Normal Percentile and Probabilistic Tolerance Calculator

Content

Evaluate normal areas, percentile values, and central tolerance intervals with one compact three-mode tool.

Use z ≥ 0 for a two-sided central area.

How to use

  1. Select Area from z, Value from probability, or Tolerance interval according to the desired output.
  2. In Area from z, choose one-sided or two-sided area and enter dimensionless z between −5 and 5; two-sided central area requires z ≥ 0.
  3. In Value from probability, enter mean μ, positive standard deviation σ, and cumulative probability P strictly between 0% and 100%. Use one consistent physical unit for μ, σ, and the result.
  4. In Tolerance interval, enter Lspec < Uspec in one unit and central probability R strictly between 0% and 100%. The specification limits are interpreted as μ ± 3σ.
  5. Select Calculate to update the single result panel. Select Reset in the active mode to restore its example and recalculate.
  6. Read the canonical result first, then supporting metrics. A green message confirms valid input; correct any red warning before interpretation.
  7. Do not use these models when normal-distribution or centred ±3σ assumptions are unjustified, for sample-based statistical inference, or for strongly skewed, multimodal, censored, or heavy-tailed data.

Equations used

Let φ be the standard-normal density, z a dimensionless critical value, and Δz = 0.001. Area mode sums from −5 to z.

φ(z) = exp(−z²/2)/√(2π)
Φ(z) = Σφ(−5 + iΔz)Δz, i = 1 … 1000(z + 5)
Aone = 100Φ(z); Atwo = 100[2Φ(z) − 1]

Area is displayed to two decimals. For cumulative probability fraction p = P/100:

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q = 0.5 − |p − 0.5|; w = √[−2 ln(q)]
zP = sgn(p − 0.5){w − (a₁ + a₂w)/[1 + w(a₃ + a₄w)]}
x = μ + σzP

Constants: a₁ = 2.30753, a₂ = 0.27061, a₃ = 0.99229, a₄ = 0.04481. The percentile value is displayed to four decimals.

μ = (Lspec + Uspec)/2; σ = (Uspec − Lspec)/6
pu = (1 + R/100)/2; α = (1 − R/100)/2
zR = Φ−1(pu)
Lprob = μ − σzR; Uprob = μ + σzR

Definitions, units, and limits: z, zP, zR, p, pu, α, and q are dimensionless; α is the probability in each excluded tail. A, P, and R are percentages. μ, σ, x, and all specification/probabilistic limits share one user-defined unit. Sign convention: positive z lies above μ; lower and upper interval offsets are −σzR and +σzR. Area mode is limited to −5 ≤ z ≤ 5. Tolerance mode assumes a centred normal process and interprets specifications as μ ± 3σ.

Method notes and reference

The normal model is useful only when the selected quantity is reasonably represented by a continuous bell-shaped distribution. Confirm distribution shape and process centring with measured data before design, acceptance, or reliability decisions.

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