3. Probability distributions in quality engineering

Every control chart limit, acceptance sampling plan, and process capability index rests on an assumed probability model. Choosing the wrong distribution does not merely shift a number — it changes the false-alarm rate, the consumer's risk, and the predicted fraction nonconforming. This chapter builds a working map of the five distributions that cover the overwhelming majority of quality-engineering data: normal, binomial, Poisson, exponential, and Weibull. The goal is not to derive them, but to recognize which one governs a given measurement or count, and to compute the quantities that drive decisions.

3.1 Choosing the Model: Continuous Measurement vs. Discrete Count vs. Time-to-Event

The first decision is structural, not numerical. Ask what the random variable actually is:

• Continuous magnitude (shaft diameter, fill volume, tensile strength) → normal model, provided the process is stable and the variation is the sum of many small independent causes.
• Count of defectives in a fixed sample of n units, each unit independently conforming or not with constant probability p → binomial model.
• Count of defects (nonconformities) per unit of exposure — flaws per panel, solder bridges per board, particles per wafer — where a unit can carry several defects → Poisson model.
• Time or cycles to failure of a component with a constant hazard rate (random failures, no wear-out) → exponential model.
• Time to failure when the hazard rate changes with age (infant mortality, wear-out, fatigue) → Weibull model, which generalizes the exponential.

A frequent error is applying the binomial to defect counts. If a printed-circuit board can have five solder defects, the count is not bounded by one per board, and the Poisson model applies. Conversely, if inspection classifies each of 200 units as pass/fail, the count of failures is binomial, not Poisson — although the Poisson is a useful approximation when n is large and p is small (np < 5).

3.2 The Normal Model in Process Control

For a stable process with mean μ and standard deviation σ, the fraction of output above an upper specification limit is 1 − Φ((USL − μ)/σ). This single calculation underlies the Z-score approach to capability. Suppose a filling line targets μ = 500 mL with σ = 4 mL and USL = 512 mL. Then Z = (512 − 500)/4 = 3.0, and the upper-tail nonconforming fraction is 0.00135, or 1350 ppm. If the process mean drifts to 504 mL while σ stays at 4 mL, Z drops to 2.0 and the tail fraction rises to 0.0228 — a 17-fold increase from a 4 mL shift. This sensitivity is why the normal model, combined with the ±1.5σ shift convention, produces the familiar 3.4 ppm figure for a centered six-sigma process.

Two cautions apply. First, the normal model requires an essentially unbounded, symmetric variable; a fill volume cannot be negative, so the model is only adequate when μ/σ is large (say, greater than 3). Second, normality should be checked, not assumed. A normal probability plot or an Anderson–Darling statistic on 50–100 stable observations is sufficient for most shop-floor work. If the data are skewed, a Box–Cox transformation or a switch to a Weibull/lognormal model is more defensible than forcing a normal fit.

3.3 Binomial and Poisson Models for Defect Data

Acceptance sampling and p-charts use the binomial. With n = 200 and p = 0.02, the expected number of defectives is np = 4 and the standard deviation is √(np(1−p)) = 1.98. The probability of observing 8 or more defectives is found by summing the binomial terms from d = 8 to 200; in practice this is computed with a spreadsheet function such as 1 − BINOM.DIST(7, 200, 0.02, TRUE), which returns approximately 0.051. That value is the producer's risk of rejecting a lot whose true fraction defective is exactly 2% when the acceptance number is 7.

The Poisson model describes defects per unit. If a coating line averages λ = 1.5 defects per panel, the probability of a panel passing a zero-defect requirement is e^(−1.5) = 0.223. Raising the average to λ = 3.0 drops that to 0.050 — a small change in the mean produces a large change in yield when the acceptance criterion is zero defects. For a c-chart, the 3-sigma upper control limit is c̄ + 3√c̄; with c̄ = 1.5 this gives 1.5 + 3(1.225) = 5.17, so any panel with 6 or more defects signals an assignable cause.

3.4 Exponential and Weibull Models for Failure Data

Reliability engineering replaces the defect count with time to failure. The exponential model assumes a constant hazard rate λ, so the reliability at time t is R(t) = e^(−λt) and the mean time to failure is 1/λ. If a power supply has MTTF = 50,000 hours, the probability of surviving 10,000 hours is e^(−0.2) = 0.819. The memoryless property means a unit that has already run 10,000 hours has the same remaining MTTF as a new one — a reasonable assumption for random electronic failures, but wrong for mechanical wear.

The Weibull distribution removes that restriction. Its reliability function is R(t) = exp[−(t/η)^β], where η is the scale parameter (characteristic life, the time at which 63.2% have failed) and β is the shape parameter. The hazard rate h(t) = (β/η)(t/η)^(β−1) decreases for β < 1, is constant for β = 1, and increases for β > 1. To estimate the parameters, take the natural logarithm twice: ln[−ln R(t)] = β ln t − β ln η. Plotting ln[−ln R(t)] against ln t yields a straight line whose slope is β and whose intercept gives η. This is the Weibull probability plot used in life testing.

Consider a bearing test with 10 units run to failure. The ordered failure times (hours) are 120, 210, 280, 340, 410, 500, 620, 780, 950, 1300. Using median ranks for R(t), a Weibull plot of these points gives an approximate slope β ≈ 1.8, indicating wear-out, and η ≈ 700 hours. The practical consequence: because β > 1, preventive replacement before 700 hours reduces field failures, whereas a β ≈ 1 result would make scheduled replacement pointless.

3.5 A Compact Decision Table

The five models in this chapter are not competing alternatives; they are complementary tools matched to the structure of the data. The next chapter applies the normal model directly to variable control charts, where the assumption of independence and stability is tested before any limit is computed.