TAHA Engineering Toolsby TAHA Knowledge Lab

Mathematics & Statistics

Normal Distribution Calculator

P(X < x), P(X > x), P(a < X < b), outside a range, or the value for a probability, for any mean and σ.

Inputs

Calculate from
Bell curve with the area to the left of x shaded, showing P(X less than x).μxP(X < x)
The shaded area left of x is P(X < x). The whole area under the curve is 1. Diagram not to scale.

Result

Probability
0.9331927987
As a percentage
93.31928%
z-score(s)
z = (53 − 50) ÷ 2 = 1.5
Density f(x) at x
0.064758798
Working
P(X < x) = Φ(z) = Φ(1.5) = 0.9331927987

For a normal distribution with mean 50 and σ = 2, the probability is 0.9331927987 (93.31928%).

Educational estimate — check the assumptions below.

About this calculator

The normal (Gaussian) distribution describes many measured quantities: dimensions of machined parts, filling weights, measurement errors. Enter its mean μ and standard deviation σ to find the probability that a value falls below, above, between or outside given limits — for example the share of shafts outside a tolerance — or to find the value that a given probability corresponds to.

Formula

P(X < x)

P(X < x) = Φ(z),   z = (x − μ) ÷ σ

Φ(z) = ½ [1 + erf(z ÷ √2)]    f(x) = e−z²/2 ÷ (σ √(2π))

P(X > x)

P(X > x) = 1 − Φ(z),   z = (x − μ) ÷ σ

The upper tail is computed directly (not as 1 minus a number close to 1), so very small probabilities stay accurate.

P(a < X < b)

P(a < X < b) = Φ(zb) − Φ(za)

Outside a and b

P(X < a or X > b) = Φ(za) + 1 − Φ(zb)

Value for a probability

x = μ + z σ,   z = Φ⁻¹(p)

For a central probability p, each tail holds (1 − p) ÷ 2, so a, b = μ ∓ Φ⁻¹(1 − (1 − p)/2) σ (for 95%: ±1.959964 σ).

Worked example

A filling process with mean μ = 50 g and standard deviation σ = 2 g:

  • P(X < 53 g): z = 1.5 → Φ(1.5) = 0.9331928
  • P(47 g < X < 53 g): z = ±1.5 → 0.9331928 − 0.0668072 = 0.8663856; outside: 0.1336144
  • Central 95%: z = 1.959964 → between 46.08007 g and 53.91993 g

Assumptions and limitations

  • The quantity follows a normal distribution with the mean and standard deviation entered.
  • Φ and Φ⁻¹ agree with SciPy within 2.3 parts in 10¹³ (worst case on a grid of 4,601 z-values from −38 to 8 and 1,599 probabilities from 10⁻³⁰⁰ to 0.999); probabilities are shown to 10 significant figures.
  • Probabilities “less than” and “less than or equal” are the same for a continuous distribution.

References

  • NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1 (normal distribution).
  • Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions, 26.2.
  • Acklam, P. J. An algorithm for computing the inverse normal cumulative distribution function (2003).