Normal Distribution Calculator
Compute cumulative P(X≤x) and right-tail P(X>x) for a normal distribution.
Input Data
Observation to find probability for.
Distribution mean.
Standard deviation (must be > 0).
Results
1
0.841345
0.158655
At a glance:For a normal distribution, P(X≤x) = Φ((x−μ)/σ), where Φ is the standard normal CDF.
Formula
z = (x − μ) / σ
P(X ≤ x) = Φ(z)
P(X > x) = 1 − Φ(z)
$$P(X \le x) = \Phi\!\left(\frac{x-\mu}{\sigma}\right)$$How to Use
- Enter the observation x, mean μ, and standard deviation σ.
- The calculator outputs the z-score z.
- It outputs the cumulative probability P(X ≤ x) and the right-tail probability P(X > x).
- Example: x=85, μ=75, σ=10 → z=1, P(X≤85)≈0.8413.
FAQ
What is the 68–95–99.7 rule?
A normal distribution has about 68% of values within μ±σ, 95% within μ±2σ, and 99.7% within μ±3σ. You can verify this with the cumulative probability from this calculator.
Is the approximation accurate enough?
It uses the standard erf approximation (Abramowitz & Stegun 7.1.26) with a maximum error around 1.5×10⁻⁷, sufficient for general statistics and teaching.
References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.