The Standard Error (σp) and Data Precision
Calculate a proportion’s standard error and recognise when dependence or study design changes its interpretation.
Updated:
Standard error describes the variability of an estimator across samples under a model. It is not the full margin of error, confidence level or a measure of every bias. A direct-work share is not, by itself, efficiency or productivity.
1. Formula and assumptions
For n independent, equal-probability Bernoulli observations with parameter p:
$$SE(\hat p)=\sqrt{p(1-p)/n}$$
Because p is unknown, an estimate commonly substitutes the observed share p̂. The expression cannot be applied unchanged to correlated rounds, unequal selection probabilities or clusters. A visit covering several workers does not necessarily create several independent units.
The approximate normal or Wald interval is:
$$\hat p\pm z_{1-\alpha/2}\sqrt{\hat p(1-\hat p)/n}$$
The margin is z×SE; multiplying by 100 expresses it in percentage points. A 95% level concerns repeated-procedure coverage under assumptions, not a probability that a fixed parameter lies in this realized interval. The 68.27% rule describes an ideal normal distribution about its mean, not an exact guarantee for every proportion.
2. Arithmetic example
Assume 500 independent visits, including 160 direct-work states: p̂=0.32.
- SE≈√(0.32×0.68/500)=0.02086.
- Wald95 margin≈1.96×0.02086=0.04089.
- Approximate interval: [27.91%, 36.09%].
This teaching calculation does not measure misclassification, reactivity or other-shift coverage. Wald may perform poorly near zero or one and in small samples; n>30 or checking only np is insufficient. NIST describes proportion-interval methods. WorkSamp’s documented advanced reporting uses Wilson; not every product interval should be called Wald.
3. Plan precision without invented benchmarks
For absolute margin e, an initial approximation is n=ceil(z²p₀(1−p₀)/e²). With e=0.05, p₀=0.5 and z=1.96, it gives 385 independent observations. Halving the margin requires approximately quadrupling n when other conditions remain fixed.
The following figures are simulated and recalculated, not sector averages:
| Case | Assumed p | n | Approximate SE | Wald95 margin |
|---|---|---|---|---|
| A | 0.34 | 850 | 0.01625 | ±3.18 pp |
| B | 0.28 | 620 | 0.01803 | ±3.53 pp |
| C | 0.31 | 550 | 0.01972 | ±3.87 pp |
| D | 0.37 | 780 | 0.01729 | ±3.39 pp |
This does not establish a Spanish industry benchmark or a mandatory 5% rule. Precision and confidence depend on the decision, cost and design. See NIST sample-size planning.
4. Changing categories does not guarantee lower error
MECE can improve consistent classification, but splitting five categories into eight changes the parameters being analyzed. It does not demonstrate a 15–22% uncertainty reduction for the same estimator. With n and p fixed, binomial SE is unchanged. Better classification may reduce measurement error without changing that formula.
A documented unknown state is preferable to guessing. Compare observer agreement and reasons for disagreement, not just the resulting percentage.
5. Reactivity and stopping rules
No five-meter observation distance, concealed uniform or ±0.08 burn-in threshold is prescribed. There is no supporting evidence here for those rules. A nonsignificant early/late comparison does not prove that Hawthorne effects are controlled: power may be inadequate or other conditions may differ.
Larger n reduces random error under a model, not systematic bias. Specify exclusions before inspecting outcomes. Repeatedly checking an interval and stopping at a favorable width can affect nominal coverage; sequential designs need appropriate procedures. An app updating its plan does not convert ordinary intervals into confidence sequences.
6. Compare shifts through the difference
Assume A=0.35 and B=0.38, each with SE=0.02, and independent estimates. For B−A:
$$SE_{diff}=\sqrt{0.02^2+0.02^2}\approx0.02828$$
The difference is 0.03 and an approximate normal95 interval is [−0.0254, 0.0854], or −2.54 to +8.54 percentage points. This example supplies insufficient evidence of a difference at that level, but does not prove equivalence. Overlap of two separate 95% intervals is not a general no-difference test. Paired or correlated data require covariance.
7. OEE, flow maps and decisions
Small SE describes one part of uncertainty under a model; it does not establish flow-map validity or ISO compliance. OEE needs planned/run time, ideal cycle and total/good unit counts. Slow-state or inspection shares do not replace those measurements. Study duration depends on required coverage and precision. See OEE formulas.
8. Reporting
Include estimand, unit, denominator, coverage, planned/actual n, dependence, p̂, interval method, absolute margin and limitations. Keep causal and economic analysis separate. WorkSamp supports observations and reports; Cronometras time studies and Induly working-time records. ASETEMYT and ProdCont are specialist-support options. A product name does not itself guarantee reliability.