Normal Distribution Calculator

Probabilities under any normal curve — below, above, between or outside given values — and the inverse: the value at a given percentile, with z-scores.

The Normal Distribution Calculator evaluates the cumulative distribution function of N(μ, σ²) to give P(X < a), P(X > a), P(a < X < b) and two-tailed probabilities, and inverts it to find the value x at a given cumulative probability.

Free, runs in your browser. Nothing you enter is uploaded.

How to use it

Enter μ and σ.
Choose the probability type.
Enter the cut-off(s).
Read the probability or the inverse value.

What you enter

μ, σ, cut-off value(s) or probability

What you get

Probability
z-scores
Percentile value
Central interval

Formula

P(X < a) = Φ((a − μ)/σ)   |   x_p = μ + σ·Φ⁻¹(p)

Φ is the standard normal CDF, computed from the error function; the inverse uses a rational approximation refined by a Newton step.

Worked example

Input: μ = 100, σ = 15, P(X < 115)

Result: z = 1.0, P = 0.8413 (84.13th percentile)

About 84 % of a normal population lies below one SD above the mean.

Questions

What is the 68-95-99.7 rule?

For a normal distribution about 68 % of values lie within ±1σ, 95 % within ±2σ and 99.7 % within ±3σ of the mean.

How do I find the value at the 90th percentile?

Use the inverse mode with p = 0.90: x = μ + 1.2816·σ.

Limitations

Only valid when the variable is (approximately) normally distributed.

References

Altman DG, Bland JM (1995). The normal distribution. BMJ 310:298.

More Statistics

All free tools · By Dr. Zubair Khalid, DVM, MS, PhD.