Attribute measurement requires a much larger sample size than variable measurement to obtain the same amount of statistical information.

Question

BlackTom AI · Accepted Answer

Attribute data record only a yes/no (defective/nondefective) outcome, while variable data record a continuous measurement (length, weight, etc.). This difference in information content is the key reason attribute measurement typically requires a larger sample size to reach the same statistical precision or power.

1) Relevant knowledge points
- Attribute data are Bernoulli: for proportion p, Var(p̂) = p(1−p)/n.
- Variable data allow estimation of mean μ and standard deviation σ; by using the continuous measurements (and the delta method) you can infer proportions beyond specification limits with smaller variance because each observation contributes magnitude information.
- Sample-size formulas: for a proportion (margin E at confidence z), n_attr = z^2 p(1−p)/E^2. For a mean, n_var = z^2 σ^2 / E_μ^2, where E_μ is allowable error in the mean; when estimating a tail probability, the propagation of error often leads to much smaller required n for variable data.

2) Step-by-step reasoning (illustrative numerical comparison)
Take a process with μ = 100, σ = 2, spec limit at 105. True defective proportion p = P(X>105) = P(Z>2.5) ≈ 0.0062. Suppose we want to estimate p within ±0.003 (E = 0.003) at 95% (z ≈ 1.96).

- Attribute sampling: n_attr = z^2 p(1−p)/E^2
  = 1.96^2 × 0.0062×0.9938 / 0.003^2
  ≈ 3.8416 × 0.00616 / 9e-6
  ≈ 0.0237 / 9e-6 ≈ 2,633 samples.

- Variable sampling (use delta method): p = 1 − Φ((105−μ)/σ). dp/dμ = −φ(z)/σ with z = (105−μ)/σ = 2.5, φ(2.5)≈0.0175 ⇒ |dp/dμ| ≈ 0.0175/2 = 0.00875. To have uncertainty in p of 0.003, we need sd(μ̂) ≤ 0.003 / 0.00875 ≈ 0.343. Since sd(μ̂)=σ/√n = 2/√n, solve 2/√n = 0.343 ⇒ √n ≈ 5.83 ⇒ n ≈ 34.

3) Intermediate steps shown above for both formulas and the delta-method derivative.

4) Why the correct answer is correct
The variable measurements retain magnitude information; when converted to a proportion they produce much more precise estimates per observed unit. The example demonstrates that to achieve the same precision about the tail probability, attribute sampling needed thousands of samples while variable sampling needed only a few dozen. Thus attribute measurement generally requires a much larger sample size to obtain the same amount of statistical information.

5) Why the other option (False) is incorrect
Saying False would imply attributes are as or more informative per sample than variables. That contradicts both information theory and practical formulas: collapsing continuous data to binary discards information and increases estimator variance, so more samples are needed.

6) Conclusion
Therefore the statement is True, which matches the provided correct answer.

Attribute measurement requires a much larger sample size than variable measurement to obtain the same amount of statistical information.判断题

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