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خطاهاي اندازه گيري در تحقيق مواددرس دوره كارشناسي ارشد مهندسي مواد

مسعود محبياستاديار گروه مهندسي مواد

دانشگاه بين المللي امام خميني

:مطالب

مقدمات

عمليات اعداد

مفاهيم آماری

روشهای آماری

روشهای کامبيناتوريالبرنامه ريزی آزمايش

روشهای آماری

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References:

آمار مهندسی مبانی روشهای آماری

Horst Czichos Tetsuya Saito Leslie Smith (Eds )Horst Czichos, Tetsuya Saito, Leslie Smith (Eds.)Springer Handbook of Materials Measurement Methods

Paolo FornasiniThe Uncertainty in Physical Measurements

Jiju AntonyDesign of Experiments for Engineers and Scientists

Category Stage Name Summary of stage Basic Example Student Task

Dualism/ Received Knowledge

1 Basic Duality All problems are solvable. The authorities know. The tutor knows what is right and wrong.

to learn the Right Solutions

2 Full Dualism There are Right Solutions, the true authorities are right, the others are frauds.

My tutor doesn't know what is right and wrong but others do.

to learn the Right Solutions and ignore the others

3 Early Multiplicity

There are 2 kinds of problems:those whose solutions we know those whose solutions we don't know yet.There are some uncertainties and the authorities are working

My tutors don't know, but somebody out there is trying to find out.

to learn the Right Solutions and ignore the others

Perry's scheme of Intellectual and Ethical Development in the College Years: student’s nine‐stage progression

Multiplicity/ Subjective Knowledge

on them to find the truth.

4 Late Multiplicity

Most problems are of the second kind.(a)Everyone has right to their own opinion.(b) The authorities don't want the right answers. They want us to think in certain way.

Different tutors think different things.There is an answer that the tutors want and we have to find it.

to shoot the bull

Relativism/ Procedural Knowledge

5 Contextual Relativism

All proposed solutions are supported by reasons.Some solutions are better than others, depending on context.Everything is relative but not equally valid.

There are no right and wrong answers, it depends on the situation, but some answers might be better than others.

to learn to evaluate solutions

6 Pre-Commitment

Student sees the necessity of: a- making choices

What is important is not what the tutor thinks but what I think.

to make his/her own decisions

b- committing to a solution

Commitment/ Constructed Knowledge

7 Commitment First commitment For this particular topic I think that.....

8 Challenges to Commitment

Several Commitments.Student experiences implications of commitment.Student explores issues of responsibility.

For these topics I think that.....

9 Post-Commitment

Student realizes commitment is an ongoing, unfolding, evolving activity.

I know what I believe in and what I think is valid, others may think differently and I'm prepared to reconsider my views.

Believe own values, respect others, be ready to learn

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علم؟

Science?

Physical science?

Scientific method?

Physical quantity?

Methods of Observation and Measurement

Morphological Method: sensorial detection of some properties by drawings, photographs, etc.

Classicatory Method: partitioning a set of objects or phenomena into classesinto classes

Comparative Method: introducing an order relation

A property for which a comparative method can be dened is a physical quantity.

Quantitative Method: correspondence between the degrees of the physical property and the set of real numbers; introducing an additive structure

Statistical Methods: statistical methods play a fundamental role in the treatment of uncertainties of physical quantities

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Physical quantity Numbers (Unit)(Dimension)

اعداد

Integerطبيعی

)اعشاری(کسری )(

نامحدود

Exact numerical value (exact numbers)اعداد دقيق

sin 30 = 0.5تعداد دانشجويان يک کالس

Approximate numerical values اعداد تقريبی

1 foot ≡ 12 inches )شمارش و تبديل واحدها(

c ≈ 3 × 108 m.s-1

cos 30 ≈ 0.8660254 ≈ 0.866

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Significant Figures

Significant Figures SfSignificant Digits SdNumber of significant figuresg g

The number of significant figures is the minimum number of digits needed to write a given value in scientific notation

47000 ?

of digits needed to write a given value in scientific notation without loss of accuracy.

1236 1.236x103 (4 significant figures)0.0123 1.23x10-2 (3 significant figures)

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آيا صفر معنی دار است؟

صفرهای بين ارقام ديگرThe number 25.04 has 4 signicant digits: 2 5 0 4

)Leading zeros(صفرهای پشت ارقام The number 0.0037 has 2 signicant digits: 3 7

) Trailing zeros(صفرهای سمت راست اعداد The number 0.50 has 2 signicant digits: 5 0

The first digit on the left:The most significant digit MSD

The last digit on the right:g gThe least significant digit LSD

The first uncertain digit is the last significant figure.

(12.30(±0.01) mL)

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گرد کردن اعداد تقريبی در محاسبات عددی

It is best to carry extra digits through intermediate calculations and round the final answer to the correct number of significant digits.

EXAMPLES:» 83.17g+0.041g =83.211g = 83.21g» 9 253g 1 3g = 7 953g = 8 0g» 9.253g -1.3g = 7.953g = 8.0g

• EXAMPLES:1.2mol x 3.4231g/mol = 4.107720 = 4.1g0.012L x 0.013mol/L = 1.56x10-4 mol =1.6x10-4 mol6.83 x 72 = 491.76 = 4.9x102

83.642 / 72 = 1.1616944 = 1.2 = 1.16

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تعداد ارقام معنی دار در گزارش يک اندازه گيری با ميزان دقت د ش يين ت ی گي انداز آن

significant digits uncertainty

آن اندازه گيری تعيين می شود.The number of significant digits of a measure is determined by the extent of its uncertainty.

:نکات مهم

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علل ايجاد عدم قطعيت در اندازه گيری

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Error

Accuracy

Precision

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Instrument Resolution and Measurement Resolution

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accuracy, resolution, precision and errorAccuracy is the extent to which a measurement approaches the true value of the measured quantity. Precision is the quality of being exactly or sharply defined or stated.Resolution is the smallest increment that can be distinguished and acted upon [i],[ii]. “Error” is inversely related to “accuracy” (and “precision” ??) and defines how far a series of experimental result vary from the expected or programmed value.

ecisionPr1Error

Accuracy1

∝∝

[i] MORRIS C., Academic Press Dictionary of Science and Technology, Academic Press, San Diego, 1992.[ii] PARKER S. P., McGraw-Hill Dictionary of Scientific and Technical Terms (Fourth Edition), McGraw-Hill, New York, 1989.

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Precision repeatability of measurement.

Accuracy agreement between measurement and the true value.

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consistent

inconsistent

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Outliersديتاهاي پرت و تشخيص آنها(extreme values)

: ريشه هاي ايجاد

خطاهاي حين انجام آزمايش -1

نمونه هاي غير عادي يا متفاوت -2

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گرافيكي: روشهاي تشخيص

آماري

هيستوگرام

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:اصول تشخيص و حذف

خارج باشد، ديتا حذف ميشود مگر % 99اگر ديتا از حد اطمينان - 1.براي حفظ ديتا وجود داشته باشد) غير آماري(اينكه دليلي

خارج باشد، ديتا فقط در صورتي حذف % 95اگر ديتا از حد اطمينان - 2.براي حذف ديتا وجود داشته باشد) غير آماري(ميشود كه دليل ديگري

در هر صورت نبايد حذف ديتاهاي پرت منجر به حذف تعداد قابل - 3.وجهي از داده ها شود

1 , 2 , 3 , 4 , 5 , 6 , 7

Example:

Mean= 28 / 7 = 4Median = 4

1 , 2 , 3 , 4 , 5 , 6 , 60

Mean= 81 / 7 = 11.6Median = 4

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Quartiles

Example:

Range:

:داده ها ي مرتب شده

Range = 48 – 27 = 21

Quartiles?

Lower Quartile Upper QuartileLQ UP

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Interquartile range?

46 - 32 = 14

Box and whiskers method

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M N

داده هاي مياني 50%

Median ± 1.5 × interquartile range

M N

MN ≡ [ Mean ± 2.7 S.D.] ≡ 99% confidence interval

.زياد داده های پرت هستندداده های خارج از اين محدوده به احتمال

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Normal distribution

εεδ

dP ⎥⎤

⎢⎡−

∫2

exp1 εσσπδ

dP ⎥⎦

⎢⎣

= ∫−

22exp

2

Student t distribution

N<20?N<20?student t distribution

shape depends on N and approaches the normal distribution as N gets large.distribution as N gets large.

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-σ σ0 -1.96σ 0 1.96σ

standard deviation error limits 95% confidence limits

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(1-2*0.025) * 100 = 95%

σNσttSm

Δ

==δ

(Δ = 95% confidence level in the mean.)

NσtSt m 95.095.0 ==Δ

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1

1.5

2

2.5

Cou

nt

N=20Mean=12.26 mLStd. deviation=1.01

22.020/01.1

26.12

95.0=Δ==

=

m

m

StS

mLMean

0

0.5

9 10 11 12 13 14 15mL EDTA added

46.0460.0)22.0(09.2 ===Δ

mean and Sm : 12.26 (0.22)mL

12.26 ± 0.46 mL (95%, N=20)

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10

15

20C

ount

N=255Mean=12.42 mLStd. deviation=0.76

Mean = 12.42mLSm = 0.76 / 255 = 0.048 = 0.05Δ = t0 95Sm

0

5

9 10 11 12 13 14 15mL EDTA added

Δ t0.95SmΔ = 1.96(0.048) = 0.094 = 0.09

mean and Sm : 12.42 (0.05)mL

12.42 ± 0.09 mL (95%, N=255)

Experimental Planningبرنامه ريزي آزمايشات

باشد ساده باشدسادهقابل تكرار باشد

عاري از خطاي سيستماتيك باشداز دقت بااليي برخوردار باشد

اشد داشته د كا ت گستره كاربردي وسيعي داشته باشدگهدفمند باشد

ارزان باشد

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P

At b=40

P = f ( a , b )

6.5665.554.543.53

a40 50 60 70 80 90

2.521.5

Pتاثير پارامتر دوم در بهترين مقدار پارامتر اول

6.56

At a=65

65.554.543.53

b10 20 30 40 50 60

2.521.5

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b70

60

50

6.5

5.5

a40 50 60 70 80 90

5.5 4.5 3 2.5 3.5 5 6

50

40

30

3.5

2.5

2 7

20

10

2.7

4

5.5

A

AB

B C

AB

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روشهای انجام آزمايشاتروش سعي و خطا

با تغيير تصادفي متغيرها و بررسي نشانه ها و بدنبال نتيجه شانسي بودن

OVATبا تغيير يك متغير با ثابت نگه داشتن بقيه بدون در نظر گرفتن اثرات تركيبي متغيرها

Full Factorial روش فاكتوريل كاملانجام تست درتمام حاالت ممكن براي متغيرهاهمراه با بررسي اثرات تركيبي متغيرها

Fractional Factorial روش فاكتوريل جزئيبررسي بخشي از حاالت ممكن همراه با بررسي اثرات تركيبي متغيرها كه اطالعات

.كمتري نسبت به فاكتوريل كامل حاصل مي كند

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ACombinatorial methods

B C

Joseph Hanak (1970)The present approach to the search fornew materials suffers from a chronicil t th t f h dli l tailment, that of handling one sample at a

time in the process of synthesis, chemicalanalysis and testing of properties. It is anexpensive and time-consuming approach,which prevents highly-trained personnelp g y pfrom taking full advantage of its talentsand keeps the tempo of discovery of newmaterials at a low level.J.J.Hanak(Journal of Materials Science 5, 964-971, 1970)

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B

A

C

Combinatorial Methods in Drug Discovery

• Bruce Merrifield (1963) on solid-phase peptide synthesis• Bruce Merrifield (1963) on solid-phase peptide synthesis(Nobel prize winner 1984)

• Early 1980s, Mario Gaysen; libraries of matrix 8x12

• 1985, Richard Houghten synthesised millions of peptidesas mixtures and partially screened (first comprehensivecombinatorial approach)

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Potentials in Materials ScienceA

B C

Taking 50 metals (metal oxides)

19 600 ternary systemsB C

n=10 N=66

19,600 ternary systems

537 million octernary2

)2n)(1n(N ++=

Artificial Intelligence

Search refinement and steering (finding localSearch refinement and steering (finding local optima)

Relational Database

A ifi i l l k- Artificial neural network

- Machine learning algorithm

- Steering software

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Combinatorial Techniques in Materials Science

• Thin Film Techniques• Thin Film Techniques

• Thick Film Method

CVDPVD with epitaxial growth

Thick Film MethodInk-jet Printing

Part of the phosphors library under 254 nm

UV

Epitaxial MethodEpitaxial MethodEpitaxial MethodEpitaxial Method

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Ceramic mixtures(sample library)

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Design of Experiments

DOEDOE = Structured Experimental Approach

برنامه ريزيانجام آزمايشاتجمع آوري ديتا

تا د ز آناليز ديتاآنالنتيجه گيري

DOE Full Factorial روش فاکتوريل کامل

2-level experiment

k2k

k = number of factors

تعداد آزمايشات=

TRIAL FACTOR 1 FACTOR 21 - -12 + -3 - +4 + +

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Practical

بررسی و آناليز نتايج آزمايشات

Practicalآيا اختالف داده ها معنی را می رساند؟ و آيا روند خاصی در اختالفها ديده می شود؟ •

Graphicalآيا با ترسيم نتايج الگويی در روند داده ها ديده می شود؟•

AnalyticalAnalyticalکدام متغير يا متغيرها بيشترين تاثير را دارند؟•

+

B

5 10High

- A +

- 6 9

Low High

Low

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StrongEffect

- +Factor

- +Factor

NoEffect

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Effect of A-B Interaction

StrongInteraction

B-1

B+1

- +Factor A

C-1No

Interaction

- +Factor A

C 1

C+1

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The 23 Factorial Design

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Fractional Factorial روش فاكتوريل جزئي

2k-p 3k-p

2-level standard experiments:

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