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**Statistical Process Control**

Key points A process may in control (capable) or out of control (not capable). All processes have some natural variation 3. The data used for SPC are the mean and standard deviation of the process (found by sampling). 4. The central limit theorem is the underlying theory. 5. SPC only applies if the process samples show a bell curve distribution.

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Process Control (a) In statistical control and capable of producing within control limits Frequency Lower control limit Upper control limit (b) In statistical control but not capable of producing within control limits This slide helps introduce different process outputs. It can also be used to illustrate natural and assignable variation. (c) Out of control (weight, length, speed, etc.) Size Figure S6.2

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**Statistical Process Control**

A process control system is set up to monitor a process and signal when assignable (or real) causes of variation are present. (This means there is a cause for the variation that can be fixed)

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**Statistical Process Control (SPC)**

Measures the variability found in every process, from Natural or common causes Special or assignable causes SPC gives a statistical signal when assignable causes are present Quality control work is to detect and eliminate assignable causes of variation Points which might be emphasized include: - Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance. - Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units - While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later.

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**Natural Variations Also called common causes**

Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

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**Natural variation Data range UCL mean LCL Time / number of samples**

Resulting normal distribution UCL mean LCL Time / number of samples

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Reading SPC charts

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**Control Charts for Variables**

Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables x-chart tracks changes in the central tendency (mean) R-chart indicates a change in spread (range) Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process. These two charts must be used together

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**Mean and Range Charts (a)**

These sampling distributions result in the charts below (Sampling mean is shifting upward, but range is consistent) x-chart (x-chart detects shift in central tendency) UCL LCL R-chart (R-chart does not detect change in mean) UCL LCL Figure S6.5

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**Mean and Range Charts (b)**

These sampling distributions result in the charts below (Sampling mean is constant, but dispersion is increasing) x-chart (x-chart indicates no change in central tendency) UCL LCL R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Normal behavior. Process is “in control.” Figure S6.7

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

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**Patterns in Control Charts**

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Erratic behavior. Investigate. Figure S6.7

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**Each square represents one sample of five boxes of cereal**

Samples To measure the process, we take output samples and analyze the sample statistics following these steps Each square represents one sample of five boxes of cereal (a) Samples of the product, (boxes of cereal) taken off the filling machine line, vary in weight Frequency Weight # Figure S6.1

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**The solid line represents the distribution**

Samples To measure the process, we take output samples and analyze the sample statistics following these steps The solid line represents the distribution Frequency Weight (b) After enough samples are taken from a stable process, they form a pattern called a distribution Figure S6.1

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Samples To measure the process, we take output samples and analyze the sample statistics following these steps (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Figure S6.1 Weight Central tendency Variation Shape Frequency

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Samples To measure the process, we take samples and analyze the sample statistics following these steps (d) If only natural causes of variation are present, the output of a process forms a stable distribution that is predictable Prediction Weight Time Frequency Figure S6.1

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Samples To measure the process, we take samples and analyze the sample statistics following these steps Prediction ? Weight Time Frequency (e) If assignable causes are present, the process output is not stable and is not predicable Figure S6.1

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**Central Limit Theorem (theoretical basis for SPC)**

In any population, the distribution of means of samples from the population will follow a normal curve The mean of the sampling distribution will be the same as the population mean m The standard deviation of the sampling distribution ( ) will equal the population standard deviation (s ) divided by the square root of the sample size, n This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues.

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**Sampling Distribution**

Figure S6.4 Sampling distribution of means Process distribution of means = m (mean) It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population.

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**Population and Sampling Distributions**

Population distributions Beta Normal Uniform Distribution of sample means Standard deviation of the sample means Mean of sample means = | | | | | | | 99.73% of all fall within ± 95.45% fall within ± Figure S6.3

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**Central Limit Theorem (theoretical basis for SPC)**

So for any population (process) that we have sampled, and taken the means of the samples, we will create a normal distribution. This allows us to use properties of the normal distribution to understand our samples (and therefore the population (process)). The key feature to use is the standard deviation of a normal population

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Central Limit Theorem As the properties of normal distributions are known we can say that a process with only natural variation will have results that show 68.26% of the process inside ± 1 std dev of the mean 95.45% of the process inside ± 2 std dev of the mean 99.73 % of the process inside ± 3 std dev of the mean (Turn it around) If a result is outside the ± 3 std dev limit we can be 99.73% sure that there is a special cause at work – and we have a problem.

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**Central Limit Theorem (as sure as 1+1 = 2)**

Z = 1 Z = 2 Z = 3 ( z = the number of std dev allowed. (see table s6.2, page 229)

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Key Points Sampling any population using small samples and taking the mean of each sample. The population of ‘means-of-samples’ will be a bell curve. This allows us to use central limit theorem rules. Which is what the SPC charts are based on using mean and std deviation to define the process.

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**Setting Chart Limits For x-Charts when we know std dev(s)**

Where = mean of the sample means or a target value set for the process z = number of normal standard deviations sx = standard deviation of the sample means s = population (process) standard deviation n = sample size

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**Setting Control Limits**

Randomly select and weigh nine (n = 9) boxes each hour Average weight in the first sample WEIGHT OF SAMPLE HOUR (AVG. OF 9 BOXES) 1 16.1 5 16.5 9 16.3 2 16.8 6 16.4 10 14.8 3 15.5 7 15.2 11 14.2 4 8 12 17.3

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**Setting Control Limits**

Average mean of 12 samples

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**Setting Control Limits**

Average mean of 12 samples

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**Setting Control Limits**

Control Chart for samples of 9 boxes Variation due to assignable causes Out of control Sample number | | | | | | | | | | | | 17 = UCL 15 = LCL 16 = Mean Variation due to natural causes Out of control

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**Setting Chart Limits For x-Charts when we don’t know std dev(s)**

where average range of the samples A2 = control chart factor found in Table S6.1 = mean of the sample means

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**Control Chart Factors TABLE S6.1**

Factors for Computing Control Chart Limits (3 sigma) SAMPLE SIZE, n MEAN FACTOR, A2 UPPER RANGE, D4 LOWER RANGE, D3 2 1.880 3.268 3 1.023 2.574 4 .729 2.282 5 .577 2.115 6 .483 2.004 7 .419 1.924 0.076 8 .373 1.864 0.136 9 .337 1.816 0.184 10 .308 1.777 0.223 12 .266 1.716 0.284

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**Setting Control Limits**

Super Cola Example Labeled as “net weight 12 ounces” Process average = 12 ounces Average range = .25 ounce Sample size = 5 From Table S6.1 UCL = Mean = 12 LCL =

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**R – Chart Type of variables control chart**

Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean

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Setting Chart Limits For R-Charts where

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**Setting Control Limits**

Average range = 5.3 pounds Sample size = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

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**Steps In Creating Control Charts**

Collect 20 to 25 samples, often of n = 4 or n = 5 observations each, from a stable process and compute the mean and range of each Compute the overall means ( and ), set appropriate control limits, usually at the 99.73% level, and calculate the preliminary upper and lower control limits If the process is not currently stable and in control, use the desired mean, m, instead of to calculate limits.

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**Steps In Creating Control Charts**

Graph the sample means and ranges on their respective control charts and determine whether they fall outside the acceptable limits Investigate points or patterns that indicate the process is out of control – try to assign causes for the variation, address the causes, and then resume the process Collect additional samples and, if necessary, revalidate the control limits using the new data

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