The Binomial Distribution in Work Study
The binomial model in work sampling: observational units, assumptions, proportions and estimation uncertainty.
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What the binomial distribution models
In work sampling, define X=1 when a chosen observable category is present and X=0 otherwise. This does not mean the plant has only two activities. A study can record several mutually exclusive categories; each category can be analyzed against its complement.
If n readings are independent and share probability p, their sum is binomial, with mean np and variance np(1−p). These are model assumptions, not automatic consequences of software or random-number generation. NIST states the conditions.
What increasing the sample changes
The sampling distribution of p̂=X/n can approach a normal distribution when both outcomes have sufficient representation and the design supports the approximation. Plant behavior itself does not become normally distributed. A rule based only on n>30 is inadequate for rare states.
More observations can reduce sampling variation; they do not repair biased routes, poor categories, or systematic behavior changes. Examine independence, particularly when several people share a round or stoppage.
Planning formula and rounding
The initial approximation is n=ceiling(z²p₀(1−p₀)/e²), where p₀ is the anticipated proportion, e an absolute margin, and z the two-sided confidence quantile.
Mathematical examples with p₀=0.5:
| Confidence | z used | Absolute margin | n rounded upward |
|---|---|---|---|
| 95% | 1.96 | 5 points | 385 |
| 95% | 1.96 | 3 points | 1,068 |
| 95% | 1.96 | 1 point | 9,604 |
| 99% | 2.576 | 1 point | 16,590 |
Using z=1.96, the three-point calculation is 1,067.111… and rounds upward to 1,068. A table showing 1,067 may be presenting an approximation or rounding to the nearest integer; it is not rounding upward. State the convention rather than mixing them. There is no universal sample count or compulsory industrial precision.
From calculation to interval
The Wald interval p̂±z√[p̂(1−p̂)/n] is approximate and can perform poorly for small samples or extreme proportions. Wilson is often a better option for a binomial proportion, but still depends on the design. See NIST's formulations.
A 95% interval does not guarantee that a particular sample represents the operation or assign a 95% posterior probability to a fixed parameter. Nor does it establish power to compare two methods; that needs a comparison-specific design.
Application boundaries
An activity proportion is not OEE, cycle duration, or individual productivity. Estimating labor time requires a valid relationship between the population of instants and observations. OEE also requires run time, ideal cycle time, and good and total output.
Sampling and time study answer different questions. Neither automatically eliminates observation reactivity or privacy obligations. Do not present the binomial model as scientific or legal certification of an application.