TMA4255 Applied Statistics V2016 (19)

TMA4255 Applied Statistics V2016 (19)Part 5: Statistical process controlGeneral [17.1-17.3]Control charts for variables [17.4]

Anna Marie Holand

March 31, 2016, wiki.math.ntnu.no/tma4255/2016v/start

www.ntnu.no [email protected], TMA4255V2016

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Part 5: Statistical Process Control

Methods for quality control— Before production: Statistical design of experiments.— In production: Statistical process control.— After production: Sampling inspection.

Related: Statistical surveillance: much used in medicine(hospitals).


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SPC: historically

— Pioneered by Walter A. Shewhart in the early 1920s.— W. Edwards Deming later applied SPC methods in the United

States during World War II, thereby successfully improvingquality in the manufacture of ammunitions and otherstrategically important products.

— Deming was also instrumental in introducing SPC methods toJapanese industry after the war had ended.

— Taguchi methods in Japan from the 1980s.— In 1988, the Software Engineering Institute introduced the

notion that SPC can be usefully applied to non-manufacturingprocesses, such as software engineering processes.


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Tensile strength example

— Process: manufacture missile (steel) components.— Aim: control of tensile strength.

• Wikipedia: Tensile strength is the maximum stress that amaterial can withstand while being stretched or pulled beforenecking, which is when the specimen’s cross-section starts tosignificantly contract. Tensile strength is the opposite ofcompressive strength and the values can be quite different.

— The process may be influenced by• natural variation (due to uncontrollable and unimportant

sources), and• nonrandom assignable causes (operator errors, improper

adjustments, ...).


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Tensile strength: example chart


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Tensile strenghts example— We want to construct a chart to monitor the process.— The characteristic we want to monitor is the tensile strength.— The centerline represents the average value of the

characteristic when the process is in control.— At each time point more than one observation is taken. The

number of observations taken is called a rational subgroupand is of the order n = 5.

— The points depicture the sample averages (over the rationalsubgroup) of this characteristic, with samples taken over time.

— The upper and lower control limits are chosen in such a wayso that one would expect all sample points to be covered bythese boundaries if the process is in control.

— This is called a control chart.


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Setting up a control chart— Control limits are set up using a part of the process that is

known is stable.— Control limits (U=upper, L=lower) are established to control the

probability of making the error of concluding that the processis out of control when in fact it is not (correpond to Type I errorin hypothesis testing).

— But, we must also be attentive to an error of the second kind(not finding the process out of control when it infact is, Type II).

— In general (and in the book):• UCL=expected value + 3 standard deviations, and• LCL=expected value - 3 standard deviations.

But, the number 3 can be substituted by a larger number.

We will focus on setting up control limits.


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Using a control chart

— When a point fall inside the control limits the process is belivedto be in statistical control and the variation observed can beexplained as natural variation.

— Then a point fall outside the control limits this is taken to beevidence of a process that is out of control, and a search forthe assignable cause is suggested.


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Control charts for variables andattributes

Quality characteristics fall into two categories:— Variables: a continuous measurement (diameter, strength,

weight). Assume normal distribution and make control chartsboth to control central tendency and variability.

Xbar mean, andR range orS standard deviation

— Attributes: does the product conform to standards?p Probability (defective of not)

C and U Count (number of mistakes in an area).We will work with these types of control charts.


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Control charts for variables [17.4]


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Simple exampleProcess to monitor engineparts has (kwown) average at µ = 50mmand standard deviation σ = 0.01mm in control.— Groups of size n = 5 are sampled every hour, and X̄ is

calculated and plotted.— If we assume that the variation in the engine parts

measurement is normally distributed we know thatX̄ ∼ N(µ, σ2/5).

— We know that we may construct a 100(1− α)% interval withlimits

LCL = µ− zα/2σ√5

UCL = µ+ zα/2σ√5

and expect that on the long run 100(1− α)% of the X̄ values willfall within the limits.


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Simple example (cont.)

LCL = µ− zα/2σ√5

UCL = µ+ zα/2σ√5

— In SPC the popular choice is three-sigma limits.— If we know that data are normal with known µ and σ then the

probability is 0.0026 that the X̄ value falls outside the controllimits when the process is in control. (Why? z 0.0026

2= 3.)

— But, when we only assume approximate normality and do notknow the expected value or the variance, then we can notknow the true coverage of the inteaval.


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Control charts

— Out of control (outside control limits)— in statistical control (inside control limits).

Basic principleUCL=Upper control limit= expected value + 3 standard deviationsLCL=Lower control limit= expected value - 3 standard deviationsThe number 3 kan be substituted by a general k


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Xbar, range and standard deviation

Xij ∼ N(µ, σ2)

X̄i = average of n observations at time i =1n

n∑j=1

Xij

Ri = range at time i = max Xi −min Xi

Si =

√√√√ 1n − 1

n∑j=1

(Xij − X̄i)2


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R or S?

— The empirical range is much easier to compute manually thanthe empirical standard deviation.

— For this reason charts based on R have been used.— However, for sample sizes (for the rational subgroups) of at

least n = 10 (or 5) it is preferred to use S.— We now proceed by first looking at charts for X̄ and S, and X̄

based on S to estimate variability (called X̄ − S chart),— (I will not show charts forX̄ and R, and X̄ based on R to

estimate variability (called X̄ − R chart)).


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Charts for X̄ and S

Assume µ and σ known.— X̄ -chart

µ± 3σ√n

— S-chartbased on E(S) = c4σ and Var(S) = (1− c2

4)σ2.

c4σ ± 3√

1− c24σ


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Appendix A22 (A23 in 8th edition)


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Charts for X̄ and SAssume µ and σ unknown, and estimate µ by

µ̂ = ¯̄X =1m

m∑i=1

X̄i

σ can be estimated from the Si ’s:

S̄ =1m

m∑i=1

Si

E(S̄) = c4σ

σ̂ =S̄c4


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X̄ − S-chartsWe found:

µ± 3σ√n

Now we insert estimates for µ and σ.

µ̂ = ¯̄X =1m

m∑i=1

X̄i

σ̂ =S̄c4

X̄ − S-chart

¯̄X ± 3S̄

c4√

n= ¯̄X ± A3S̄


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S chart

We found:

c4σ ± 3√

1− c24σ

σ̂ =S̄c4

which gives

S̄ ± 3S̄c4

√1− c2

4 = [B3S̄,B4S̄]


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Sample size for X̄ -chart


TMA4255 Applied Statistics V2016 (19)

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