Control - history

Control Chart • Developed in 1920’s

• By Dr. Walter A. Shewhart

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Process in control

A production process is said to be in control when the “A phenomenon is said to be controlled characteristics of a product are subject only to when, through the use of past experience, we random variation (or common cause variation) that is can predict, at least within limits, how the variation in process phenomenon may be expected to vary in the performance due to normal future.” or inherent among process components Walter A. Shewhart, 1931 (people, machine, material, environment, and methods).

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Sara Gradara 1 Process out-of-control

A production process is said to be out-of-control when the quality characteristic of a product are subject also to variation due to assignable causes that is variation in process performance Control are useful to establish when a due to events that are process has endured a meaningful not part of the normal modification; control charts separate the two process and represents types of variation in a product quality sudden or persistent characteristic. abnormal changes to one or more of the process components.

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Control Chart Control Chart

All control charts have three basic components:

• a centerline that represents the value for the in- control process

• two horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL) that define the limits of common variation causes. Common Cause In the process (normal noise) • performance plotted over time. Special Cause Outside the process (extraordinary)

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Sara Gradara 2 Control Chart Control Chart

Control charts let you know Control charts help to separate what your processes can do: signal from noise, so that you you can set achievable goals. can recognize a process change when it occurs.

They represent the “voice of the process”. Control charts identify unusual events. They pinpoint fixable problems and potential process Control charts provide the evidence of stability improvements. that justifies predicting process performance.

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Control Chart - assumptions Control Chart – 3 sigma

Control limits on a control chart are commonly drawn at 3- The two important assumptions are: sigma from the center line because 3-sigma limits are a good balance point between two types of errors: • The measurement-function (e.g. the mean), that is used to monitor the process parameter, follows a •Type I or alpha errors occur when a . In practice, if your data seem point falls outside the very far from meeting this assumption, try to even though no special cause is transform them. operating. • Measurements are independent of each other. •Type II or beta errors occur when you miss a special cause because the chart isn't sensitive enough to detect it.

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Sara Gradara 3 Control Chart – 3 sigma Instabilities and Out-of-Control Situations

All process control is vulnerable to these two types of errors. The reason that 3-sigma control limits balance the risk of error To test for instabilities in processes, we examine control is that, for normally distributed charts for instances and patterns that signal non-random data, data points will fall inside 3- behavior. sigma limits 99.73% of the time when a process is in control. Values falling outside the control limits and unusual patterns within the running record suggest that assignable causes exist. The limits are chosen so that it is 99.73% likely that unusual causes of "in control" implies that all points are between the control variation will be detected. limits and they form a random pattern. -4 -3 -2 -1 0 1 2 3 4

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Instabilities and Out-of-Control Situations Instabilities and Out-of-Control Situations

Test 1: A single point falls outside the 3-sigma control Test 3: At least four out of five successive values fall limits. on the same side of, and more than one sigma unit away from, the center line. Test 2: At least two of three successive values fall on the same side of, and more than two sigma units away Test 4: At least eight successive values fall on the same from, the center line. side of the center line.

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Sara Gradara 4 Instabilities and Out-of-Control Situations Control Chart

Is there a Best Control Chart?

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Variable and attribute data Variable and attribute data

Two broad classes of control charts: It is important to understand the distinction between variables data and attributes data •variable data, which is continuous •attribute data, which is discrete Because control limits for attributes data are often Choice of what control chart to use should be based computed in ways quite different from control limits for on knowing the right assumptions! variables data. Unless you have a clear understanding of the distinctions Use the correct formulas for the kind of control chart between the two kinds of data, you can easily fall victim selected! to inappropriate control charting methods.

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Sara Gradara 5 Variable and attribute data Variable and attribute data

Attributes data occur when is recorded Variables data (sometimes called measurement data) are only about whether an item conforms or fails to conform usually measurements of continuous phenomena. to a specified criterion or set of criteria. Attributes data almost always originate as counts. Examples: measurements of length, weight, volume and speed. Examples: the number of defects found, the number of source statements of a given type, the number of lines of Software examples: elapsed time, effort expended, years of comments in a module of n lines, the number of people experience, memory utilization and cost of rework. with certain skills or experience on a project or team, and the percent of projects using formal code inspections.

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Control Chart selection Types of Variable Control Chart

• X- • R chart • s chart • Individual chart • Moving chart

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Sara Gradara 6 Types of Variable Control Chart Types of Variable Control Chart

• X-bar chart: based on the average of a subgroup. • Individual chart: displays each value. A subgroup size is Subgroups of 2 to 30 samples may be used when used to compute the limits, with value of 2 being most common, computing the control limits for the X-bar chart when although the subgroup size may be as large as 30. based on the range. • Moving Range chart: takes into account the moving range of • R chart: takes into account the range of a subgroup. a process. It is used to control variability of processes which do Subgroup sizes may be as small as 2 or as large as 30. not form natural subgroups.

• S chart: takes into account the of a subgroup. There is no limit to the subgroup size.

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Notation for Variable Control charts Notation and Values

n: size of the sample (collection of observations, Ri = range of the values in the i-th sample sometimes called a subgroup) chosen at a point in time Ri = max(X i ) − min(X i ) m: number of samples selected R= average range for all m samples

xi = average of the observations in the i-th sample µ is the true process mean, usually unknown but it can be (where i = 1, 2, ..., m) estimated by averaging a large number (for example 20) of samples mean obtained when the process is in control x = grand average or “average of the averages (this value is used as the center line of the control chart) σ is the true process standard deviation, usually unknown but it can be estimated from a large sample of data 27 collected while the process is in control 28

Sara Gradara 7 X-bar charts X-bar charts

• Let X = { X 1 ,...... , X m } be the set of observations

divided into samples. “The process mean is changed during •For each sample X = { x ,...... , x } compute the average the observation period?” i i1 in x + x + + x X = i1 i2 L in i n and the mean average X + X + + X X = 1 2 L m m

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X-bar charts R charts

X i is normally distributed with mean, µ, and standard deviation, . () “Is the dispersion of the values observed in the σx =σ/ n samples due to the presence of exceptional Lower Control Limit: X −3σx ≅ X − A2R causes?” Center Line: X

Upper Control Limit: X +3σx ≅ X + A2R

A2 is a constant based on the subgroup size. 31 32

Sara Gradara 8 R charts R charts

• Let be the set of observation X = {X1,...... , X m}

divided into samples. Lower Control Limit: R − 3σR ≅ D3R

Center Line: R •For each sample X i the range is

Upper Control Limit: R +3σ ≅ D R Ri = max(X i ) − min(X i ) R 4

and the range average is

D3 and D 4 are constants based on the subgroup size. R + R + + R R = 1 2 L m m 33 34

X-bar charts and R charts for a process out-of-control X-bar charts and R charts

X-bar chart is typically used in conjunction with R chart. In fact, since the sample range is used to construct the X-bar chart, it is essential to examine an R chart first (to be sure that the process variation is stable).

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Sara Gradara 9 X-bar charts and R charts X-bar charts and R charts: example

As an example, suppose you misure those 4 software module It is important to construct and interpret an sizes each month for 6 months. Our example collected data R chart before the X-bar chart. looks like the following:

When the R chart indicates that process variation is in control, analyze the X-bar chart otherwise X-bar chart are not meaningful

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X-bar charts and R charts: example s charts

Process variability can be controlled by either a R chart or a Standard Deviation chart (s chart) depending on how the population standard deviation is estimated.

S chart is used to determine whether the standard deviation has changed.

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Sara Gradara 10 s charts X-bar charts and s charts

• Let X = { X ,...... , X } be the set of observations 1 m Lower Control Limit: X −3σx ≅ X − A3S divided into samples. Paremeters for X-bar Center Line: X char { •For each sample X i the standard deviation is n Upper Control Limit: X +3σx ≅ X + A3S 2 ∑(xij − xi ) 2 j=1 Si = Lower Control Limit: S −3σ ≅ B S n −1 S 3 and the average is Paremeters for s char Center Line: S 1 m { S = S ∑ i Upper Control Limit: S +3σ ≅ B S m i=1 S 4 41 42

XmR: Individual and Moving Range charts XmR chart

An individual chart is equivalent to X-bar but reported to single observation, not to samples. The solution is to artificially create subgroups from the data and then calculate the range of Used when the nature of the process is such that it is each subgroup. difficult or impossible to group measurements into subgroups This is done by creating rolling groups (most often pairs) of data through time and using the This occurs frequently in low volume production and pairs to determine the range R. in situations in which continuously varying quantities within the process are process-related variables. The resulting ranges are called moving ranges.

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Sara Gradara 11 XmR chart XmR chart

Lower Control Limit: X −3σx ≅ X −E2MR Paremeters The moving range is defined as MR = x −x for i i i−1 Individual Center Line: X char { which is the absolute value of the delta between Upper Control Limit: X +3σx ≅ X +E2MR two consecutive data points. Lower Control Limit: max [0 , M R − 3 σ ] = max [0,D MR] Paremeters MR 3 for Moving Center Line: MR Range char { Upper Control Limit: MR +3σMR ≅ D4MR 45 46

XmR chart: example XmR chart: example

The following table contains a set of sample data, for the KLOC generated each month for one software module.

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Sara Gradara 12 Attribute Control Chart Attribute Control Chart

Attribute control charts arise when items are The argument can be made that a LCL should not compared with some standard and then they are exist, since rates of nonconforming product outside classified as to whether they meet the standard or the LCL is in fact a good thing; we WANT low rates not. of nonconforming product.

The control chart is used to determine if the rate of However, if we treat these LCL violations as simply non-conforming products is stable and detect when a another search for an assignable cause, we may learn deviation from stability has occurred. from the drop in nonconformities rate and possibly permanently improve the process.

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Types of Attribute Control Chart Types of Attribute Control Chart

• p chart: a chart of the percent defective in each sample set. The sample size may vary. • p chart •npchart • np chart: a chart of the number defective in each • c chart sample set. The samples have the same size. • u chart • c chart: a chart of the number of defects per unit in each sample set

• u chart: a chart of the average number of defects in each sample set

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Sara Gradara 13 p charts p charts

Suppose y is the number of defective units in a random

sample of size n. The fraction: pˆ i = y i / n for each sample is plotted on the chart. k ˆ We assume that y is a binomial with ∑pj unknown parameter p. The mean sample proportion is p = j=1 k Let k be the number of samples. p(1− p) The of the p is n

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p charts p charts: example

The variable defect no. contains the number of defective units, Using p to estimate the process proportion defective p and the variable lot sample size contains the lot sample size. the center line and upper and lower control limits for the p chart are: p(1− p) Lower Control Limit: p −3σ = p−3 p n Center Line: p p(1− p) Upper Control Limit: p+3σ = p+3 p n

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Sara Gradara 14 p charts: example np charts

It can be used when the sample are of equale size, n. k ˆ ∑pj In the same way of the p chart: n p = j = 1 and k np(1− p) Lower Control Limit: n p −3σ =np−3 p n Center Line: n p np(1− p) Upper Control Limit: n p +3σ =np+3 or 0 if UCL<0 p n 57 58

np charts: example np charts: example

The variable defect no. contains the number of defective units, and the lot sample size is 500.

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Sara Gradara 15 c charts c charts

c charts are used to chart count of defects where the area The Poisson distribution provides a good of opportunity for a defect is constant. model for the for the number c of defects. Es: defects per 1000 feet of base material in a roll of plastic film produced. The area of opportunity could be a If c possesses a Poisson probability distribution with parameter λ , then E ( c ) = λ and . physical area (defect/1000ft), a product such as scratches σ c = λ per monitor, an amount of time such as broken spindles c per day or any combination of these area of Observe over a reasonably large number, k , of equally c opportunities. spaced points in time and use c , the average value of to estimate λ .

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c charts c charts: example

k c ∑ i Assume that the following table contains defect data for The average value of c is: c = i=1 k one of the system design document.

So the center line and upper and lower control limits for the c chart are:

Lower Control Limit: c −3σ c = c −3 c

Center Line: c

Upper Control Limit: c + 3σ c = c + 3 c 63 64

Sara Gradara 16 c charts: example u charts

If the area of opportunity is not constant, use the u chart instead of the c chart.

Let beui the number of defect over a i the i-th area of opportunity then the average number of defect is k ∑ui i=1 u = k ∑ai i=1 65 66

u charts u charts: example

Assume that the following table contains defect data for The center line and upper and lower control limits for the system design documents of 8 sofware applications the u chart are:

Lower Control Limit: u −3 u / ai

Center Line: u

Upper Control Limit: u + 3 u / ai

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Sara Gradara 17 u charts: example Control Chart - conclusions

First Step: Determine what type of data you are working with.

Second Step: Determine what type of control chart can be used with your data set.

Third Step: Calculate the average and the control limits. Fourth Step: Detect Instabilities and Out-of-Control Situations

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