variation and quality (2.008x lecture slides)

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Variation and QualityMIT 2.008x

Prof. John HartProf. Sanjay Sarma

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Quality: the relentless pursuit of perfection

Lexus, 1992: https://www.youtube.com/watch?v=AktHnnA9QIM

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Quality

Variation

Tolerance

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Quality: Conformity to requirements or specifications. In other words, the ability of a product or service to consistently meet customer needs.

Variation: A change in outcome of a process.

Tolerance: Permissible limit of variation of a process.

2.008xWhat are the measures of

Lego quality?

Drawing from Clipstone, C. J., Hahn, S., Sonnenberg, N., White, C., and Zhuk, A., 2004, “Razor blade technology.”Blade edge: https://scienceofsharp.files.wordpress.com/2014/05/astra_stainless_x_05.jpg

and for Gillette razors?

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Car body measurement using a CMM (Nikon)

Excerpt from: https://www.youtube.com/watch?v=A5zXdSv60Ag

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Car body build variation: production launch

Figure 4 from Ceglarek D, Shi J. "Dimensional Variation Reduction for Automotive Body Assembly." Manufacturing Review Vol. 8, No. 2, 1995:139-154.

2 mm body project: http://www.atp.nist.gov/eao/gcr-709.htm

6 standard deviations from the mean: 3.4 defects per million!

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Car body assembly hierarchy

Figure 5 from Ceglarek D, Shi J. "Dimensional Variation Reduction for Automotive Body Assembly." Manufacturing Review Vol. 8, No. 2, 1995:139-154.

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What do we need to know?

§ What the customer wants (i.e. what is ‘good quality’) and how to relate this to our specifications.

§ How to quantify variation (statistically).

§ What causes process variation, and how to minimize variation as needed.

§ How to monitor variation and maintain process control.

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Agenda: Variation and Quality

§ The normal distribution§ Error stackup and simple fits§ The lognormal distribution§ Process sensitivity§ Principles of measurement§ Statistical process control§ Conclusion

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Variation and Quality:

2. The Normal Distribution

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Measured variation: hex nuts

Mean = 5.58 mmStdev = 0.033

Hex nut thickness [mm]

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Hex nut thickness: observations

§ What do we learn from the distribution of values?

§ Would the values be different if we measure freehand versus on the bolt? Why/not?

§ What is the meaning of the variation we measured?

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The normal distribution

Figure 36.3b, Kalpkjian and Schmid, Manufacturing Engineering and Technology

n→∞The histogram of x with n samples approaches the normal distribution as

Denoted by: mean (à shift): standard deviation (à flatness)

sometimes denoted s; e.g., 2s= 2 standard deviations

2( )21( )

f x e σ

−−

x ∈ N x,σ x( )

σ x =1N

xi − x( )2

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Normal probability density function (PDF)

f (x) = 12π s

⋅e−x−x( )2

From https://en.wikipedia.org/wiki/Normal_distribution (public domain)

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Cumulative distribution function (CDF)

⎥⎥

⎢⎢

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −− 2

From https://en.wikipedia.org/wiki/Normal_distribution (public domain)

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Probability: { }

{ } 1)(

==∞≤≤∞−

=≤≤

∫∞

∞−

dxxfxP

dxxfbxaPb

Normalized to “Z-scores”

{ } ∫−

=≤≤

f (x) = 12π s

⋅e−x−x( )2

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Z-scores

z = x − xs

Z 0 0.02 0.04 0.06 0.08

-3 0.0013 0.0013 0.0012 0.0011 0.0010

-2.5 0.0062 0.0059 0.0055 0.0052 0.0049

-2 0.0228 0.0217 0.0207 0.0197 0.0188

-1.5 0.0668 0.0643 0.0618 0.0594 0.0571

-1 0.1587 0.1539 0.1492 0.1446 0.1401

-0.5 0.3085 0.3015 0.2946 0.2877 0.2810

0 0.5000 0.5080 0.5160 0.5239 0.5319

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Z-scores

z = x − xs

Z 0 0.02 0.04 0.06 0.08

0 0.5000 0.5080 0.5160 0.5239 0.5319

0.5 0.6915 0.6985 0.7054 0.7123 0.7190

1 0.8413 0.8461 0.8508 0.8554 0.8599

1.5 0.9332 0.9357 0.9382 0.9406 0.9429

2 0.9772 0.9783 0.9793 0.9803 0.9812

2.5 0.9938 0.9941 0.9945 0.9948 0.9951

3 0.9987 0.9987 0.9988 0.9989 0.9990

2.008xZ 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

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Example: manipulating the normal distributionCar tires have a lifetime that can be modeled using a normal distribution with a mean of 80,000 km and a standard deviation of 4,000 km.

à What fraction of tires can be expected to wear out within ±4,000 miles of the average?

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Solution: how many wear out between 76,000 and 84,000 miles?

à Area under the curve between these pointsz(1) – z(-1) = 0.8413 – 0.1587 = 0.6826

= 68% will wear out

0.84130.1587

+1.00-1.00

0.6826

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Example: manipulating the normal distributionCar tires have a lifetime that can be modeled using a normal distribution with a mean of 80,000 km and a standard deviation of 4,000 km.

à What fraction of tires can be expected to wear out within ±4,000 miles of the average?

à 68% will wear out

à What fraction of tires will wear out between 70,000 km and 90,000 km?

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Solution: failures within 70,000-90,000 miles

à % of tires that will wear out = z(2.5) – z(-2.5) = 0.9938 – 0.0062 = .9876

à 98%

0.99380.0062

0 +2.5-2.5

0.9876

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3. Error stackup and simple fits

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Measured variation: hex nutsSingle hex nut

Stack of two hex nuts

Mean = 5.58 mmStdev = 0.033

Mean = 11.15 mmStdev = 0.049

Stack thickness [mm]

Hex nut thickness [mm]

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Modeling ‘stackup’: superposition of random variables

Proof: http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

1 2y x x= ±

2 2y x xσ σ σ= +

( )σ,1 xNx ∈ ( )σ,yNy∈( )σ,2 xNx ∈

1 2 3 n

In general, if we define a new random variabley = c1x1 + c2x2 + c3x3 + c4x4 + …

• ci are constants • xi are independent random variables

It can be shown that: µy = c1µ1 + c2µ2 + c3µ3 + c4µ4 + ...

σy2 = c1

2σ12 + c22σ2

2 + c32σ3

2 + c42σ4

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What is the the probability of a successful assembly?

c = D− d

The new critical dimension is the clearance (c):

c = D− d

σ c = σ D2 +σ d

The distribution of clearances is defined by: DD t±

dd t±

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ANSI hole-shaft fit classification

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4. The lognormal distribution

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Lognormal distributionà The logarithm of x is distributed normally

From http://en.wikipedia.org/wiki/Log-normal_distribution (public domain)

Probability density function (PDF)

Example: size distribution of particles in a powder, size distribution of grains within a metal

N (ln x;µ,σ ) = 1xσ 2π

e−(ln x−µ )2

µ = ln m1+ v /m2

%&& σ = ln 1+ v /m2( )

m, v = mean and variance of raw data

Cumulative distribution function (CDF)

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Lognormal distribution: metal powder for 3D printing

GE fuel nozzle: http://www.gereports.com/post/116402870270/the-faa-cleared-the-first-3d-printed-part-to-fly/SEM image: http://advancedpowders.com/our-plasma-atomized-powders/products/ti-6al-4v-titanium-alloy-powder/#15-45_m

Ti6Al4VSpecification: 15-45 um

Selective Laser Melting (SLM)

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5. Process sensitivity

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How hex nuts are made

Excerpt from: https://www.youtube.com/watch?v=MR6q_nXH2IQ

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What can cause process variation?

§ The process: inherent capability; change of settings.

§ Material: raw material variation, defects.

§ Equipment: tool wear, equipment needs maintenance/calibration

§ Operator: procedure, fatigue, distraction, etc.

§ Environment: temperature, humidity, vibration, etc.

§ Measurement: Capability of measurement tool; change of performance (à calibration needed)

§ …

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Climb milling (first cut) versus conventional milling (second cut)

6061-T6 Aluminum with ¼” endmill

Spindle Speed: 4000 rpmFeed: 20.0 in/minDepth of cut: 0.400”Width of cut: 0.070”

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Example: climb versus conventional millingExpected width of material .610”

Conventional cut width (red): Top edge .609”, Bottom of cut .611”

Climb cut width (green): Top edge .612”, Bottom of cut .619”

Conventional§ Chip from thin à thick§ Lower forces but rougher

surface

Climb§ Chip from thick à thin§ Higher forces but

smoother surface

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Reference frame

The ‘structural loop’

The machine, tool, and workpiece are flexible

ErrorWork

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Injection molding process window

à We also must understand the sensitivities to process variables within the window.

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Injection molding: varying process parametersNote the mean shifts compared to the variation

0 10 20 30 40 50 60

Number of Run

Run Chart for Injection Molded Part

Width (mm) Average

Holding Time = 5 sec Injection Press = 40%

Part%radius%[mm]

Run%number

Hold = 5 secPressure = 40% of max

Hold = 5 secP = 40% max

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à Systematic (“special cause”) variation: influences of process parameters or external disturbances that can be isolated and possibly predicted or removed.

à Random (“common cause”) variation: caused by uncontrollable factors that result in a steady but random distribution of output around the average of the data. In other words, this is the ‘noise’ of the system.

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A general model of process variation

ProcessInput (u) Output (Y)

Disturbances, such as:§ Equipment performance changes§ Material property changes§ Temperature fluctuations

Control inputs (process parameter settings)

SensitivityDisturbance (α)

ΔY = ∂Y∂α

Δα +∂Y∂u

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Some example sensitivities (if all other parameters are held constant)

Injection molding§ Relationship between molecular weight of polymer

(determines viscosity) and accuracy (final part dimension compared to mold)

§ Relationship between injection pressure and accuracy

Machining§ Relationship between depth of cut and surface

roughness (= spatial frequency of tool marks)§ Relationship between tool life (sharpness) and accuracy

(= workpiece deformation via higher force and temperature rise)

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All together, this determines the amount of variation, and thus a reasonable tolerance that can be specified!

When the process is ‘under control’:

If tolerances are too tight:§ Extra cost (slower rate)§ More process steps (e.g.

finishing)§ Lots of scrap (rejects)§ Manufacturer “no quote”

(unreasonable expectations)

ΔY = ∂Y∂α

Δα +∂Y∂u

Figure 13.30 from Ashby, Material Selection in Mechanical Design

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6. Principles of measurement

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à Where must the Resolution be on this chart?

True (exact) value

Repeatability

Accuracy

2.008xAccuracy = “the ability to tell the truth”à Difference between the measured and true value

Repeatability = “the ability to tell the same story many times”à Difference between consecutive measurements intended to be identical

Resolution = “the ability to tell the difference”à Minimum increment that can be measured

A. Slocum, Precision Machine Design

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A. Slocum, Precision Machine Design

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Mitutoyo high performance micrometer

§ A highly rigid frame and high-performance constant-force (7-9 N) mechanism enable more stable measurement* *Patent pending in Japan, the United States of America, the European Union, and China.

§ Body heat transferred to the instrument is reduced by a (removable) heat shield, minimizing the error caused by thermal expansion of the frame when performing handheld measurements.

http://ecatalog.mitutoyo.com/MDH-Micrometer-High-Accuracy-Sub-Micron-Digimatic-Micrometer-C1816.aspx

Range = 0-25 mm

Resolution = 0.0001 mm (0.1 micron)Accuracy = 0.0005 mm (0.1 micron)

Flatness: 0.3 micron (across ‘jaws’)Parallelism: 0.6 micron

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Machine vision (Keyence)

Photos taken at IMTS 2014

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Robot-mounted 3D scanner (Creaform)“70 micron accuracy over the “size of a pickup truck” à correcting for low robot accuracy by imaging dots on the sphere

At IMTS 2014

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At IMTS 2014

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At IMTS 2014

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7. Statistical Process Control

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Monitoring a process: CONTROL CHARTSinvented by Walter A. Shewhart (Bell Labs, 1920’s)§ Needed to improve reliability of telephone transmission systems§ Stressed the need to eliminate all but “common cause” variation, and

minimize this variation

à “a process under surveillance by periodic sampling maintains a constant level of variability over time”

0 10 20 30 40 50 60 70 80 90 100Run number

eter Upper control limit

Lower control limit

Step disturbance

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OPERATION CHARACTERISTIC DATES

MACHINE SAMPLE SIZE SAMPLE FREQUENCY REMARKS

ACTIONINSTRUCTIONS

NOTES:

DATE OR TIME

SUMXRS

BASE PREDEPOSITION SHEET RESISTANCE 2/80 - 2/24, 1988

FCE #5 4 EVERY 5th LOT NOTE UNUSUAL OCCURANCES

2/8 2/8 2/9 2/9 2/10 2/10 2/11 2/11 2/11 2/12 2/12 2/15 2/16 2/16 2/17 2/17 2/18 2/18 2/19 2/22 2/22 2/23 2/24 2/242/932 28 31 32 34 33 30 33 35 39 37 33 34 29 32 30 34 33 29 30 29 28 30 292927 25 29 26 32 26 27 29 31 32 31 27 26 25 27 25 27 28 27 28 20 26 23 312227 29 27 25 29 25 25 31 26 30 35 28 30 22 24 22 25 26 27 26 22 25 25 262234 30 25 30 28 33 23 27 27 34 30 25 31 25 26 20 28 25 27 25 25 24 26 2527

120 112 112 113 123 117 105 120 119 135 133 113 121 101 109 97 114 112 110 109 96 103 104 11110030 28 28 28.3 30.8 29.3 26.3 30 29.8 33.8 33.3 28.3 30.3 25.3 27.3 24.3 28.5 28 27.5 27.3 24 25.8 26 27.8257 5 6 7 6 8 7 6 9 9 7 8 8 7 8 10 9 8 2 5 9 7 7 65

LOW ON SOURCE -MORE ADDED*

UCL X = 33.16

LCL X = 23.08

UCL R = 15.97

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What might be going on here?à “a process under surveillance by periodic sampling maintains a constant level of variability over time”

LCL0 10 20 30 40 50

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Basic types of control charts

Average chart: plot of mean values of each sample , centered around

the grand average (mean of all samples)

Range chart: plot of range of each sample (max - min), centered

around the average range.

à Why do we need both charts?

Figure 36.5 from "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

Controlchartsareconstructedfrommeasurements ofsamples (eachwithnparts)fromthepopulation(N,allpartsmanufactured).

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Reveals shift

Process mean is shifting upward

Does not reveal shift

When the mean shifts:

SamplingDistribution

x-Chart

R-chart

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Does not reveal increase

Process variability is increasing

Reveals increase

When the mean shifts:

SamplingDistribution

x-Chart

R-chart

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How do we choose the sample size (n) and frequency of sampling?§ Likelihood of unexpected disturbances§ Importance (cost) of defects§ Cost of measurementàTypically based on experience and knowledge of the above

(sometimes trial and error)

How do we define the control limits (LCL, UCL)?§ Based on pre-tabulated statistics of sample variation versus

sample size

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Calculating the control limitsAverage chartGrand average:

Control limits:

Range chartAverage range:

Control limits:

Figure 36.5 from "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

LCL = X − A2R

UCL = X + A2R

LCL = D3R

UCL = D4R

X =X i

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Factors for calculating control limitsà These constants are for a 3-sigma approach, i.e., control limits are placed at +/- 3 standard deviations from the estimated process mean

Table 36.2 from "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

AveragechartGrandaverage:

Controllimits:

RangechartAveragerange:

Controllimits:

LCL = X − A2R

UCL = X + A2R

LCL = D3R

UCL = D4R

X =X i

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Process control vs. capabilityà Even if a process is in control (i.e., constant mean and variation), it may not be capable (i.e., giving what we want as set by the specifications a.k.a. the tolerances)

Upper control limit (UCL)

Lower control limit (LCL)

In Control and Capable(Variation from common cause reduced)

In Control but not Capable(Variation from common causes excessive)

Lower specification limit (LSL) Upper specification

limit (USL)

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Control limits vs. tolerances (specification limits)

Control limits are:§ Based on process mean and variability.§ Dependent on the sampling parameters.à Thus, control limits are a characteristic of the process and measurement method.

Tolerances (specification limits) are:§ Based on functional considerations.§ Used to establish a part’s conformability to the design

intent.à Thus, we must have a formal method of comparison.

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Process capability: compares process variation to tolerances

usewhicheverissmaller,becauseà

Cp =USL− LSL6σ x

Cpk =USL−µx3σ x

Cpk =µx − LSL3σ x

Generalrule:Cp shouldbeatleast1.33

LSL,USL=tolerancelimitsσx =processstdev

LSL USL

9.80 10.00 10.05 10.20 (mm)

DesignIntent

True process

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0 10 20 30 40 50 60 70 80 90 100Run number

Example: calculating Cp, Cpk

Assume:µx =1.000”σx =0.001”

Specification=0.999”+/- 0.005”

Upper control limit

Lower control limit

Step disturbance

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Recommended values of process capability

à How do we really judge what’s good enough? Knowledge of the ‘cost’ of defects in our product, thereby defining a ‘quality loss function’ (beyond scope today).

Recommendedprocesscapabilityfortwo-sidedspecifications

Defects(partsoutofspec) permillionoperations

Existing (stable)process 1.33 63

Newprocess 1.50 8

Existing process,safety-critical

1.50 8

Newprocess, safety-critical 1.67 1

Six-sigma quality 2.00 0.002

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8. Conclusion

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The big picture

‘Pilot’ production

This is a control chart

Design for Manufacturing (DFM)

Does not conform

Conforms (good!)

Change design? Modify process (know what to do)

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Embracing the variation: Apple

Excerpt from: https://youtu.be/7cIRpmgYBJw?t=274

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Embracing the variation: Intel§ The speed of each processor made is

measured, and this sets the specification and price

§ As production improves, faster processors are released for sale

Image © Intel Corporation 2016 http://www.intel.com/content/www/us/en/embedded/products/bay-trail/atom-processor-e3800-platform-brief.htmlhttp://ark.intel.com/compare/78416,80270,80269,80268

Product Name

Intel® Atom™ Processor Z3740D (2M Cache, up to

1.83 GHz)

Intel® Atom™ Processor Z3745 (2M Cache, up to

1.86 GHz)

Intel® Atom™ Processor Z3775D (2M Cache, up to

2.41 GHz)

Intel® Atom™ Processor Z3775 (2M Cache, up to

2.39 GHz)Performance# of Cores 4 4 4 4# of Threads 4 4 4 4Processor Base Frequency 1.33 GHz 1.33 GHz 1.49 GHz 1.46 GHz

Burst Frequency 1.83 GHz 1.86 GHz 2.41 GHz 2.39 GHzScenario Design Power (SDP) 2.2 W 2 W 2.2 W 2 W

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Reflection: learning objectives§ Recognize how process tolerances are defined and

variation is monitored, and how a manufacturing process is established to control variation.

§ Be fluent with manipulation of normally distributed dimensions, combinations of dimensions (e.g., to predict fits, lifetimes, etc.).

§ Understand how process physics influence statistical outcomes (e.g., mean, variation). What are the sensitive parameters, and how can the variation be addressed?

§ Understand accuracy, repeatability, resolution; assess the suitability of a measurement technique to monitor a process.

§ Know how to construct and interpret control charts and evaluate process capability.

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References1 Introduction

iPhone with a cracked screen, photo by User: Philipp Zurawski (Freetagger) - PixabayCC0. This work is in the public domain.

LEGO brick assembly, photo by User: M W (Efraimstochter) - Pixabay CC0. This work is in the public domain.

Gillette razor blade section, Figure 1 from "Razor blade technology US6684513 B1" by Clipstone, et al. (2004). This work is in the public domain.

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References2 Normal Distribution

Normal distribution: Figure 36.3b in "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

Normal probability distribution function, image by User: Inductiveload via wikimedia. This work is in the public domain.

Cumulative distribution function, Image by User: Inductiveload via wikimedia. This work is in the public domain.

Automobile tire, photo by User: Robert Balog (Bergadder) - Pixabay CC0. This work is in the public domain.

3 Variation Stackup

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References4 Lognormal Distribution

Log-normal probability distribution function, image by User: Krishnavedala via wikipedia -CC0. This work is in the public domain.

Particle size distribution for Ti-6Al-4V powder stock of various size ranges from Advanced Powders and Coatings (APC), figure 5 from Title: Raymor AP&C: Leading the way with plasma atomised Ti spherical powders for MIM; Journal: Powder Injection MouldingInternational; Vol: 5; No: 4; December 2011; pages: 55-57. © Inovar Communications Ltd

5 Sensitivity

Conventional vs. climb milling: Figure 24.3 in "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

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ReferencesNormal distribution: Figure 36.3b in "Manufacturing Engineering & Technology (7th Edition)" by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

6 Measurement

Accuracy, resolution and repeatability: Figure 2.1.1 in "Precision Machine Design" by Alexander H. Slocum; Publisher: Prentice Hall; Year: 1992; ISBN: 0136909183. (c) Prentice Hall 1992.

ESPN Monday Night Football, ESPN broadcast footage (c) Disney Corporation.

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References7 SPC

Control charts of averages and ranges of sample measurements: Figure 36.5 from "Manufacturing Engineering & Technology (7th Edition)," Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

Control limit equation constants as a function of sample size: Table 36.2 from"Manufacturing Engineering & Technology (7th Edition)," by Kalpakjian, Schmid. (c) Upper Saddle River; Pearson Publishing (2014).

8 Conclusion

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