Methods and formulas for Tolerance Intervals (Normal Distribution)

General Definitions

Let X ₁, X ₂, ..., X _n be the ordered statistics based on random sample of size n from some continuous distribution.

Let the distribution function be F(x;θ) for Ω in some parameter space with dimension greater than or equal to 1.

Let L < U be two statistics based on the sample such that for any given values α and P, with 0 < α < 1 and 0 < P < 1, the following holds for every θ in Ω:

Then, the interval [L, U] is a two-sided tolerance interval with content = P x 100% and confidence level = 100(1 – α)%. Such an interval can be called a two-sided (1 – α, P) tolerance interval. For example, if α = 0.10 and P = 0.85, then the resulting interval is called a two-sided (90% , 0.85) tolerance interval.

If L = –∞ and U < +∞, then the interval (-∞, U] is called a one-sided (1 – α, P) upper tolerance bound. If L > -∞ and U = +∞, then the interval [L, +∞) is called a one-sided (1 – α, P) lower tolerance bound.

Tolerance factor for one-sided intervals

where t_n-1,1-α(δ) is the 1 – α percentile of the noncentral t-distribution with n – 1 degrees of freedom and noncentrality parameter, δ, which is given by the following formula:

Tolerance factor for two-sided intervals

The exact tolerance factor for a two-sided interval is obtained by solving the following equation for k. See Krishnamoorthy and Mathew¹.

where F_{n – 1} is the cumulative distribution function for a chi-square distribution with n – 1 degrees of freedom, and χ²_1,p is the P^th percentile of the noncentral chi-square distribution with 1 degree of freedom and noncentrality parameter z². The left-hand side of the equation can be rewritten as:

where:

where Φ(z) is the probability density function of the standard normal distribution. Minitab uses a 36-point Gauss-Legendre quadrature to evaluate I(k, n, P).

Notation

Term	Description
1 - α	the confidence level of the tolerance interval
P	the coverage of the tolerance interval (the target minimum percentage of population in the interval)
L	the lower limit of the tolerance interval
U	the upper limit of the tolerance interval
	the mean of the sample
k	the tolerance factor (also called k-factor)
S	the standard deviation of the sample
n	the number of observations in the sample
ZP	the Pth percentile of the standard normal distribution

One-sided intervals

Let k be the largest integer that satisfies the following:

where Y is a binomial random variable with parameters n and 1 – P. It can be shown (see Krishnamoorthy and Mathew⁴) that a one-sided (1 – α, P) lower tolerance bound is given by X_k . Similarly, a one-sided (1 – α, P) upper tolerance bound is given by X _{n - k +1}. In both cases, the actual or effective coverage is given by P(Y > k).

Two-sided intervals

Let k be the smallest integer that satisfies the following:

where V is a binomial random variable with parameters n and P. Thus,

where F _V ^-1(x) is the inverse cumulative distribution function of V. It can be shown (see Krishnamoorthy and Mathew⁴) that a two-sided (1 – α, P) tolerance interval may be given as ( X_r , X_s ). Minitab chooses s = n - r + 1 so that r = ( n – k + 1) / 2. Both r and s are rounded down to the nearest integer. The actual or effective coverage is given by P(V < k – 1).

Notation