Sampling Distributions for Proportions, Counts and Defects

Introduction

Quality professionals rarely observe an entire production population. They inspect a sample and use the resulting counts, proportions, yields, or event rates to infer process performance. The statistical behavior of such sample results is described by a sampling distribution. Understanding these distributions is essential for interpreting control-chart signals, estimating defect rates, designing acceptance samples, comparing suppliers, and evaluating improvement projects.

This chapter focuses on four closely related models. The binomial distribution describes the number of defectives among a fixed number of independent units when each unit is classified as conforming or nonconforming. The multinomial distribution extends this idea to more than two mutually exclusive categories. The Poisson distribution describes counts of defects or events occurring over a fixed opportunity, area, length, time, or volume. Related distributions and approximations provide practical methods when sample sizes are large, event probabilities are small, or exact computation is required.

A central quality distinction must be maintained throughout:

• A defective unit is a unit that fails at least one requirement. A unit is counted once, even if it has several faults.
• A defect is an individual nonconformity. One unit may contain several defects.

Consequently, binomial methods are usually appropriate for counts of defectives, whereas Poisson methods are often appropriate for counts of defects or events.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Distinguish populations, samples, parameters, statistics, defectives, and defects.
2. Derive and apply binomial, multinomial, Poisson, hypergeometric, and related sampling distributions.
3. Calculate expectations, variances, standard errors, and probabilities for quality data.
4. Select exact, normal, or Poisson approximations using defensible assumptions.
5. Construct and interpret confidence intervals for proportions, rates, and rare-event counts.
6. Diagnose overdispersion, dependence, changing opportunity, and other violations of standard models.
7. Apply these methods to inspection, yield, reliability, service quality, and process-improvement decisions.

1. Foundations of Sampling Distributions

Let a population contain units, opportunities, or events generated by a process. A parameter describes the population, while a statistic is computed from a sample. For a binary quality characteristic, let p denote the population probability that a randomly selected unit is defective. If X is the number of defectives in a sample of n units, the observed defective proportion is p̂ = X/n. The statistic p̂ varies from sample to sample even when the process is stable. Its probability distribution is the sampling distribution of the sample proportion.

For independent Bernoulli observations Y_i, where Y_i = 1 denotes a defective and Y_i = 0 a conforming unit,

and, when every unit has the same defective probability p,

The standard error of p̂ is therefore

In practice, p is unknown and is often replaced by p̂, giving the estimated standard error √[p̂(1 − p̂)/n]. These results assume independent observations and a common underlying probability. Stratification, clustering, serial correlation, changing process conditions, and repeated inspection of the same unit can invalidate them.

Finite populations and sampling without replacement

If a finite lot contains N units, of which K are defective, and n units are sampled without replacement, the number of defectives follows a hypergeometric distribution:

Here C(a,b) is the number of combinations of a items taken b at a time. The possible values of x are restricted by the lot composition and sample size. The hypergeometric model is exact for finite-lot inspection. The binomial model is a reasonable approximation when the sampling fraction n/N is small, commonly less than about 0.05, because removal of a few units has little effect on the remaining defective probability.

2. The Binomial Distribution: Defective Units

Model and probability function

A binomial model applies when:

1. There is a fixed number n of trials.
2. Each trial has two mutually exclusive outcomes, such as defective or conforming.
3. The trials are independent, or dependence is negligible.
4. The probability p of a defective is constant across trials.

If X is the number of defectives, then

The mean and variance are

The corresponding defective proportion p̂ = X/n has

The variance is largest, for a fixed n, when p = 0.5. It becomes smaller as the process approaches either zero or one defectives, although estimation near a boundary requires care because normal approximations can become poor.

Worked example: incoming inspection

An incoming lot is believed to have a defective-unit probability of 0.02. If 50 units are selected independently, what is the probability of finding exactly two defectives?

Here n = 50, p = 0.02, and x = 2. Thus

Since C(50,2) = 1,225, the probability is approximately 0.1858. The expected number of defectives is np = 1, and the standard deviation is √(50 × 0.02 × 0.98) ≈ 0.990. The probability of exactly two is not the same as the probability of at least two; the latter is 1 − P(X=0) − P(X=1).

Cumulative probabilities and acceptance decisions

Quality decisions often involve cumulative events:

• P(X ≤ c): accept if no more than c defectives are found.
• P(X ≥ c): investigate if at least c defectives are found.
• P(X = 0): zero-defective observation.

For an acceptance rule that accepts a lot when X ≤ c, the probability of acceptance at defective fraction p is

This function is the operating characteristic of the sampling plan. It describes the chance of accepting lots at different quality levels; it does not prove that an accepted lot contains no defectives. Sampling plans must therefore be evaluated using producer's risk, consumer's risk, and the quality levels considered acceptable or unacceptable.

3. Approximations for Binomial Data

Normal approximation

When both np and n(1−p) are sufficiently large, the binomial distribution can be approximated by a normal distribution with mean np and variance np(1−p). Rules such as both expected counts being at least 5 or 10 are practical guidelines, not universal laws. The quality of approximation depends on the degree of skewness and the required accuracy.

For an integer threshold, a continuity correction improves approximation. For example,

where Φ is the standard normal cumulative distribution function. For P(X ≥ c), use the boundary c − 0.5.

Poisson approximation

When n is large, p is small, and λ = np is moderate, the binomial distribution can be approximated by a Poisson distribution:

The Poisson approximation is useful for rare defective units, but it should not be applied merely because a count is available. It assumes a rare-event structure and approximately independent opportunities. If p is not small, the binomial model is preferable.

Exact confidence intervals for a proportion

A normal interval

is simple but can be inaccurate for small samples, rare defectives, or proportions near 0 or 1. The Wilson interval is generally more reliable. For confidence level 1−α, with z = z_α/2 and p̂ = x/n, it is

The Clopper–Pearson interval inverts exact binomial tail probabilities. It has at least the stated coverage, although it is conservative and can be wider than necessary. For zero defectives in n independent units, a two-sided exact interval has an upper endpoint obtained from (1−p_U)ⁿ = α/2, hence

For a one-sided 95% upper bound, replace α/2 by α. A zero count is therefore evidence of limited information, not proof of a zero process rate.

4. Multinomial Distributions and Multiple Quality Categories

Binary classification can hide important information. A product may be conforming, have a cosmetic defect, have a dimensional defect, or have a functional defect. If each inspected unit falls into exactly one of k mutually exclusive categories with probabilities p₁,...,p_k, where ∑p_j = 1, the vector of category counts follows a multinomial distribution.

subject to ∑x_j = n. The expected count and variance for category j are

For two different categories j and l,

The negative covariance is structural: with a fixed total sample size, more observations in one category leave fewer available for the others. The sample category proportions p̂_j = X_j/n satisfy ∑p̂_j = 1, so they are not independent.

Application to defect-mode analysis

Suppose 200 units are classified as conforming, cosmetic defect, dimensional defect, or functional defect. The counts are 170, 12, 10, and 8. The observed proportions are 0.850, 0.060, 0.050, and 0.040. A multinomial model can assess whether these counts are consistent with a specified defect-mode profile or compare profiles across suppliers.

A common global test compares observed counts O_j with expected counts E_j = np_j:

Under suitable regularity conditions and a fully specified model, the statistic is approximately chi-square with k−1 degrees of freedom. If parameters are estimated from the data, the degrees of freedom are reduced by the number of independently estimated parameters. Expected counts should not be too small; sparse categories may require exact methods, category combination based on engineering logic, or simulation.

A crucial modeling choice is whether defect categories are mutually exclusive. If a single unit may receive several defect labels, multinomial classification is inappropriate. In that situation, analysts may model each defect mode separately, use multivariate methods, or represent each unit by binary indicators while recognizing dependence among indicators.

5. The Poisson Distribution: Defects, Events and Rates

Model and assumptions

The Poisson distribution models the number of events in a fixed opportunity or exposure when:

1. Events occur singly rather than in simultaneous batches.
2. The average event rate is stable over the observation interval.
3. Events are approximately independent.
4. The probability of more than one event in a very small interval is negligible.

If X is the count in an exposure of size t and λ is the mean count per unit exposure, then the expected count is μ = λt, and

The defining property is

Examples of exposure include defects per metre of fabric, customer complaints per 10,000 transactions, failures per operating hour, or particles per square centimetre. The unit of λ must match the denominator. A rate of 0.4 defects per metre cannot be inserted into a formula whose exposure is measured in square metres without conversion.

Defects versus defectives

If each unit can have multiple defects and the number of opportunities per unit is reasonably stable, a Poisson model for defects may be appropriate. If the analysis counts whether a unit has at least one defect, the resulting statistic is binary and a binomial model is usually more appropriate. The probability that a Poisson-distributed unit has at least one defect is

Thus, if the mean number of defects per unit is λ = 0.10, the implied probability of at least one defect is approximately 1−e^−0.10=0.0952, assuming the Poisson model. The defect rate and defective-unit rate are related but are not interchangeable.

Worked example: surface defects

A coating inspection examines 80 square metres and finds 12 pinhole defects. The estimated rate is

For a future 10-square-metre panel, the expected count is μ = 0.15×10 = 1.5. The probability of no pinholes is

The probability of at least one is approximately 0.7769. This calculation is valid only if the future panel has comparable conditions and the spatial independence and constant-rate assumptions are defensible.

Poisson confidence intervals and rates

For an observed count x over exposure T, the maximum-likelihood estimate of the rate is

A large-sample approximate interval is

because the estimated variance of λ̂ is approximately x/T². This approximation is unreliable when x is small. Exact Poisson intervals are obtained by inverting Poisson tail probabilities or, equivalently, using chi-square quantiles. For count x and confidence level 1−α, the exact limits for the mean count μ are

with μ_L=0 when x=0. Divide both limits by exposure T to obtain limits for λ. Notation for chi-square quantiles must be checked against the software convention; the displayed expression uses χ²_ν,q as the q quantile with ν degrees of freedom.

6. Related Models and Advanced Quality Situations

Negative binomial and overdispersion

A Poisson model requires equality of mean and variance. Quality data often show overdispersion, where the variance exceeds the mean because of heterogeneous lots, clustering, changing conditions, or unobserved subpopulations. A negative binomial model introduces extra variation in the Poisson mean. One common parameterization has

where r>0 is a dispersion or size parameter. As r→∞, the variance approaches the Poisson variance. Parameterization varies across software, so the definition must always be stated.

Zero-inflated data

Some processes produce more zeros than a Poisson model predicts. A zero-inflated model treats observations as arising from a structural-zero state with probability π or from a count process with probability 1−π. It can be useful when many units have no opportunity for a defect, but it should not be used simply to force a better fit. Engineering explanations for structural zeros should be sought first.

Poisson process and time-to-event reasoning

Under a homogeneous Poisson process with rate λ per unit time, counts in disjoint time intervals are independent Poisson variables. The waiting time to the first event is exponentially distributed:

where T is the waiting time and t is a nonnegative duration. The exponential model has the memoryless property, which may be unrealistic for equipment subject to wear-out or preventive maintenance. Reliability analyses should distinguish random failure processes from age-dependent failure mechanisms.

Yield and rolled throughput yield

For a single independent stage with conforming probability Y_i, the yield is Y_i = 1−p_i. If stages are independent, the probability of passing all m stages is

For small defect probabilities, RTY can decline materially across many stages. Independence is a strong assumption: common materials, operators, machines, or environmental conditions can create dependence. In that case, multiplying marginal yields may misrepresent actual end-to-end performance.

7. Selecting an Appropriate Method

The method should follow the data-generating process, not merely the appearance of a table of counts.

A disciplined analysis asks the following questions:

1. What is the observational unit: product, defect, opportunity, transaction, or exposure?
2. Is the denominator fixed in advance, random, or dependent on the process?
3. Are observations independent, and is the rate stable?
4. Can one unit contribute more than one event?
5. Is sampling with or without replacement?
6. Are categories mutually exclusive and collectively exhaustive?
7. Are expected counts large enough for an approximation?
8. What decision will the result support, and what error is costly?

Exact methods are preferred when counts are small, proportions are near boundaries, or regulatory or contractual decisions require reliable tail probabilities. Approximations are valuable for speed, explanation, and planning, but their assumptions should be documented.

8. Detailed Illustrative Case: Electronic Connector Quality

The following is a fictional illustrative case designed for instructional use.

Background

A manufacturer of electronic connectors performs final inspection on a stable production line. Each connector is tested for functional conformity. During a separate visual examination, all individual surface defects are recorded. Management wants to estimate functional defective proportion, characterize defect modes, and forecast surface defects in a larger shipment.

Problem

In a random sample of 400 connectors, 6 fail the functional test. Among the 400 connectors, visual inspection records 18 scratches, 7 contamination marks, and 5 plating voids. A future shipment contains 2,000 connectors, and historical evidence suggests that the average number of visual defects per connector is stable.

Analysis of defectives

The functional result is binary, so the estimated defective proportion is

A normal interval would be questionable because the observed defective count is small. An exact binomial or Wilson interval should be reported. The estimate should not be interpreted as a guaranteed shipment fraction; it is an estimate subject to sampling uncertainty.

If the contractual question is whether the process is below 2% defectives, an exact one-sided binomial test can evaluate H₀:p ≥ 0.02 against H₁:p < 0.02. The test must be specified before treating the result as evidence of compliance, and the practical importance of the threshold should be considered alongside statistical significance.

Analysis of defect types

The visual counts total 30 defects. The observed defect-mode proportions among recorded defects are 0.60 scratches, approximately 0.233 contamination marks, and approximately 0.167 plating voids. If each defect is assigned exactly one mode, a multinomial model is appropriate for comparing the observed composition with an engineering target profile. If one connector can have both a scratch and a contamination mark, the categories are not mutually exclusive at the unit level; the analysis must then preserve the multiple-response structure.

Forecasting defects

The observed visual defect rate is

For 2,000 comparable connectors, the expected number is 2,000×0.075 = 150 defects. A Poisson forecast would give variance 150 and standard deviation approximately 12.25 only if the rate is stable and defects are approximately independent. If defects cluster by production batch or reel of material, the actual variance may be larger; a negative binomial or batch-level model would be more credible.

Approach and outcome

The quality team reports three separate indicators rather than one blended number: functional defective proportion, defect-mode composition, and defects per connector. It uses an exact interval for the rare functional failures, a multinomial analysis for exclusive defect categories, and a Poisson model as an initial forecast with an overdispersion check. Management therefore receives information about failed units, failure mechanisms, and total opportunities for improvement.

Learning points

1. Defectives and defects answer different managerial questions.
2. A single connector can be defective once but contribute several defects.
3. A low observed count requires interval methods that respect the discrete data.
4. Forecasts depend on exposure, stability, and independence assumptions.
5. Model checking and process knowledge are as important as formula selection.

9. Limitations, Governance and Contemporary Practice

Sampling distributions describe uncertainty under a model; they do not correct biased sampling. Convenience samples, inspection that removes only visibly poor units, changing product mix, and selective data deletion can produce precise but misleading estimates. Sampling frames, inclusion rules, inspection sensitivity, and missing observations should be documented.

Measurement-system error is especially important. If an inspection system misses defects, the observed count is not the true count. Attribute agreement analysis, sensitivity studies, and validated inspection procedures should precede major decisions. Automated vision and AI classification can increase inspection volume, but classification thresholds, training-data drift, false negatives, false positives, and explainability must be monitored. An algorithmic count should not be treated as objective merely because it is automated.

For regulated or safety-critical products, acceptance sampling and release decisions should follow applicable standards, contracts, and regulatory requirements. Statistical evidence should not be used to conceal known hazards. When a rare event could have severe consequences, risk-based engineering controls may be required even when the estimated probability is small.

Best practice includes:

• define the unit, opportunity, exposure, and denominator before collecting data;
• separate defectives from defects in databases and reports;
• preserve raw counts and inspection effort, not only percentages;
• use exact or boundary-respecting intervals for small counts;
• check stability, independence, and overdispersion;
• stratify by machine, shift, supplier, product, and time when scientifically justified;
• report assumptions and practical consequences with statistical results;
• avoid repeated testing until a favorable result appears;
• use process knowledge to explain unusual zeros, clusters, and category changes.

Chapter Summary

Sampling distributions provide the bridge from inspected samples to quality-process conclusions. The binomial distribution models the number of defective units in a fixed sample of independent binary trials. The multinomial distribution models mutually exclusive quality categories whose counts sum to the sample size. The Poisson distribution models defect or event counts over a fixed exposure and is characterized by equality of mean and variance. The hypergeometric distribution is exact for sampling without replacement from a finite lot.

Normal and Poisson approximations can simplify computation, but only when their conditions are credible. Exact binomial and Poisson methods are preferable for small counts, rare events, and boundary proportions. Confidence intervals communicate uncertainty more honestly than point estimates alone. Overdispersion, clustering, zero inflation, changing rates, and measurement error require expanded models or improved data collection. Most importantly, the analyst must identify whether the quality question concerns defective units, individual defects, defect modes, yield, or event rates before selecting a distribution.

Key Terms

Sampling distribution: Probability distribution of a statistic across repeated samples.

Defective: A unit that fails at least one specified requirement.

Defect: An individual nonconformity or fault.

Bernoulli trial: A trial with two possible outcomes.

Binomial distribution: Distribution of successes or defectives in a fixed number of independent binary trials.

Defective proportion: Number of defective units divided by the number inspected.

Hypergeometric distribution: Exact distribution for sampling without replacement from a finite population.

Multinomial distribution: Distribution of counts in mutually exclusive categories.

Poisson distribution: Distribution of event counts over a fixed exposure.

Exposure: Time, area, length, volume, transactions, or other opportunity base for a count.

Rate: Expected count per unit of exposure.

Operating characteristic: Probability that a sampling plan accepts a lot at a specified quality level.

Continuity correction: Adjustment used when approximating a discrete distribution by a continuous distribution.

Overdispersion: Variance greater than that predicted by a basic Poisson or binomial model.

Negative binomial distribution: Count model allowing greater variance than the Poisson distribution.

Zero inflation: Excess occurrence of zero counts relative to a reference count model.

Rolled throughput yield: Probability of passing all process stages without rework or failure.

Exact confidence interval: Interval obtained by inverting the exact sampling distribution.

Measurement-system error: Error arising from the inspection or classification process.

Review Questions

Short Answer Questions

1. What is the difference between a defective unit and a defect?
2. State the four principal assumptions of the binomial model.
3. Why are multinomial category counts negatively correlated?
4. What does exposure mean in a Poisson model?
5. Why can a zero observed count not establish a zero defect rate?

Long Answer Questions

1. Derive the mean and variance of a binomial count using sums of independent Bernoulli variables.
2. Compare binomial, hypergeometric, multinomial, and Poisson models in a quality-inspection context.
3. Explain when normal and Poisson approximations to the binomial distribution are defensible and when they are risky.
4. Discuss the distinction between defect rates and defective-unit rates, including an example where confusing them would mislead management.
5. Explain overdispersion, its possible process causes, and the role of the negative binomial model.

Analytical / Application Questions

1. A sample of 250 units contains 4 defectives. Select an appropriate confidence-interval method, justify it, and explain what additional information management needs before making a release decision.
2. A factory records 22 surface defects over 110 square metres. Estimate the defect rate and calculate the probability of observing no defects on a future 5-square-metre panel under a Poisson assumption. Critically evaluate the assumption.
3. A supplier reports 96 conforming units, 3 cosmetic defectives, and 1 functional defective in a sample of 100. Explain whether a multinomial model is appropriate, identify a suitable analysis, and discuss what changes if a unit may receive both defect labels.

Multiple Choice Questions with Answers

Question 1

Which distribution is most appropriate for the number of defective units in a fixed sample when each unit is independently classified as conforming or defective?

A. Binomial

B. Poisson

C. Multinomial with unrestricted categories

D. Exponential

Correct Answer: A. Binomial

Question 2

If a process has a defective probability of 0.02 and 50 independent units are inspected, what is the expected number of defectives?

A. 0.02

B. 0.98

C. 1.00

D. 2.50

Correct Answer: C. 1.00

Question 3

Which condition most directly supports use of the hypergeometric distribution?

A. Events occur over time at a constant rate

B. Sampling is without replacement from a finite lot

C. Each observation has several mutually exclusive categories

D. The event probability is extremely small

Correct Answer: B. Sampling is without replacement from a finite lot

Question 4

A Poisson model is generally used to describe which quantity?

A. The number of defects per fixed area or time exposure

B. The proportion of conforming units in a finite sample only

C. The rank of suppliers by average quality

D. The waiting-time median under every reliability model

Correct Answer: A. The number of defects per fixed area or time exposure

Question 5

For a Poisson random variable with mean 4, what is its variance under the standard model?

A. 0

B. 2

C. 4

D. 16

Correct Answer: C. 4

Question 6

Why are counts from different categories in a multinomial sample negatively correlated?

A. The categories have identical probabilities

B. The fixed total means more observations in one category leave fewer for others

C. Multinomial observations are always collected over time

D. Each category is modeled by a separate Poisson process

Correct Answer: B. The fixed total means more observations in one category leave fewer for others

Question 7

A coating process produces 12 defects over 80 square metres. Under a constant-rate Poisson model, what is the estimated defect rate?

A. 0.015 defects per square metre

B. 0.15 defects per square metre

C. 6.67 defects per square metre

D. 92 defects per square metre

Correct Answer: B. 0.15 defects per square metre

Question 8

Which situation is evidence that a basic Poisson model may be inadequate?

A. The observed variance is substantially larger than the observed mean

B. The count is a nonnegative integer

C. The exposure is measured in square metres

D. The sample contains at least one zero

Correct Answer: A. The observed variance is substantially larger than the observed mean

Question 9

If the mean number of defects per unit is Poisson with λ = 0.10, what is the probability that a unit has at least one defect?

A. 0.0100

B. 0.0952

C. 0.1000

D. 0.9048

Correct Answer: B. 0.0952

Question 10

Which statement about a zero-defect observation is correct?

A. It proves the process defect rate is zero

B. It provides no information about the process rate

C. It provides evidence but still has sampling uncertainty

D. It automatically justifies a normal confidence interval

Correct Answer: C. It provides evidence but still has sampling uncertainty