standard deviation (σ).
- Enter or choose area.
- Enter or choose probability (area).
- The cut-off value updates as you type. Use Copy result or Share to keep it.
Method
Left tail: x = μ + σ·Φ⁻¹(p); right tail: x = μ + σ·Φ⁻¹(1 − p); central: μ ± σ·Φ⁻¹((1 + p) ÷ 2)
- p
- the area (probability)
- Φ⁻¹
- inverse standard normal CDF, computed with Wichura’s algorithm AS 241 (about 16 significant digits)
- μ, σ
- mean and standard deviation
Worked example
- Φ⁻¹(0.95) = 1.644854: 95% of a standard normal lies below 1.6449.
- x = 100 + 1.644854 × 15 = 124.672804.
In a normal distribution with mean 100 and standard deviation 15, the 95th percentile is 124.67.
What the result means
Left tail gives a percentile. Right tail gives the threshold that only the top share exceeds. Central gives the symmetric band holding the stated share, which with μ = 0 and σ = 1 is the two-sided critical value (±1.96 for 95%).
Assumptions and limits
- The quantity follows a normal distribution with the entered mean and standard deviation.
- Extreme percentiles depend strongly on the shape of the tails, where real data often depart from normality.
Questions
How do I get the critical value for a 99% confidence interval?
Set μ = 0, σ = 1, choose Central and enter 0.99. The answer is ±2.5758.
Is this the same as Excel’s NORM.INV?
Yes for the left tail: NORM.INV(p, μ, σ) returns the same cut-off.