Normal distribution calculator
z = (x − µ) / σ
Three calculations on the normal distribution, for any µ and σ: probabilities at a point (P(X ≤ x), P(X > x), z-score, density), the probability between two bounds, and critical values. The z such that P(X ≤ x) = p, the right tail, or the central interval containing a given probability. Tails are computed separately in relative precision: P(Z > 6) comes back with its digits, not as zero.
0.975002
- Calculation
- z = (x − µ)/σ = (1.96 − 0) ÷ 1 = 1.96
Graph
0.975002 of the population lies below x, 0.0249979 above. Both tails are computed separately, in relative precision.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasz = (x − µ) / σ
z = (x − µ) / σ
Φ(z) = P(Z ≤ z) = ½·erfc(−z/√2)
P(a ≤ X ≤ b) = Φ(z_b) − Φ(z_a)
critical value: z such that Φ(z) = p; central interval: Φ(z) = (1 + p)/2
The z-score maps any normal distribution to the standard one: it is the number of standard deviations between x and the mean. Φ is evaluated through the error function with two classical representations (Maclaurin series near the centre, Legendre continued fraction in the tails, Abramowitz & Stegun, Handbook of Mathematical Functions, §7.1). Which agree to machine precision in their overlap region. The quantile is obtained by Newton’s method on Φ with its exact derivative (the density), bracketed by bisection.
- z-score
- · the gap between x and the mean, measured in standard deviations. z = 2 means “two standard deviations above the mean”, whatever the original normal distribution.
- Critical value
- · the z (or x) delimiting a given probability, the 1.96 of 95% intervals is the central critical value for p = 0.95.
- Tail
- · the probability beyond a point. The tool computes it directly (never as 1 − Φ), preserving the significant digits of tiny probabilities.
Validity domainThe tool computes probabilities under the assumption that the variable is normally distributed: it does not check that your data follows it: that is the role of a normality assessment, outside its scope.
The tool computes probabilities under the assumption that the variable is normally distributed: it does not check that your data follows it: that is the role of a normality assessment, outside its scope. The quantile requires a probability strictly between 0 and 1: 0 and 1 have no finite preimage. Probabilities smaller than ~10⁻³⁰⁰ fall below the floating-point floor and are returned as 0.
Common pitfall: P(X = x) does not existFor a continuous distribution, the probability of an exact value is zero: only probabilities of intervals are meaningful, and P(X ≤ x) = P(X < x).
For a continuous distribution, the probability of an exact value is zero: only probabilities of intervals are meaningful, and P(X ≤ x) = P(X < x). The displayed density f(x) is not a probability: it can exceed 1 when σ is small; it is a probability per UNIT of x. Another trap: 1.96 belongs to the central 95% interval; the one-sided 95% quantile is 1.645, mixing them up is the most frequent table mistake.