Binomial and Poisson Distribution in Quality Control

Binomial and Poisson: what they are and when to use each

The binomial distribution counts how many defective parts show up in a sample of fixed size, when each part is either good or bad. The Poisson distribution counts how many defects show up in one unit, area or period, when the same item can have several defects.

That choice sets the control chart and the sampling plan: binomial for the p chart and for acceptance plans, Poisson for the c and u charts.

Visual map: binomial and Poisson in quality control

The map puts both distributions side by side, with the formulas and the numeric example from this article.

How to choose between binomial and Poisson

The deciding question is: what am I counting?

A part with five scratches counts as one defective part under the binomial and as five defects under Poisson. Both views are valid but answer different questions: how many parts did I lose, or how much defect does the process produce.

Binomial distribution

It applies when there are n independent trials and the probability p of a defect is the same in all of them (stable process, sample small relative to the lot).

P(X = k) = C(n, k) × p^k × (1 − p)^{n − k}, with C(n, k) = n! ÷ (k! × (n − k)!)

Mean = n × p. Standard deviation = √(n × p × (1 − p)).

Worked example (fictional)

A line of plastic caps has 4% defective parts (p = 0.04). In samples of n = 50 parts, the expected mean is 50 × 0.04 = 2 defective and the standard deviation is √(50 × 0.04 × 0.96) ≈ 1.39.

For k = 2: C(50, 2) = 1,225; 0.04² = 0.0016; 0.96⁴⁸ ≈ 0.1409. So P(X = 2) = 1,225 × 0.0016 × 0.1409 ≈ 0.2762.

Probability of up to 2 defective: 0.1299 + 0.2706 + 0.2762 = 0.6767. Probability of 3 or more: 1 − 0.6767 = 0.3233. In other words, with the process at 4%, one sample of 50 in three contains 3 or more defective parts by chance alone, with nothing having changed.

Poisson distribution

It applies when defects occur randomly, independently, at a constant average rate per inspection unit, and can repeat on the same unit.

P(X = k) = e^−λ × λ^k ÷ k!

Mean = λ. Variance = λ. Standard deviation = √λ.

Worked example (fictional)

Electronic boards average λ = 3 solder defects per board. Then e⁻³ ≈ 0.0498.

A board with no defects: 4.98%. Up to 2 defects: 0.0498 + 0.1494 + 0.2240 = 0.4232. Five or more: 1 − (0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680) = 0.1847. A standard deviation of √3 ≈ 1.73 defects explains why boards with 5 defects appear without any new problem.

When Poisson approximates the binomial

With large n and small p, the binomial approaches a Poisson with λ = n × p. Common rule of thumb: n ≥ 20 and p ≤ 0.05. Comparison for n = 200 and p = 0.01 (λ = 2):

The difference shows up in the third decimal. That is why DPMO and defect rates in ppm, with very small p, are usually handled with Poisson. For larger p (for example 4% with n = 50) the approximation already loses accuracy: P(X = 2) is 0.2762 under the binomial and 0.2707 under Poisson. If the sample is large enough and p is not extreme, the normal distribution also approximates the binomial.

Link to control charts and sampling

• p chart: uses the binomial. With p-bar = 0.04 and n = 50, the upper limit is 0.04 + 3 × √(0.04 × 0.96 ÷ 50) ≈ 0.1231 (12.31%), and the lower limit is negative, so it is 0. More in the p chart article.
• c chart: uses Poisson. With c-bar = 3 defects per board, the upper limit is 3 + 3 × √3 ≈ 8.20 and the lower limit is 0. A board with 9 defects is a signal. See c and u charts for units of varying size.
• Acceptance plan: the probability of accepting a lot with n = 50 and acceptance number c = 2 is the binomial P(X ≤ 2). For p = 2%, 4% and 8% it gives 0.9216, 0.6767 and 0.2260. That is the principle of the operating characteristic curve, the basis of AQL sampling. To size n, see sample size.
• Yield: the probability that a unit comes out defect-free (e^−λ under Poisson) ties both to FPY and RTY.

For the overview of building and reading charts, see SPC and the companion article on the Western Electric and Nelson rules.

Common mistakes

• Counting defects and using the binomial. If a part can have several defects, the model is Poisson (or count defective parts instead).
• Mixing unit sizes. λ holds only for a defined inspection unit; change the area or time and λ changes proportionally.
• Ignoring dependence. Clustered defects (one cause hitting several consecutive parts) break independence. Warning sign: variance well above the mean.
• Assuming constant p with an unstable process. The binomial describes the sample only if p has not changed.
• Approximating without checking the rule. Poisson in place of the binomial with a high p misstates the tails.
• Sample too large relative to the lot. If the sample is large relative to the lot and drawn without replacement, the correct model is the hypergeometric.

Frequently asked questions

What is the difference between binomial and Poisson?

The binomial counts defective parts among n items (each item is good or bad). Poisson counts defects per unit, with no upper limit, and its mean equals its variance.

When should I use the Poisson distribution in quality?

When you count defects per part, board, meter or hour, and they occur independently at a constant average rate.

What is the binomial distribution formula?

P(X = k) = C(n, k) × p^k × (1 − p)^(n − k). In the example, with n = 50 and p = 0.04, P(X = 2) = 0.2762.

Which control chart uses each distribution?

The binomial underlies the p and np charts. Poisson underlies the c and u charts.

Can I approximate the binomial with Poisson?

Yes, when n is large and p is small (for example n ≥ 20 and p ≤ 0.05), using λ = n × p.

How do I compute the probability of accepting a lot?

Add the binomial probabilities from 0 up to the acceptance number c. With n = 50, c = 2 and p = 4%, the probability of acceptance is 0.6767.

About the author

Vagner Soares

Lean Manufacturing & Behavioral Management Specialist

Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.

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