Agricultural Statistics & Data Science · Time series, forecasting and modelling · also SQC, Quality Control and Reliability
Monitoring and accepting quality: Shewhart charts for variables and attributes, CUSUM and EWMA charts, process capability, acceptance sampling plans with their OC, AOQ and ATI curves, and reliability models, applied to seed testing, food processing and input quality. The Agricultural Engineering track covers control charts briefly; this is the full subject.
In the exams: UPSC ISS Statistics III (heavy, reported); SSC CGL JSO (present); AICE-JRF Dairy Microbiology syllabus names 'statistical quality control' (present); light in AICE-JRF Ag Stat
Statistical quality control (SQC) applies statistics to keep products and processes within acceptable limits and to decide whether lots of material should be accepted. Its two classical branches are statistical process control, which monitors a process over time with control charts so that problems are detected and corrected while production goes on, and acceptance sampling, which decides on the basis of a sample whether to accept or reject a lot already produced. Reliability adds the study of how long products and systems last.
Agriculture and agro-industry depend on these methods. Seed companies chart germination and purity of successive lots and sample incoming lots from growers; feed and fertiliser plants monitor bag weights and nutrient content; dairies chart fat and solids-not-fat of milk and bacterial counts; food processors control fill weights, moisture and contamination; procurement agencies sample grain for moisture and foreign matter; and machinery and irrigation equipment are judged by their reliability. Much of this is governed by national and international standards, which specify sampling plans and acceptance criteria; the standards differ between countries and products, so this manual teaches the statistics behind them rather than quoting any one standard's tables.
The chapters cover chance and assignable causes; Shewhart charts for variables (X̄, R, s) and attributes (p, np, c, u); CUSUM and EWMA charts; process capability; acceptance sampling by attributes with its OC, AOQ, AOQL, ASN and ATI curves; acceptance sampling by variables; an introduction to Six Sigma and design for quality; and reliability. The manual builds on Sampling Theory I (sampling from finite lots), Statistical Inference II (hypothesis tests, error rates and Wald's sequential probability ratio test), Probability Distributions (binomial, Poisson, hypergeometric and normal) and Bioassay, Survival and Epidemiological Statistics (hazard functions and censored life data). The Agricultural Engineering track covers control charts briefly; this is the full statistical treatment.
The Indian Statistical Service includes SQC in Statistics III, the SSC Junior Statistical Officer paper includes it, and it appears in the dairy and food-technology syllabi of ICAR examinations; questions are mainly computations of control limits, OC and AOQ values, capability indices and system reliabilities.
Every process varies. Walter Shewhart distinguished chance (common, random) causes, the many small, unavoidable sources of variation inherent in a process, from assignable (special) causes, identifiable disturbances such as a worn filling nozzle, a change of raw-material supplier, a miscalibrated moisture meter or an untrained operator. A process operating with only chance causes is in statistical control: its output behaves like random draws from a stable distribution and is predictable within limits. Assignable causes produce shifts, trends or unusual points that make the process unpredictable, and they can usually be found and removed by the people running the process.
The two kinds of variation call for different actions. Reacting to chance variation as if it had a special cause (tampering, such as resetting a machine after every slightly heavy bag) increases variation; Deming's funnel experiment shows this vividly. Ignoring assignable causes lets defects accumulate. Reducing chance variation itself requires changing the process (better equipment, raw materials or methods), usually a management decision. A control chart is the tool for telling the two apart, and it gives the people closest to the process a shared, objective signal for when to act and when to leave the process alone.
A Shewhart control chart plots a statistic (a subgroup mean, range, proportion defective or count) against time, with a centre line at its in-control mean and control limits, conventionally at three standard errors on either side. Points outside the limits, or non-random patterns inside them, signal a possible assignable cause. Each point is in effect a test of the hypothesis that the process is in control: three-sigma limits give a false-alarm probability of about 0.0027 per point for a normal statistic, so the in-control average run length (ARL₀, the expected number of points until a false alarm) is about 1/0.0027 = 370. The ARL when the process has shifted (ARL₁) measures how quickly the chart detects a change.
Supplementary run rules (the Western Electric rules), such as eight successive points on one side of the centre line or two of three points beyond two sigma, increase sensitivity to small shifts but also raise the false-alarm rate, so ARL₀ falls well below 370 when several rules are used together. Control limits are not specification limits: control limits describe what the process does, specification limits what the customer requires. Rational subgrouping, choosing subgroups so that variation within them is chance variation only (for example five consecutive bags from one filling head), is the key design decision. Charts are set up in Phase I, using historical data to estimate parameters and remove assignable causes, and used for monitoring in Phase II with the limits fixed.
Agricultural and food processes bring their own complications. Raw material varies far more than in most manufacturing: milk composition changes with season, breed and feed; grain moisture with harvest weather; fruit size with orchard and picking date. A chart whose limits were set in one season may signal constantly in the next, not because the process is out of control but because the input has changed. Practical responses are to chart the process output adjusted for input (for example the deviation of fill weight from target rather than raw weight), to set limits by season or batch type, or to chart residuals from a model that includes the known input effects.
Sampling is often destructive or slow: a germination test takes a week or more, a microbiological count two days. Results then arrive after the product has moved on, so control charts are used for process learning and supplier management as much as for real-time correction. Where rapid proxies exist (near-infrared moisture or protein readings, tetrazolium viability tests), they are charted alongside the slower reference tests, with periodic checks that the proxy remains calibrated.
For a measured characteristic (bag weight, milk fat percentage, grain moisture), samples (subgroups) of n units are taken at intervals. The X̄ chart monitors the process mean and the R (range) or s (standard deviation) chart monitors its variability. With k subgroups in Phase I, the grand mean x̿ and the average range R̄ estimate the process mean and, through σ̂ = R̄/d₂, the process standard deviation, where d₂ is a constant depending on n (the expected range of n standard normal observations: 1.128 for n = 2, 2.059 for n = 4, 2.326 for n = 5).
The X̄ chart has centre line x̿ and limits x̿ ± 3σ̂/√n = x̿ ± A₂R̄, with A₂ = 3/(d₂√n) (0.577 for n = 5). The R chart has centre line R̄ and limits D₃R̄ and D₄R̄ (for n = 5, D₃ = 0 and D₄ = 2.114). Example: 25 subgroups of five 50-kg fertiliser bags give x̿ = 50.12 kg and R̄ = 0.40 kg. The X̄ chart limits are 50.12 ± 0.577 × 0.40 = 49.889 to 50.351 kg; the R chart limits are 0 and 0.846 kg; and σ̂ = 0.40/2.326 = 0.172 kg. The R chart should be examined first: if variability is out of control, the X̄ limits, which depend on R̄, are not meaningful.
For larger subgroups (n above about 10) the sample standard deviation is more efficient than the range. The s chart has centre line s̄ and limits B₃s̄ and B₄s̄, and the X̄ chart limits become x̿ ± A₃s̄; σ is estimated by s̄/c₄, with c₄ the bias-correction constant for s (0.940 for n = 5). When only one measurement per period is available (a daily laboratory analysis of a bulk tank, a batch moisture reading), an individuals (X) chart is used with a moving-range (MR) chart, σ being estimated by MR̄/1.128.
Practical points. The normal-theory three-sigma interpretation of the X̄ chart is fairly robust because subgroup means are close to normal by the central limit theorem; individuals charts are much more sensitive to non-normality. Autocorrelated data, common in continuous processes such as milk pasteurisation temperatures, produce many false alarms on standard charts; modelling the autocorrelation (time-series residual charts) or widening the sampling interval helps. The power of an X̄ chart to detect a shift of δ standard deviations in one subgroup is P(|Z + δ√n| > 3) approximately; for n = 5 and δ = 1 this is about 0.22, so a one-sigma shift takes on average about 4.5 subgroups to detect, which motivates CUSUM and EWMA charts for small shifts.
When each unit is classified as conforming or non-conforming (a seed sample failing germination, a packet underweight, a fruit with a blemish), the proportion non-conforming p̂ in subgroups of size n is binomial. The p chart has centre line p̄, the average proportion over Phase I, and limits p̄ ± 3√[p̄(1 − p̄)/n]; a negative lower limit is set to zero. With varying subgroup sizes the limits vary with nᵢ, or a standardised chart is plotted. The np chart plots the number non-conforming, with centre np̄ and limits np̄ ± 3√[np̄(1 − p̄)], and is used when n is constant.
Example: 20 lots of 200 seeds each are tested, with an average of 4% abnormal seedlings. The p chart has limits 0.04 ± 3√(0.04 × 0.96/200) = 0.04 ± 0.0416, i.e. 0 (truncated) to 0.0816. A lot with 18 abnormal seedlings (9%) is outside the limits and should be investigated. Because the lower limit is zero, improvements (fewer defects) cannot be signalled by a point below the limit unless n is large enough for a positive lower limit, which requires n > 9(1 − p̄)/p̄ = 216 here.
When defects are counted on each inspection unit (blemishes per fruit tray, weed seeds per kilogram of seed, bacterial colonies per plate, defects per metre of drip tape), the count is often well described by a Poisson distribution. The c chart, for a constant inspection unit, has centre c̄ and limits c̄ ± 3√c̄. The u chart, for counts per unit when the amount inspected varies (nᵢ units), plots uᵢ = cᵢ/nᵢ with centre ū and limits ū ± 3√(ū/nᵢ).
Example: weed seeds in 1-kg samples of a wheat seed lot average c̄ = 4. The c chart limits are 4 ± 3 × 2 = −2 (set to 0) and 10. A sample with 12 weed seeds signals contamination. If defects cluster (overdispersion), the Poisson limits are too narrow and produce false alarms; the remedy is to estimate the variance empirically or use negative-binomial limits, the same overdispersion issue met in the categorical-data subject. Attribute charts need much larger samples than variable charts for the same sensitivity, because a pass/fail classification carries less information than a measurement; when a measurement is available, a variables chart is usually preferable.
Designing an attribute chart means choosing how many units to inspect and how often. Larger subgroups detect smaller changes in p but cost more; more frequent sampling detects changes sooner. A useful rule for the p chart is to choose n large enough that a shift from p̄ to a worse level p₁ has a reasonable chance of a signal on the next subgroup: the probability is about P(Z > [p̄ + 3√(p̄(1 − p̄)/n) − p₁]/√(p₁(1 − p₁)/n)). For p̄ = 0.04 and p₁ = 0.08, n = 200 gives a detection probability of about 0.47 per subgroup, while n = 400 raises it to about 0.78.
Counts of rare events are a particular problem: if defects occur at one per thousand units, subgroups of a hundred contain no defects most of the time and the chart is uninformative. Charts of the time or number of units between events (geometric or exponential charts, and the 'g chart') are then more effective, a link with the Poisson-process ideas of the stochastic-processes subject.
Shewhart charts use only the current point and are slow to detect small, sustained shifts. The cumulative sum (CUSUM) chart accumulates deviations from the target and detects shifts of about one standard error or less much faster. The tabular CUSUM for detecting an upward shift in the mean computes C⁺ᵢ = max[0, xᵢ − (μ₀ + K) + C⁺ᵢ₋₁], and for a downward shift C⁻ᵢ = max[0, (μ₀ − K) − xᵢ + C⁻ᵢ₋₁], both starting at 0, where xᵢ is the observation or subgroup mean. A signal occurs when either exceeds the decision interval H. The reference value K is usually half the shift to be detected, K = δσ/2, and H is commonly 4σ or 5σ (σ being the standard deviation of the plotted statistic); with K = 0.5σ and H = 5σ, ARL₀ is about 465 and the ARL for a one-sigma shift about 10, against about 44 for a Shewhart chart on individual values.
Example: target fill weight μ₀ = 500 g, σ = 2 g, K = 1 g, H = 10 g. Observations 501, 503, 502, 504 and 503 give C⁺ = 0, 2, 3, 6, 8; one more value of 504 gives C⁺ = 11 > 10, a signal that the mean has drifted upwards by about a standard deviation. The CUSUM is closely related to Wald's sequential probability ratio test (see the hypothesis-testing manual): it is a sequence of SPRTs restarted whenever the statistic returns to zero.
The exponentially weighted moving average (EWMA) chart plots zᵢ = λxᵢ + (1 − λ)zᵢ₋₁, starting at z₀ = μ₀, with a weight 0 < λ ≤ 1 on the newest observation. Older observations receive geometrically declining weights, so the EWMA remembers recent history. Its control limits are μ₀ ± Lσ√{[λ/(2 − λ)][1 − (1 − λ)²ⁱ]}, which settle to μ₀ ± Lσ√[λ/(2 − λ)]. Small λ (0.05 to 0.25) with L near 2.7 to 3 gives good performance for small shifts; λ = 1 reduces the EWMA to a Shewhart chart. With λ = 0.2, the steady-state limits are μ₀ ± 3σ√(0.2/1.8) = μ₀ ± σ.
EWMA charts are robust to moderate non-normality and suit individual measurements, which makes them popular for laboratory and continuous-process data in dairies and food plants. The EWMA is also a simple forecaster (exponential smoothing), which links process monitoring with forecasting in the time-series subject. Both CUSUM and EWMA charts trade a little sensitivity to large, sudden shifts for much better sensitivity to small, persistent ones, so they are often run alongside a Shewhart chart.
The choice depends on the shifts that matter and on how data arrive. Large, sudden changes (a broken component, a wrong raw-material batch) are caught fastest by Shewhart charts with three-sigma limits. Small, gradual drifts (nozzle wear, slow loss of calibration, gradual change in raw material) are caught much faster by CUSUM or EWMA charts tuned to the size of shift worth detecting. When only individual readings are available and the distribution is skewed, the EWMA's averaging makes it more reliable than an individuals chart. When several correlated characteristics are monitored together (fat, protein and lactose in milk), multivariate charts based on Hotelling's T² or multivariate EWMA control the overall false-alarm rate and detect changes in combinations of variables that separate charts miss; these build on the multivariate-analysis subject. Whatever chart is used, its ARL properties should be checked for the actual sampling plan before it is adopted. Average run lengths for CUSUM and EWMA designs are tabulated in standard texts or computed by Markov-chain approximation or simulation; a design should be tuned so that ARL₀ is acceptably long for the cost of a false alarm (stopping a packing line, re-testing a batch) and ARL₁ short for the shift that would cause real loss.
Process capability compares the natural spread of an in-control process with its specification limits. With lower and upper specification limits LSL and USL, process mean μ and standard deviation σ, the potential capability index is Cp = (USL − LSL)/6σ, the ratio of the specification width to the natural tolerance width 6σ. It ignores centring. The actual capability index Cpk = min[(USL − μ)/3σ, (μ − LSL)/3σ] accounts for the mean being off-centre; Cpk ≤ Cp, with equality when the process is centred. One-sided specifications use only the relevant half: Cpu = (USL − μ)/3σ or Cpl = (μ − LSL)/3σ.
Example: a seed-packet filler has specifications 98 to 102 g, mean 100.5 g and σ = 0.5 g. Cp = 4/3 = 1.33 and Cpk = min(1.5/1.5, 2.5/1.5) = 1.00. A centred normal process with Cp = 1 produces about 0.27% outside specifications (2,700 parts per million); with Cp = 1.33 about 63 ppm. Common customer requirements are Cpk ≥ 1.33 for existing processes and higher for critical or new ones, but requirements vary between industries and contracts.
Capability indices are meaningful only for a process in statistical control: an unstable process has no single capability. They also assume normality when converted into fractions non-conforming; for skewed characteristics (contaminant counts, moisture near a bound) percentile-based or transformed indices are used. σ should be estimated from within-subgroup variation (R̄/d₂ or s̄/c₄) for Cp and Cpk; indices based on the overall standard deviation, Pp and Ppk ('performance' indices), include between-subgroup variation and are smaller when the process drifts.
Estimated indices are random: with 50 observations a sample Cpk of 1.33 has an approximate 95% lower confidence bound near 1.10, so contracts should specify sample sizes or confidence bounds. The Taguchi-inspired Cpm index, (USL − LSL)/[6√(σ² + (μ − T)²)], penalises deviation from the target T, not only from the specification limits. In agricultural processing, capability studies guide decisions such as whether a grading machine can meet a size specification, whether a packing line needs adjustment to cut give-away (over-filling), and which supplier's process can meet a buyer's moisture or purity specification.
Capability analysis also quantifies give-away. If a 1-kg pack must not be under-weight beyond a legal tolerance, a filler with σ = 8 g must run its mean well above 1,000 g to keep under-weight packs rare; halving σ to 4 g allows the mean to be lowered by about 12 g (three times the reduction in σ) while keeping the same protection, saving 12 g of product per pack. Over millions of packs this saving usually pays for better filling equipment, which is why capability, not only conformance, is tracked in packing operations. Legal metrology rules for pre-packed goods differ between countries and specify how tolerances and averages are assessed.
Acceptance sampling decides whether to accept or reject a lot from a sample, without inspecting every item. A single sampling plan by attributes (n, c) draws n items at random from the lot, counts the non-conforming items d, and accepts the lot if d ≤ c (the acceptance number), rejecting it otherwise. For a lot fraction non-conforming p, the probability of acceptance is P_a(p) = P(d ≤ c), computed from the hypergeometric distribution for small lots, the binomial when the lot is large relative to n (n/N below about 0.1), and the Poisson approximation with mean np when p is small.
Example: a seed buyer uses the plan n = 50, c = 1 on lots from growers. If 2% of seeds in a lot are off-type, P_a = (0.98)⁵⁰ + 50(0.02)(0.98)⁴⁹ = 0.364 + 0.372 = 0.736 by the binomial; at 5% off-type, P_a = 0.279; at 10%, 0.034. Sampling plans are described by two points: the acceptable quality level (AQL), a quality that should be accepted with high probability 1 − α (α being the producer's risk, often 0.05), and the lot tolerance percent defective (LTPD, also called the rejectable or limiting quality), a quality that should be rejected with high probability 1 − β (β being the consumer's risk, often 0.10). Choosing n and c to meet both points is a two-point design problem solved with binomial or Poisson tables.
A double sampling plan takes a first sample of n₁ and accepts if d₁ ≤ c₁, rejects if d₁ > c₂ (or ≥ r₁), and otherwise takes a second sample of n₂, accepting if the total d₁ + d₂ ≤ c₂. Good and bad lots are often decided on the first sample, so the average sample number (ASN) is lower than for a single plan with the same OC curve, at the cost of administrative complexity and a variable workload. Multiple sampling extends the idea to several stages, and item-by-item sequential sampling, based on Wald's SPRT with parallel acceptance and rejection lines d = h₁ + sn and d = −h₂ + sn, gives the smallest expected sample size for given risks. The SPRT boundaries for p₁ (good) and p₂ (bad) quality are exactly those of the sequential pest-sampling plans in the hypothesis-testing manual.
In practice plans are taken from standards. ISO 2859-1 (derived from the former MIL-STD-105) indexes attribute plans by AQL, lot size and inspection level, with switching rules between normal, tightened and reduced inspection that reward good suppliers and protect against poor ones; its protection comes from the switching system, so using single plans without the switching rules weakens it. Dodge-Romig tables give plans that minimise average total inspection for a given LTPD or AOQL under rectifying inspection. National seed-testing and grain-grading standards specify their own sampling intensities and tolerances; they differ between countries and should be read from the current official texts.
The operating characteristic (OC) curve of a plan plots P_a(p) against the lot fraction non-conforming p. An ideal plan would accept every lot better than a chosen quality and reject every worse lot, a vertical step; real plans give an S-shaped curve. Increasing n with c/n fixed makes the curve steeper (more discriminating); increasing c with n fixed shifts it to the right (more lenient). Plans with c = 0 are popular because they look strict, but their OC curves are convex and drop sharply even at very low p, rejecting good lots often, so they are not as protective of producers as they seem. An OC curve should be read at both ends: the producer looks at P_a near the AQL, the consumer at P_a near the LTPD, and a plan is acceptable only if both risks are acceptable to both parties. Type A OC curves (for isolated lots, hypergeometric) and type B curves (for a continuing process, binomial or Poisson) differ when lots are small.
Under rectifying inspection, rejected lots are inspected 100% and non-conforming items replaced or removed, and in accepted lots the sampled non-conforming items are replaced. The average outgoing quality is then AOQ(p) = P_a(p) p (N − n)/N, approximately P_a(p) p when N is large relative to n. AOQ is low for very good incoming quality (little to remove) and for very bad quality (lots are rejected and screened), and reaches a maximum, the average outgoing quality limit (AOQL), at an intermediate p: the worst long-run average quality that can pass the scheme, whatever the incoming quality. For the plan n = 50, c = 1 with large lots, AOQ at p = 0.02 is 0.736 × 0.02 = 0.0147, and the AOQL is about 0.0167 at p ≈ 0.032.
The average total inspection per lot is ATI(p) = n + [1 − P_a(p)](N − n): the sample plus 100% inspection of the rejected fraction. With N = 2,000, n = 50 and P_a = 0.736, ATI = 50 + 0.264 × 1,950 = 565 items per lot. For double and sequential plans the average sample number ASN(p) replaces n: for a double plan, ASN = n₁ + n₂ × P(second sample needed). Curves of ATI and ASN against p show the inspection cost of a plan across incoming qualities and are used, alongside the OC curve, to choose among plans with similar protection.
A worked double plan: take n₁ = 30; accept if d₁ = 0, reject if d₁ ≥ 3, otherwise take n₂ = 30 and accept if d₁ + d₂ ≤ 2. At p = 0.02, P(d₁ = 0) = 0.545 and P(d₁ = 1 or 2) = 0.433, so the ASN is 30 + 30 × 0.433 = 43.0 items, against 60 if both samples were always taken; the overall probability of acceptance is 0.893. At very good or very bad quality the decision is usually made on the first sample and the ASN approaches 30.
When the quality characteristic is measurable and approximately normal, acceptance can be based on the sample mean and standard deviation instead of a count of non-conforming items. For an upper specification limit U and known σ, a single variables plan draws n items and accepts the lot if (U − x̄)/σ ≥ k, where the acceptability constant k and n are chosen to meet AQL and LTPD requirements. With σ unknown, s replaces σ and the plan accepts if (U − x̄)/s ≥ k, needing a larger n for the same protection; lower limits and two-sided specifications are handled similarly, or through an estimate of the fraction non-conforming (the M-method of ISO 3951).
Example: for moisture in grain with upper limit U = 14% and σ known to be 0.5%, a plan with n = 10 and k = 1.8 accepts a lot when x̄ ≤ 14 − 1.8 × 0.5 = 13.1%. A sample mean of 13.3% leads to rejection.
Variables plans need much smaller samples than attribute plans for the same OC curve, because each measurement carries more information than a pass/fail record; this matters when testing is destructive or expensive, as with germination tests, chemical residue analysis or tensile tests of packaging. Their limitations are that the normality assumption is critical, since the plan in effect estimates a tail probability, and a skewed or contaminated distribution can make it badly wrong; that each characteristic needs its own plan; and that a lot can be rejected even though no sampled item is outside specification, which suppliers can find hard to accept. Checking normality on historical data, or using attribute plans for characteristics that are clearly non-normal, is standard practice. As a rough guide, for the same OC curve a known-σ variables plan may need only a fraction of the attribute sample size, with the saving largest at low AQLs, where attribute plans need very large samples to observe any non-conforming items at all. ISO 3951 provides variables plans matched to the ISO 2859 attribute system.
Six Sigma is a management approach to quality improvement developed in manufacturing in the 1980s and widely adopted in services and agro-food businesses. Its statistical target is a process whose nearest specification limit is six standard deviations from the mean. By convention, allowing for a long-term drift of the mean by 1.5σ, such a process is said to produce 3.4 defects per million opportunities (the tail beyond 4.5σ); without the shift the rate would be about 0.002 per million. The 1.5σ shift is a convention, not a law, and has been criticised, but the 3.4 DPMO figure is widely quoted.
Projects follow the DMAIC cycle: Define the problem and the customer requirement, Measure the current performance (including measurement-system analysis, since a gauge with large variation can hide or invent process changes), Analyse the causes of variation using data, Improve the process (often by designed experiments), and Control the improved process with control charts and procedures. The toolkit is largely the classical one of this and the design-of-experiments subjects: Pareto charts, cause-and-effect diagrams, capability analysis, regression, ANOVA, factorial and response-surface designs, and control charts.
Applied in agro-food firms, Six Sigma projects typically target measurable losses: reducing broken grains in rice milling, cutting rejection of export consignments for residues or size grading, lowering milk spoilage in the collection chain, or reducing variation in feed composition. The statistical content is modest; the value lies in insisting on measured baselines, verified causes and sustained control rather than one-off fixes.
Genichi Taguchi emphasised that any deviation from target, not only falling outside specification, causes a loss, often modelled by the quadratic loss function L(y) = k(y − T)². The expected loss is k[σ² + (μ − T)²], so reducing variation and centring the process both reduce loss even within specifications; the Cpm index embodies the same idea. Taguchi's robust design seeks settings of controllable factors that make performance insensitive to noise factors (raw-material variation, ambient temperature), using designed experiments with inner (control) and outer (noise) arrays and signal-to-noise ratios. Statisticians have criticised some of his analysis methods and recommend response-surface and combined-array approaches, but the engineering principle of robustness is widely accepted.
In agro-processing, robust design appears in choosing drying or pasteurisation conditions that give consistent quality despite variable raw material, and in packaging that protects produce under variable storage conditions. Quality by design in pharmaceuticals and veterinary products formalises the same approach. Designed experiments for these purposes are treated in the experimental-design and response-surface subjects.
Measurement matters before any of this. If a moisture meter's own variation is large, a process can appear out of control when it is not, or appear capable when it is not. A gauge repeatability and reproducibility study partitions observed variance into process variance and measurement variance (repeatability within an operator and reproducibility between operators) using a crossed random-effects design (see the mixed-models subject); a common rule is that measurement variation should be a small fraction of the specification width, but thresholds differ between guidelines.
Reliability is the probability that an item performs its function without failure for a stated time under stated conditions: R(t) = P(T > t), the survival function of the bioassay-survival subject. The hazard (failure) rate h(t) = f(t)/R(t) is often bathtub-shaped over an item's life: decreasing during early life (infant mortality from manufacturing defects), roughly constant during useful life (random failures) and increasing during wear-out. The exponential distribution, with constant hazard λ, describes the useful-life phase: R(t) = e^(−λt) and the mean time to failure (MTTF) is 1/λ. The Weibull distribution covers all three phases through its shape parameter.
Systems are analysed from their components. In a series system every component must work: R_s = ΠRᵢ, and for exponential components the system hazard is the sum of the component hazards, so MTTF_s = 1/Σλᵢ. In a parallel (redundant) system at least one component must work: R_p = 1 − Π(1 − Rᵢ). A k-out-of-n system works if at least k of n identical components work, with reliability given by a binomial tail. Example: a drip-irrigation controller has a pump (R = 0.95 over a season), a valve (0.98) and a timer (0.99) in series: R = 0.95 × 0.98 × 0.99 = 0.922. Adding a standby pump in parallel raises the pump stage to 1 − 0.05² = 0.9975 and the system to 0.968. Redundancy is most effective at the weakest stage.
Life tests estimate reliability from items run until failure or until the test ends. Because waiting for all failures is slow, tests are usually censored: Type I tests stop at a fixed time and Type II tests at a fixed number of failures, the censoring schemes of survival analysis. For exponential lifetimes the maximum-likelihood estimate of the MTTF is the total time on test divided by the number of failures, θ̂ = T_total/r, where T_total adds the running times of failed and surviving items; under Type II censoring, 2T_total/θ has a χ² distribution with 2r degrees of freedom, giving exact confidence limits. Example: 20 sprayer nozzles are tested for 500 hours; 4 fail at 120, 250, 310 and 430 hours and 16 survive. T_total = 1,110 + 16 × 500 = 9,110 hours and θ̂ = 9,110/4 = 2,278 hours.
Accelerated life tests run items at higher stress (temperature, pressure, voltage) to obtain failures quickly and extrapolate to normal conditions through a model such as the Arrhenius relation for temperature; the extrapolation is the weak point. Reliability demonstration tests are acceptance-sampling plans for lifetimes: accept the design if no more than c failures occur in a stated total test time. Weibull analysis of field failure data (with probability plotting or maximum likelihood, allowing for censoring) guides warranty and maintenance policies for tractors, pumps and processing equipment, linking to the renewal and replacement models of the stochastic-processes subject.
Two further worked results. A 2-out-of-3 system of identical sensors, each with reliability 0.9, works with probability 3(0.9)²(0.1) + 0.9³ = 0.972, which is why voting arrangements are used for critical measurements such as storage temperature alarms. For a Weibull lifetime with shape 2 and characteristic life 1,000 hours, R(500) = exp[−(500/1,000)²] = 0.779: because the hazard increases, the reliability over the first half of the characteristic life is much higher than an exponential model with the same characteristic life would give (e^(−0.5) = 0.607).
A student needs to compute X̄, R, s, p, np, c and u chart limits from Phase I data, interpret out-of-control signals and run rules, compute tabular CUSUM and EWMA statistics, calculate Cp and Cpk, compute P_a, AOQ, AOQL and ATI for single plans and ASN for double plans, design a variables plan, and calculate series and parallel system reliabilities and exponential MTTF estimates from censored life tests. ISS papers and JSO tests ask exactly these.
A statistician five years into work at a seed company, dairy or food processor sets up and maintains control charts for weights, moisture, fat and microbial counts, chooses CUSUM or EWMA for slow drifts, runs capability studies for buyers, designs incoming-inspection plans for grower lots and packaging, interprets national and international standards correctly (including switching rules), and analyses equipment failure data for maintenance planning.
A senior statistician with twenty-five years of experience designs quality systems for an organisation or a regulator: sampling and tolerance rules in seed certification and grain grading, monitoring programmes for residues and contaminants, supplier-quality agreements based on capability and AOQL, and reliability requirements for procured machinery. Such a statistician audits how standards are applied in practice and explains to managers the risks each plan accepts.
A PhD with long experience develops monitoring and sampling methods: charts for autocorrelated and multivariate process data, risk-adjusted and Bayesian monitoring, acceptance sampling for heterogeneous lots with composite samples (as in mycotoxin testing), reliability models for repairable agricultural systems, and the statistical properties of standards' sampling schemes. Such a researcher proves properties, checks them by simulation and supervises doctoral work.
Tables of control-chart constants, sampling plans and standards are not reproduced beyond the values used in examples; the constants quoted are standard for normal data. Not covered in depth: multivariate control charts (Hotelling T² charts), charts for autocorrelated data, adaptive and Bayesian charts, measurement-system analysis in detail, design of Six Sigma programmes, accelerated-life-test modelling and repairable-system reliability. National seed-certification, grain-grading and food standards differ by country and edition and are not quoted. Statistical tables are not reproduced; questions state the values they need.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions. Every numerical answer was computed with a script; the values needed are stated in the question.
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