PharmaCalc

A statistical tolerance interval (STI) is a range [L, U] such that at least P% of the population falls within [L, U] with C% confidence. This is distinct from confidence intervals (which describe the mean) and prediction intervals (which describe the next observation). Tolerance intervals answer the regulatory question: "Will at least 99% of future batches fall within specification?"

FDA requirement: Process Validation Guidance (2011) Stage 2 requires statistical evidence that the process is capable of meeting specifications. Tolerance intervals provide this evidence. For example, if P=99% (coverage) and γ=95% (confidence), then a tolerance interval demonstrates with 95% confidence that 99% of the population falls within [L, U].

Typical applications: Specification verification (confirm 99% of tablets meet weight specs), cleaning validation (ensure 99% of surfaces pass ATP limits), potency assay (ensure 99% of batches meet ICH Q3A/Q3B limits), and dissolution testing (ensure 99% of tablets meet Q criteria).

For data assumed to follow a normal distribution, a two-sided tolerance interval is calculated as:

where k is the tolerance factor (from standard statistical tables), X̄ is the sample mean, and s is the sample standard deviation.

The tolerance factor k depends on three parameters:

• n (sample size) — larger samples give smaller k (tighter intervals)
• P (coverage percentage) — typically 99% for pharma (want 99% of population in interval)
• γ (confidence level) — typically 95% (95% confident the interval contains P% of population)

One-sided intervals: For upper confidence bounds only, the formula is [−∞, X̄ + k×s] or lower bound only [X̄ − k×s, ∞], using one-sided k values. One-sided intervals are narrower and sometimes used when only upper limit (e.g., impurity) or lower limit (e.g., yield) matters.

Key interpretation: If the calculated tolerance interval [L, U] falls entirely within the specification limits [LSL, USL], then the process PASSES validation — at least P% of future production is guaranteed (with γ% confidence) to meet specifications.

• Scenario: An API synthesis process is under process validation. The specification limit for yield is 90–110%. Data from n=20 pivotal batches (Stage 2) shows X̄=97.2% and s=1.8%. Calculate a 99%/95% tolerance interval to assess whether at least 99% of future production will meet specification with 95% confidence.
• Data and Parameters:
  Process Validation Data

  n = 20 batches
  X̄ = 97.2% (sample mean)
  s = 1.8% (sample SD)
  Specification: LSL = 90%, USL = 110%
  Desired tolerance interval: P = 99%, γ = 95% (two-sided)
• Step 1: Obtain tolerance factor k from statistical tables.
  For n=20, P=99%, γ=95% (two-sided):
  k = 3.615
  
  (This value comes from published tolerance factor tables or statistical software. For reference, k increases as n decreases and as P or γ increase.)
• Step 2: Calculate lower tolerance bound.
  L = X̄ − k × s
  L = 97.2 − 3.615 × 1.8
  L = 97.2 − 6.507
  L = 90.69%
• Step 3: Calculate upper tolerance bound.
  U = X̄ + k × s
  U = 97.2 + 3.615 × 1.8
  U = 97.2 + 6.507
  U = 103.71%
• Step 4: Compare tolerance interval to specifications.
  Tolerance Interval (99%/95%): [90.69%, 103.71%]
  Specification Limits: [90%, 110%]
  
  Check: Is 90.69% > 90%? YES ✓
  Check: Is 103.71% < 110%? YES ✓
  
  Tolerance interval is completely within specs.
• Step 5: Interpretation and Regulatory Conclusion.
  Process PASSES Validation (Stage 2).
  
  With 95% confidence, at least 99% of future API batches will have yield between 90.69% and 103.71%, which is entirely within the specification window of 90–110%. The process is statistically capable of meeting specifications and is approved for continued production. Recommend ongoing Stage 3 monitoring with control charts to maintain this capability.

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