Relative vs. Absolute Precision in Sample Design
Convert relative into absolute precision and plan a sample consistent with the proportion to be estimated.
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Two ways to state the precision objective
Absolute precision e is expressed in proportion units: e=0.03 means three percentage points. Relative precision s expresses that margin as a fraction of a reference proportion p₀. For planning, e=s×p₀.
These are not rival methods; they can describe exactly the same objective. Choose according to the decision-relevant difference, not a mandatory p<0.15 threshold.
Planning formulas
For a binary category under an appropriate model:
n₀ = z²p₀(1−p₀)/e².
Substituting e=s×p₀, for p₀>0:
n₀ = z²(1−p₀)/(s²p₀).
Round upward. These are approximations based on an initial proportion and design, not guarantees for every realized sample. NIST describes planning for proportions.
Comparable example: p₀=0.30 and z=1.96
| Objective | Equivalent absolute margin | n rounded upward |
|---|---|---|
| Absolute three points | 0.03 | 897 |
| Relative 10% of 0.30 | 0.03 | 897 |
| Relative 5% of 0.30 | 0.015 | 3,586 |
| Relative 1% of 0.30 | 0.003 | 89,638 |
Planning ranges around 30% would be 27–33%, 28.5–31.5%, and 29.7–30.3%, respectively. They are not intervals calculated from actual samples or customer outcomes.
The table does not establish that 89,638 readings are worthless or that 897 suffice for a particular investment. Assess cost, availability, coverage, and the consequence of a wrong decision.
Small proportions
With p₀=0.05 and e=0.03, the margin is 60% of the reference, not “300% variance.” It may be too broad for one decision and adequate for another. Detecting a state at least once is a different question from estimating its proportion.
A zero pilot proportion does not establish the state's absence or justify n=0. Consider pilot uncertainty and a defensible conservative assumption where information is lacking.
The reporting interval and assumptions
After collection, calculate the observed proportion and an appropriate interval. Wilson can give asymmetric limits; do not impose the symmetric planned margin on the result. NIST compares interval methods.
Confidence describes procedural coverage under assumptions, not a promise that the current sample is representative. Within-round correlation, mixed populations, and classification error are not repaired by choosing relative precision.
Design for the actual decision
Comparing two methods requires a target difference and comparison design; a single-proportion formula does not by itself determine power. When reporting many categories or strata, distinguish each one's precision from aggregate precision.
Do not automatically assign one thousand readings to an initial phase and a second count based only on rarity. Document adaptive changes and their analytical consequences.
Sampling is not equivalent to cycle timing and does not turn percentages into OEE. Randomness does not guarantee eliminated reactivity or privacy compliance. A useful result combines a defined quantity, sufficient evidence, and explicit limits.