Standard Deviation and Tolerance Interval Estimator

How to use

1. This calculator has one mode: a symmetric central normal interval derived from a specified ±3σ dimensional tolerance.
2. Enter nominal size D and choose its length unit. D may be any finite datum appropriate to the dimension.
3. Enter positive tolerance half-width A in the same unit. A is the distance from nominal to either specified tolerance limit, not the full tolerance width.
4. Enter central probability R strictly between 0% and 100%.
5. Select Calculate to update σ, z, interval half-width T, and the lower and upper limits. Select Reset to restore D = 20 mm, A = 0.02 mm, and R = 90%.
6. Read all dimensional outputs in the selected unit. A green message confirms a valid result; correct any red warning before interpretation.
7. Do not use this model when the ±3σ convention is not justified, or for asymmetric, shifted, mixed, censored, or strongly non-normal processes. It is not a sample-based confidence interval or a finite-sample statistical tolerance interval.

Equations used

Let D be nominal size, A the positive tolerance half-width assumed to equal 3σ, σ the implied process standard deviation, R the requested central probability in percent, and p the probability in each tail.

The approximation constants are a₁ = 2.30753, a₂ = 0.27061, a₃ = 0.99229, and a₄ = 0.04481. The displayed z value is rounded to four decimals; interval calculations retain the unrounded value.

Definitions, units, and limits: D, A, σ, T, L_lower, and L_upper share the selected length unit; R is percent; p and z_R are dimensionless. Sign convention: A, σ, z_R, and T are positive magnitudes; the lower and upper offsets are −T and +T. The model assumes an independent normal process centred at D and a tolerance half-width equal to three standard deviations.