In previous classes we mentioned statistics on two occasions: when we discussed control charts among the basic quality tools and when we discussed the beginnings of quality. Statistical process control (SPC) was born with Walter Shewhart, at Bell Laboratories, in 1924, when he proposed the first control chart. The big shift in thinking was this: instead of separating good parts from bad ones after they are finished, as traditional inspection did, SPC follows the process while it is running, based on samples, so that the problem is corrected before it produces bad parts.Objectives of statistical process controlOriginally, SPC aimed to monitor a product or service during its production process: if the process showed problems, it would be stopped so the failures could be fixed and everything could return to normal.Today, more than a statistical tool, SPC is seen as a management philosophy and a set of techniques and skills, drawn from statistics and industrial engineering, aimed at ensuring the stability and continuous improvement of a process.So the main purpose of SPC is to control and improve the process, and its fundamental principles are:think and decide based on data and facts;separate cause from effect, always looking for the root cause of problems;recognize that every production process has variability, and manage it;quickly identify where failures are and correct problems in time.SPC can be applied to any process, in manufacturing or in services. For that, it relies on a set of tools: histograms, check sheets, Pareto charts, cause-and-effect (Ishikawa) diagrams and, above all, control charts.What is the main difference between SPC and traditional final inspection?Answer: SPC follows the process during production in order to act before defective parts appear — final inspection sorts out what has already come out bad; SPC uses samples throughout the process to prevent it. It also applies to services, not only to factories.Control chartA control chart is a record, in chronological order, of a characteristic calculated from samples taken from production, such as the average of a batch of parts. The values are plotted on the chart, which has a center line and two limits, called control limits: the upper control limit (UCL) and the lower control limit (LCL).In the chart in the image (labels in Portuguese), each point is the average of one sample. The center line (Linha Central) is at 74.0012, the UCL (Limite Cont. Superior) at 74.0143 and the LCL (Limite Cont. Inferior) at 73.9881. Samples 14 and 18 fell outside the limits, and samples 3 to 9, seven consecutive points above the center line, were also flagged as a sign of a special cause (Causas Especiais), as we will see in phase IV.All variation in a process comes from common causes or from special causes.Common causes of variationThese are the small random variations that are inherent to the process, have no single identifiable cause and cannot be removed without changing the process itself. When only common causes are present, the process is stable (in control) and its results vary in a predictable way, often roughly following a normal distribution. For example, the weight of rice packages from a packing machine varies a little from package to package; if this variation comes only from common causes, the sample averages will stay within the control limits, with no unusual patterns.Special causes of variationThese happen for identifiable reasons that can be eliminated. A special cause changes the parameters of the process (the mean, the standard deviation or both), and that is why points fall outside the control limits or form abnormal patterns. For example, a piece of equipment with incorrect calibration causes special variation; once it is adjusted, that variation disappears.In the chart in the image, samples 3 to 9 are all above the center line but within the limits. Why were they flagged as a sign of a special cause?Answer: Because seven consecutive points on the same side is very unlikely with only common causes, about 0.78% — in a stable process each point has a 50% chance of being above the mean: 0.5 to the power of 7 = 0.0078. This suggests the process mean has probably shifted, even with no point outside the limits.How to use the control chartWhat is a control chart for? To reveal special causes of variation and allow action on them when needed, avoiding losses in the production process. Setting up a chart follows five phases:Choose the type of control chart to use.Establish a plan for taking samples from what is being produced.Calculate the center line and control limits for each characteristic being controlled.Plot the values on the chart and check whether the process is stable, that is, in control.Analyze the results and decide whether any action is needed.Phase I – Choose the type of control chartThere are two basic types of control charts: for variables and for attributes.Control charts for variablesThese are used when the characteristic can be measured on a continuous scale (weight, height, length, temperature, etc.). The best known are:X̄ and R (mean and range): the most widely used, for small samples (usually 2 to 10 units). The two charts complement each other and must be used together: the X̄ chart tracks the average level of the process and any change in it; the R chart tracks the spread, which can also increase because of special causes, such as a poorly adjusted machine.X̄ and s (mean and standard deviation): an alternative to X̄ and R, preferred for larger samples (more than 10 units), where the standard deviation measures spread better than the range.I and MR (individual values and moving range): used when it makes more sense to control the process with individual measurements than with samples. This happens when inspection is expensive, the test is destructive, production is slow or the characteristic is homogeneous (such as the pH of a chemical solution). The moving range is the difference between two consecutive measurements.Control charts for attributesThese are used when the quality characteristic is not measured numerically but counted or classified. For example, a light bulb is classified as "works" or "does not work"; in fabric, the defects per meter are counted. The best known are:p chart: controls the proportion of defective units in each sample;np chart: controls the number of defective units per sample (samples of constant size);c chart: controls the number of defects per sample (for example, scratches on a sheet);u chart: controls the number of defects per unit of product, useful when the sample size varies.Note the difference between defective and defect: one defective part can have several defects. The p and np charts count defective parts; the c and u charts count defects.Phase II – Establish the sampling planBefore collecting data, you need to choose the sample size (the samples of a control chart are called rational subgroups), the sampling frequency and the number of samples to collect.The rule for rational subgroups is: each subgroup should be as homogeneous as possible inside, so that differences show up between one subgroup and another. In practice, the units in a subgroup are produced almost at the same time, on the same machine, by the same operator and from the same batch of raw material. That way, if a special cause appears, it shows up as a difference between subgroups.This is different from an opinion poll. In a poll of voting intentions for the governor of the state of Rio de Janeiro, you would mix people from several neighborhoods and age groups in the same sample, so that it represents the whole population. In SPC, mixing parts from two machines or two shifts in the same subgroup would hide exactly the difference we want to detect. What you do is take subgroups from all machines and shifts over time, but each subgroup from a single source, or keep a separate chart for each machine.Phase III – Calculate the control limitsThe idea is similar to identifying outliers. We have a class on what outlier means and how to calculate it.A point outside the limits is treated as something that would hardly happen if the process had only common causes. According to the normal distribution rule, 99.73% of values lie within 3 standard deviations of the mean, so only 0.27% lie beyond (0.135% on each side).That is why Shewhart's limits are: center line equal to the mean of the statistic plotted on the chart; upper limit equal to the mean plus 3 standard deviations of that statistic; lower limit equal to the mean minus 3 standard deviations. The key point is that you use the standard deviation of the plotted statistic (the sample mean, the proportion, etc.), not necessarily that of a single part.X̄ and R chartsIn practice, the limits of the X̄ and R charts are calculated from the grand mean of the samples (x̿) and the average range (R̄), with tabulated constants that depend on the size n of each sample:\[ \text{UCL}_{\overline{x}} = \overline{\overline{x}} + A_2 \cdot \overline{R} \qquad \text{LCL}_{\overline{x}} = \overline{\overline{x}} - A_2 \cdot \overline{R} \]\[ \text{UCL}_{R} = D_4 \cdot \overline{R} \qquad \text{LCL}_{R} = D_3 \cdot \overline{R} \]nA2D3D4d221.88003.2671.12831.02302.5741.69340.72902.2822.05950.57702.1142.32660.48302.0042.53470.4190.0761.9242.70480.3730.1361.8642.84790.3370.1841.8162.970100.3080.2231.7773.078The constant d2 is used to estimate the process standard deviation from the average range: σ ≈ R̄ / d2. A2 already includes the 3 standard deviations of the mean: A2 = 3 / (d2 · √n).Example: a machine fills packages, and every hour we take 5 consecutive packages. After 25 samples, the grand mean is 500 g and the average range is 10 g. With n = 5:\[ \text{UCL}_{\overline{x}} = 500 + 0.577 \cdot 10 = 505.77 \text{ g} \qquad \text{LCL}_{\overline{x}} = 500 - 0.577 \cdot 10 = 494.23 \text{ g} \]\[ \text{UCL}_{R} = 2.114 \cdot 10 = 21.14 \text{ g} \qquad \text{LCL}_{R} = 0 \cdot 10 = 0 \]And the estimated standard deviation of each package is 10 / 2.326 ≈ 4.30 g.np chart: number of defectives in the sampleWhen the proportion p of defective parts in the process is known and samples of n units are taken, the number of defectives in the sample follows the binomial distribution. Its mean is n · p and its standard deviation is:\[ σ = \sqrt{np(1-p)} \]For example, if the process produces 10% defective parts and each sample has 100 units, on average we will find 10% of 100, that is, 10 defective parts per sample. The standard deviation will be:\[ σ = \sqrt{100 \cdot 0.1 \cdot (1 - 0.1)} \]\[ σ = \sqrt{10 \cdot 0.9} \]\[ σ = \sqrt{9} \]\[ σ = 3 \]So the control chart limits for this sample are:\[ \text{Upper limit} = n p + 3 \cdot σ \]\[ \text{Upper limit} = 10 + 3 \cdot 3 \]\[ \text{Upper limit} = 19 \]This means defective parts are already expected in every sample. We will only really worry about the equipment if a sample of 100 has more than 19 defective parts.\[ \text{Lower limit} = n p - 3 \cdot σ \]\[ \text{Lower limit} = 10 - 3 \cdot 3 \]\[ \text{Lower limit} = 1 \]The same reasoning applies to the lower limit: we will only be suspicious if a sample has fewer than 1 defective part, that is, none.Both situations (more than 19 defectives, or none) are so rare when the process is stable that they indicate a special cause. Above the UCL, the machine is probably out of adjustment. Below the LCL, the news may be good (the process has improved, and you need to find out why in order to keep the improvement) or it may be an inspection failure (someone failed to record the defects). In both cases, you investigate. When the lower limit calculation gives a negative number, zero is used.p chart: proportion of defectivesIf, instead of the number, we want to track the proportion of defective parts in each sample, the standard deviation changes: we divide the np chart's standard deviation by n, which gives:\[ σ = \sqrt{\frac{p(1-p)}{n}} \]Note that the larger the sample, the smaller this standard deviation: the observed proportion becomes more and more concentrated around p, and the central limit theorem allows us to treat it as approximately normal.Recalculating the previous example (p = 0.10 and n = 100):\[ σ = \sqrt{\frac{0.1 \cdot 0.9}{100}} = 0.03 \]\[ \text{Upper limit} = 0.10 + 3 \cdot 0.03 = 0.19 \qquad \text{Lower limit} = 0.10 - 3 \cdot 0.03 = 0.01 \]That is, 19% and 1%: the same limits as the np chart divided by 100, as expected.c and u charts: number of defectsIn defect charts, the count approximately follows the Poisson distribution, in which the variance equals the mean. So, with an average of c̄ defects per sample:\[ \text{UCL} = \overline{c} + 3\sqrt{\overline{c}} \qquad \text{LCL} = \overline{c} - 3\sqrt{\overline{c}} \]If a sheet has 4 scratches on average, the UCL is 4 + 3 · 2 = 10 scratches, and the LCL would be 4 − 6 = −2, so zero is used. In the u chart, with ū defects per unit and samples of n units, the limits are ū ± 3 · √(ū / n).Phase IV – Plot the points and check stabilityStability concerns how the process variation behaves: whether it shows only common causes or whether there are signs of special causes.A process is considered stable when its control charts show no signs of abnormality. Processes in control have "well-behaved" charts that follow a known pattern.To check stability, charts are always analyzed in pairs (mean and range, or mean and standard deviation), looking for signs of abnormality.The characteristics of a natural, stable pattern on a control chart are:most points are close to the center line (about 68% within ±1σ of it), without, however, an excessive concentration in that band;about 95% of the points (19 out of 20) are within ±2σ of the center line;no point falls outside the control limits (the probability of that, with a stable process, is only 0.27% per point);the points are spread more or less evenly above and below the center line;there are no long runs on the same side and no upward or downward trends. For example, the probability of seven consecutive points above the mean is 0.5 to the power of 7, or 0.78%; when this happens, it is interpreted as an upward shift of the mean. Some rule sets use 8 or 9 consecutive points to reduce false alarms;there are no cyclical oscillations.Phase V – Analyze and actOnce a signal is found, the team investigates the special cause (machine, material, method, manpower, measurement, environment), corrects it and records what was done. Points attributed to a special cause that has been removed can be dropped and the limits recalculated. If there is no signal, the process should not be adjusted: tweaking a stable process at every small variation only increases the variation.An X̄ and R chart uses samples of 4 parts. The grand mean is 20.0 mm and the average range is 0.4 mm. What are the limits of the X̄ chart?Answer: 19.71 mm and 20.29 mm — with n = 4, A2 = 0.729: 20.0 ± 0.729 x 0.4 = 20.0 ± 0.29. The option 19.08 and 20.91 used D4 (2.282 x 0.4 = 0.91), which is the constant for the upper limit of the R chart, not the X̄ chart.A process has 4% defective parts and samples have 200 units. What are the limits of the np chart?Answer: UCL of about 16.3 and LCL equal to zero — mean np = 200 x 0.04 = 8; σ = square root of 8 x 0.96 = 2.77; 8 + 3 x 2.77 = 16.3. The calculated LCL is 8 minus 8.3, a negative value, and since a count cannot be negative, zero is used.ConclusionProcess control lets us take corrective action before deviations appear that cause rework, scrap or other losses.Always keep in mind that there is no way to achieve higher productivity, lower costs and, consequently, higher profit margins without having production processes in control, that is, free of special causes. If they exist, it will be up to you, a future industrial engineer, to apply your knowledge to eliminate them.And remember that there do not need to be points outside the control limits for you to act on the process: a process trending toward the upper or lower limit already shows that it needs intervention. When we see a car stopped ahead, we do not wait for the collision to take action: we brake or swerve to avoid the impact. The action was taken before the event actually became a problem.Exercises1) In the manufacture of toilet paper, each roll must be 30 meters long. The cutting machine operates with an average of 30 meters, but from time to time it makes a roll a little longer or a little shorter. Each roll is measured individually, and the standard deviation of the length, estimated from 50 rolls, is 30 centimeters. Calculate the upper and lower limits. For which of the following values should you stop production to check the machine: 30.63 meters; 29.85 meters; 29.06 meters; 30.89 meters?2) There is a 5% probability that a product is defective. You take samples of 100 units to check the condition of the machinery. What are the upper and lower limits of the np chart?
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