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EMCH 361 – Fall 2001 Measurements and Instrumentation

http://web.engr.sc.edu/emch361/

EMCH 361 MWF 12.20 – 1:10 Measurements and Instrumentation

Handout Packet 2 8/21/2001

1. GENERAL INFORMATION ......................................................................................................................... 4 1.1 SCHEDULE .................................................................................................................................................. 4 1.2 INSTRUCTOR............................................................................................................................................... 5 1.3 GRADING.................................................................................................................................................... 5 1.4 COURSE CONTENT ...................................................................................................................................... 5

2. ABET SYLLABUS........................................................................................................................................... 6 2.1 CATALOG COURSE DESCRIPTION............................................................................................................... 6 2.2 PREREQUISITE(S)........................................................................................................................................ 6 2.3 TEXTBOOK(S) AND/OR OTHER REQUIRED MATERIAL ............................................................................... 6 2.4 COURSE OBJECTIVES.................................................................................................................................. 6 2.5 TOPICS COVERED ....................................................................................................................................... 6 2.6 ASSESSMENT METHODS.............................................................................................................................. 7 2.7 CLASS/LABORATORY SCHEDULE ............................................................................................................... 7 2.8 CONTRIBUTION OF COURSE TO MEETING THE PROFESSIONAL COMPONENT .............................................. 7 2.9 RELATIONSHIP OF COURSE TO PROGRAM OBJECTIVES............................................................................... 7

3. HOMEWORK AND EXAMS......................................................................................................................... 9 3.1 HOMEWORK ............................................................................................................................................... 9 3.2 EXAMS........................................................................................................................................................ 9

3.2.1 Mid Term Exam.................................................................................................................................. 9 3.2.2 Course evaluation and Final Exam.................................................................................................... 9

4. LABS ............................................................................................................................................................... 10 4.1 TA ASSIGNMENTS.................................................................................................................................... 10 4.2 LAB GUIDELINES AND PROCEDURE .......................................................................................................... 10 4.3 REPORT GUIDELINES ................................................................................................................................ 11 4.4 CONTENTS ................................................................................................................................................ 11 4.5 EQUATIONS .............................................................................................................................................. 12 4.6 TABLES AND FIGURES .............................................................................................................................. 12 4.7 UNITS ....................................................................................................................................................... 12 4.8 LATE REPORTS AND HOMEWORK ............................................................................................................. 13 4.9 GRADING.................................................................................................................................................. 13 4.10 CHEATING POLICY ................................................................................................................................... 13 4.11 CLASS MATERIAL ..................................................................................................................................... 13

5. TECH TALK GUIDELINES........................................................................................................................ 14 5.1 REQUIREMENTS........................................................................................................................................ 14 5.2 THINGS TO REMEMBER:............................................................................................................................ 14 5.3 AVAILABLE TECH TALK TOPICS ............................................................................................................... 15 5.4 GRADING.................................................................................................................................................. 15

6. STATISTICS/PROBABILITY OVERVIEW ............................................................................................. 18 6.1 INTRODUCTION......................................................................................................................................... 18 6.2 STATISTICS ............................................................................................................................................... 18 6.3 PROBABILITY............................................................................................................................................ 18

6.3.1 Example 1: ....................................................................................................................................... 19 6.3.2 Example 2: ....................................................................................................................................... 20



6.3.3 Example 3: ....................................................................................................................................... 21 6.4 CONFIDENCE INTERVALS ......................................................................................................................... 21

6.4.1 Confidence intervals for large samples............................................................................................ 21 6.4.2 Example 4......................................................................................................................................... 22 6.4.3 Confidence intervals for small samples ........................................................................................... 22

6.5 EXCEL STATISTICAL FUNCTIONS.............................................................................................................. 23



1. GENERAL INFORMATION

1.1 Schedule Date Lecture topic(s) Text Laboratory/Homework Perform Submit

Aug. 24 Orientation, basic concepts 1 Lab 1 – Take Home M 27 Statistics and probability 3 29 Uncertainty, basic concepts 3 31 Propagation of uncertainty Sept. M 3 Labor Day Break – No classes 5 Curve fitting 3 7 Basic circuits Lab 1 – Take Home M 10 Electrical measurements 11 Lab 2 - Electricity (A018) 12 Strain measurements and strain gages 14 Writing technical reports (PCC) Hmwk1: 3.1, 2, 4, 5, 6, 9, 10, 17, 38, 40 M 17 Study time - no lecture 3 19 Study time - no lecture 13 21 Wheatstone bridge 6 M 24 Signal conditioning - filters 6 Lab 3A - Strain Gages I (A018) 26 Calibration 12 28 Other electrical sensors 7 Oct. M 1 Time dependent measurements 7 Lab 3B - Strain Gages II (A018) Lab 2 - Electricity report 3 Temperature measurement, thermocouples 5 5 Analog to digital conversion, digital signals 6 M 8 Study time - no lecture 5 Hmwk2: 12.1, 8, 10, 11, 12, 30 10 Mid-term review 12 Mid-term Exam Tech Talk topic M 15 Fall Break – No classes 17 Study time - no lecture 7 19 Linear measurements 16 M 22 Study time - no lecture Lab 4 – Thermocouples (A018) Lab 3 - Strain Gages report 24 Study time - no lecture 26 Force and torque measurements M 29 Presentations (PCC) 8 31 Digital signals Nov. 2 Design of experiments I 15 M 5 Design of experiments II Hmwk3: 5.3, 4, 5, 6, 11, 12 7 Study time - no lecture 9 Review M 12 Help session/Recitation Tech Talk (A018) Lab 4 - Thermocouples report 14 Study time - no lecture 16 Study time - no lecture M 19 Help session/Recitation 21 Thanksgiving – No classes 23 Thanksgiving – No classes M 26 Help session/Recitation 28 Study time - no lecture 30 Study time - no lecture Hmwk4: 16.1, 3, 13, 15, 16, 18Dec. M 3 Help session/Recitation 5 Study time - no lecture 7 Study time - no lecture M 10 Exam Review 12 13 Course evaluation, Final Exam - 14:00 to 17:00



1.2 Instructor

Victor Giurgiutiu ( jurjutzu ), PhD, Associate Professor

Office A222, 300 Main Street

Office Hours MTWThF 8:30 to 9:30, or by appointment

Telephone (803) 777-8018

FAX (803) 777-0106

Email [email protected]

1.3 Grading

Lab Reports 40%

Homework 20%

Tech Talk 10%

Mid-Term Exam 15%

Final Exam 15%

A 94 – 100

B+ 87 - 93

B 81 - 86

C+ 77 - 80

C 71 - 76

D 61 - 70

F � 60

1.4 Course content

Instrumentation 30%

Measurement skills 30%

Written presentation skills 25%

Oral presentation skills 15%



2. ABET SYLLABUS

2.1 Catalog Course Description

EMCH 361-Measurements and Instrumentation. (3) (Prereq or coreq: ENGR 260) Principles of measurement, probability of statistics, analysis of data, and experimental planning. Measurement of parameters in mechanical engineering systems.

2.2 Prerequisite(s)

ENGR 260 - Solid Mechanics

2.3 Textbook(s) and/or Other Required Material

Mechanical Measurements, 5th edition, Beckwith, Marrangoni and Lienhard

A Guide to Writing as an Engineer, Beer and McMurrey

2.4 Course Objectives

{Assessment Methods shown in Braces}

1. Students will demonstrate the ability to organize and write a laboratory report {2}.

2. Students will demonstrate the ability to organize and give an oral presentation {1}.

3. Students will demonstrate the ability to explain the operating principles of common instrumentation and interpret the output {2, 3}.

4. Students will demonstrate the ability to apply statistical skills in creating an experiment and interpret the results {2, 3}.

2.5 Topics Covered

1. Statistics and probability

2. Organizing and writing the laboratory report

3. Presentation of data and uncertainty analysis

4. Linear measurements

5. Electrical measurements

6. Force, stress, strain and torque measurements

7. Measurement of dynamic systems

8. Linear regression and curve fitting

9. Organizing and making the technical oral presentation

10. Thermal and fluid measurements

11. Designing experiments for measuring specified parameters



2.6 Assessment methods

1. Oral presentations

2. Written lab reports

3. Written tests

2.7 Class/Laboratory schedule

Lecture: Two 50 minute sessions per week

Recitation: Four 1.0 hour sessions over the semester

Laboratory: One 3.0 hour session per week according to schedule

2.8 Contribution of course to meeting the professional component

Engineering Topics 65 %

Math and Science 2 %

General Education 33 %

2.9 Relationship of course to program objectives

The importance of each course objective to meeting the program outcomes is indicated with the following scale:

3 = major importance

2 = moderate importance

1 = minimal importance

Blank if not related



Course Objectives Program Outcomes

(see list for complete description)

CO1 CO2

CO3 CO4

1.1. analyze, design and realize

1.2. computation techniques 1

1.3. design and interpret experiments 2 1 1

1.4. apply linear algebra, calculus

1.5. apply statistical methods 3

1.6. understand chemistry and physics 3

2.1. engineering economic analyses

2.2. plan and execute projects

2.3. oral and written communications 3 3

2.4. professional responsibility

2.5. multi-disciplinary teams

2.6. life-long learning

3.1. engineering in modern society 1 1 1 1

3.2. literature, arts, humanities.

3.3. foreign language

Person Who Prepared This Description and Date of Preparation:

Dr. Victor Giurgiutiu, Associate Professor, 2/24/99

amended by J.Lyons, 5/28/99



3. HOMEWORK AND EXAMS

3.1 Homework

During the semester, 4 homework will be assigned. In order to pass the class you must turn in all the homework.

In case you are not turning in the homework before the due day, you are late and points will be taken from your grade.

In order to get the maximum credit, you need to SHOW ALL YOUR WORK.

For more information please see the Late reports and homework policy.

3.2 Exams

3.2.1 Mid Term Exam

1. Location: B202 (computer lab).

2. Time and duration: as scheduled

3. The exam contains questions and bonus. The questions add up to 100%. The bonus can add up to 20%. To get max grade you need to achieve 100%.

4. The exam is held in the computer lab such that you can make maximum use of the software.

5. The exam is close-book, close-notes but you are allow to bring one sheet of paper with what do you want written on it.

6. SHOW ALL YOUR WORK.

3.2.2 Course evaluation and Final Exam

1. Location: B202 (computer lab).

2. Time and duration: 14:00, 3 hours.

(We will allow 30 min at the beginning for logging in and preparation. During this time, you will also have the opportunity to fill in the course evaluations. Please be prepare, one of you, to volunteer to take them to the ME office at the end of the exam).

3. The exam contains questions and bonus. The questions add up to 100%. The bonus can add up to 20%. To get max grade you need to achieve 100%.

4. The exam is held in the computer lab such that you can make maximum use of the software.

5. The exam will be given to you at the beginning of the period in both paper and electronic formats. The exam in electronic format (Word document) will be on a floppy disc. As soon as you receive the paper exam and the disc, you should write your name on both.

6. You will be able to write by hand on the paper copy whatever you consider necessary to be done manually.

7. You will also be able to use the Word document to paste-in pictures from the Excel and Mathcad files, where required.

8. The paper exam and the electronic Word document, together, will make your submission. (If you chose to do everything electronically in the Word document, it's OK, just say so on the paper copy such that the grader knows).

9. You do not need to submit the Excel and Mathcad files. THE GRADER WILL NOT LOOK AT THE EXCEL AND MATHCAD FILES, BUT ONLY ON THE WORD DOCUMENT.

10. SHOW ALL YOUR WORK.



4. LABS

Labs to be done this semester:

�� Lab 1: Statistics/Probability (take home lab);

�� Lab 3: Electrical Measurements;

�� Lab 4: Strain Gauges;

�� Lab 5: Temperature Measurements

4.1 TA Assignments

Day TA email Office Phone Office Hours

Mon 10:00-11:00 AM Tues Yil Kim [email protected] A136 777-0753

Fri 10:00-11:00 AM

Tues 4:00-5:00 PM Wed Adrian Cuc [email protected] A236 777-0619

Thurs 4:00-5:00 PM

Please utilize with maximum benefit the TA’s office hours. All other requests to see a TA, outside the allocated time, are up to the TA’s availability. TA’s will try to accommodate the students but you must be aware that you are requesting a meeting outside the office hours and TA’s may not be available at the time you requested being busy with other work.

4.2 Lab guidelines and procedure

�� In the lab be aware of equipment safety and people working around you.

�� All lab reports are due at the beginning of the lab session. If you are late for lab, your report is late.

�� Everyone must attend the lab sessions. Failure to attend will result in a lab grade of zero. Emergencies will be handled on a case-by-case basis.

�� Equipment will be ready when you enter the lab. If you need additional equipment, please let the lab instructor know this.

�� Special rules will be posted and included in the lab procedure handouts.

�� If equipment breaks down or does not function properly, please inform the lab instructor.

�� Report ANY injuries to the TA immediately.

�� MSDS's (Material Data Safety Sheets) are in a yellow binder with red stripes, and can be obtained from the TAs

�� Any unsafe or deliberately harmful act can result in loss of grade and dismissal from the lab.

�� No entry is allowed into the lab more than 5 minutes after the beginning of the period.

�� If attendance at a different session is required, approval from both TAs is required before the lab session.



4.3 Report guidelines

Fill-in templates are provided, and can be downloaded from course website http://…….. The following general guidelines apply:

�� Use Report Template that can be downloaded from website.

�� 8½ x 11 unlined paper must be used, except for graph paper as needed.

�� Lines must be double-spaced, with ½ inch indent for paragraphs.

�� All pages, except the title page, must be numbered in the bottom center.

�� All margins are 1 inch.

�� Text in the body of the report must be 12 point Times New Roman.

�� Sections headings use 12 point bold all caps. Sub-headings use 12 point bold, upper and lower case.

�� References must be given when appropriate, and must be complete.

�� Grammar and spelling must be correct. Use MS Word Spell checker and Grammar checker with Technical style selected in Tools/Options.

�� Numbers less than 1 must have a leading zero (put zero before decimal point!).

�� Leave only one space between sentences.

�� Always put comma between items in all enumeration, even before ‘and’.

�� Report folders are required. Best are folders with a clear front or window that leaves the title and author's name visible. Head TA has report covers that you can borrow.

4.4 Contents

The following items must be present in ALL lab reports in this order:

Title page

Table of contents (on a separate page)

List of figure captions and list of table captions (on a separate page)

1. INTRODUCTION.

2. THEORY.

3. APPARATUS.

4. PROCEDURE

5. RESULTS.

6. DISCUSSION.

7. CONCLUSION.

8. ACKNOWLEDGMENTS.

9. REFERENCES – INCLUDE ONLY WHEN NEEDED.

10. APPENDICES – INCLUDE ONLY WHEN NEEDED.



�� If multiple groups of data exist, such as raw data and graphs of the results, put them in separate appendices. Multiple appendices may be lettered (Appendix A, Appendix B...), but all must appear in the table of contents.

�� Information within each appendix must follow all the general rules for formatting. They must be organized into tables, graphs, or figures. Graphs, tables, and figures must be numbered and captioned in the same manner as the body of the report.

�� A sample of a good lab report, from previous years is posted on the web. You will need Adobe Acrobat Reader to access it.

4.5 Equations

Equations should follow the conventional rules of equation formatting. Using notation tricks, such as "^" for exponents is not acceptable.

All equations must be numbered, with the number on the far right with parentheses. The text must refer to equations by number, not location (e.g., "results are found using Equation (3)", not "results are found using the following equation").

All variables in the equations must be explained in the text. If a variable is used the same way in several equations, it only needs explanation for the first usage.

4.6 Tables and figures

All tables and figures must have a number and a caption (e.g., Table 1 - list of equipment). The captions for figures are placed under the figures. The captions for tables are placed above the tables. The same figure or table number and caption must also appear in the list of figures and tables. All tables and figures must be referred to in the body of the report before they appear.

Graphs must follow the conventional rules of technical drafting. Both axes must be labeled, including units. When multiple sets of data are on a graph, they must be identified on the graph legend. Do not use the text of the report for identifying lines. Whenever there are multiple sets of similar data on the same graph use different symbols. This is the only easy way to allow comparison. If data is split across multiple graphs, they all need to have the same axis ranges.

Tables are appropriate for listing four or more data items to list. Do not make a table with two entries, or try to list 10 items in paragraph form. All tables must have column headers, and most will need row headers. Headers must include units when appropriate; do not put units inside the table. Significant digits must also be watched here. Do not give more digits than the accuracy of the data justifies. Display the same number of digits for all the items in a column.

4.7 Units

Keep units consistent; don't switch between SI and US units. The SI units are preferred. You may give SI units as primary and US units in parenthesis.



4.8 Late reports and homework

Reports and homework are due as specified in the schedule. Lab reports must be turned in when you enter the lab, before the lab starts. If the report or the homework is not handed in on time, it is late. Points are taken as follows:

�� Up to 24 hours from the due date, 10 points

�� 24 to 48 hours from the due date, 20 points

�� 48 to 72 hours from the due date, 30 points

�� More than 72 hours from the due date, 40 points

Late lab reports will not be accepted after graded reports have been returned. In most cases, lab reports are returned 1 week after they are turned in, but late labs will not necessarily be completed as quickly.

4.9 Grading

A copy of the grading guidelines sheet is attached to each lab instructions.

4.10 Cheating Policy

No cheating will be tolerated in this course. Although the students are encouraged to work together on labs, each report that is turned in must be your own work, your own writing. If cheating is suspected, the TAs and the instructor will discuss the problem immediately. Each incident will be dealt with in accordance to the University of South Carolina’s cheating policies found in the student handbook.

4.11 Class material

All class material (tools, books, etc) that will be handled to you during the semester must be returned in good shape by the end of semester. You will not be assigned a final grade if you did not return the material.



5. TECH TALK GUIDELINES

For the Tech Talk, you choose a subject from the list in section 5.3. You may use the presentation template available on the website. If you want to propose a new subject, contact your TA.

The Power Point file containing your presentation should be named as:

�� Last Name_First Name_TT.ppt

You must drop the file in the drop folder using the following link: \\me-giurgiutiu2\361DROP

5.1 Requirements.

�� The subject of the first talk must be from the list given in class.

�� Subjects must be given by the deadlines given in class.

�� A page listing the talk outline and at least 3 references must be provided at the beginning of the second talk.

�� Talks should target 8 minutes.

�� Talks completed in less than 6 minutes are penalized.

�� Talks WILL end at 10 minutes.

�� A question and answer session will follow each talk.

�� The Q&A session does not count toward the time limit.

�� Visual aids must be used, with overhead transparencies suggested.

�� Talks will include the following 3 parts :

1. Introduction, including the speaker's name and topic

2. Body of the talk

3. Conclusion

5.2 Things to remember:

�� Organize!

�� Show enthusiasm - if the speaker is not interested, the audience won't be.

�� Use the introduction to capture the interest of the audience.

�� Use clear SIMPLE visual aids. Don't allow them to be a distraction.

�� Give the audience adequate time to view visual aids, and no more.

�� Proof visual aids twice, then have someone else check them.

�� Don't apologize - if something is too poor to show, don't.

�� Talk to the audience – make eye contact.

�� Keep units consistent; don't switch between SI and US units. The SI units are preferred. You may give SI units as primary and US units in parenthesis.

�� Use cue cards - don't read the presentation.



�� Adjust the scope of your topic to meet the time requirements.

�� Target the intended audience - Mechanical Engineering juniors.

5.3 Available tech talk topics

1. Explain how to measure Thermal Conductivity

2. Explain how to measure Viscosity

3. Explain how to measure pH

4. Explain how seismic instruments work

5. Explain what is stress and how is it measured

6. Explain how to detect and measure nuclear radiation

7. Explain how octane ratings are obtained

8. Explain how a dynamometer is used to obtain a power curve

9. Explain how air pollution is measured

10. Give and explain Newton's three laws - with examples

11. Explain Heisenberg's Uncertainty Principle - relate it to EMCH 361

12. Explain insulation and how it is measured

13. Explain fatigue strength and how it is measured

14. Explain data acquisition on a PC using analog to digital converters

15. Explain Kirchoff’s voltage and current Laws.

16. How accurate are automobile speedometers?

17. Other technical topics of interest to you may be choose, but must be cleared with your TA

5.4 Grading

Note: This is the scoring method used for ASME Regional Student Conference Presentations.

Content

�� To what extent was the subject of interest to a technically oriented audience?

�� Was credit given to others when their work was cited or otherwise used?

�� Was the work reported independent and original?

�� Was the subject presented at an appropriate technical level for the audience?

Organization

�� Was there any novel approach to the subject?

�� Was the background information presented sufficient to properly introduce the subject?

�� Were facts developed in a logical and continuous sequence?

�� Did the speaker reach a definite conclusion and was it adequately based on the facts or data presented?



Delivery and Effectiveness

�� Were words pronounced clearly and distinctly?

�� Did the speaker use proper English?

�� Was the speaker's vocabulary sufficient?

�� Was personal appearance appropriate?

�� Did the speaker demonstrate any distracting mannerisms?

�� Was the manner of delivery satisfactory (memorized or read from a script)?

�� Were visual aids used effectively?

�� Did the speaker adhere to the prescribed time limits?

Discussion

�� Did the presentation evoke spontaneous questions from the audience?

�� Did questions indicate the need for clarification of the facts or seek additional information?

�� How readily and with what self-assurance did the speaker answer questions?

�� Did the answers indicate knowledge of the subject beyond the original presentation?

�� Did the speaker clearly demonstrate the ability to think?

A copy of the grading guidelines sheet follows on the next page.


Measurements and Instrumentation

Scoring Sheet for EMCH 361 Tech Talks Name __________________________________________________ Lab Section Subject _________________________________________________ Content Subject Matter: General or Technical (10%) ________ Personal Contribution: Library Research,

Independent Creative Project (10%) ________ Knowledge of Subject: Limited or Complete (10%) ________ Organization Introduction: Background (eliciting audience

interest), Objectives, Outline of Presentation (10%) ________ Continuity: Essential Facts Developed in Logical

Sequence (10%) ________ Conclusion: Definite and Based on Facts (10%) ________ Delivery and Effectiveness Vocal Delivery: Conversation vs. Memorized,

Proper Volume, Clarity, Pronunciation, Timing (10%) _________

Body Language: Eye Contact with Audience, Distracting Mannerisms (10%) _________

Visual Aids: Legibility, Effectiveness (10%) _________ Discussion (10%) _________ Total Score _________

Considerations in Judging 1. Content - To what extent was the subject of interest to a

technically oriented audience? Was credit given to others when their work was cited or otherwise used? Was the work reported independent and original? Was the subject presented at an appropriate technical level for the audience?

2. Organization - Was there any novel approach to the subject?

Was the background information presented sufficient to properly introduce the subject to the audience? Were facts developed in a logical and continuous sequence? Did the speaker reach a definite conclusion and was it adequately based on the facts or data presented?

3. Delivery and Effectiveness - Were words pronounced clearly and

distinctly? Did the speaker use proper English? Was the speaker's vocabulary sufficient? Was personal appearance appropriate? Did the speaker demonstrate any distracting mannerisms? Was the manner of delivery satisfactory (conversation memorized or read from the manuscript)? Were visual aids used effectively? Did the speaker adhere to the prescribed time limits?

4. Discussion - Did the presentation evoke spontaneous questions

from the audience? Did questions indicate the need for clarification of the facts presented or were they merely of the type seeking additional information? How readily and with what self-assurance did the speaker answer questions? Did the answers indicate knowledge of the subject beyond that disclosed in the original presentation? Did the speaker clearly demonstrate an ability to think?



6. STATISTICS/PROBABILITY OVERVIEW

TEXTBOOK REFERENCE:

Chapter 3, sections 1-5, 10-11, 14

6.1 Introduction

During experiments, some variation is inevitable. Operator variability, procedural variability, and equipment settings, may contribute to variation in the measurements. In the case of paper clip bending test, non-uniformity of the material, variability in the way you hold the specimen and perform the test, and other factors may contribute to the variability.

6.2 Statistics

For a sample of finite length, i.e. a data set (in our case, N=20), one calculates:

Median, which is the number in the middle of an ordered data set, i.e., half the members have values that are greater than the median, and half have values that are less. To calculate the median, one has to put its elements in increasing order. For an odd number of elements, the median is the element situated in the middle of the ordered set. For an even number of elements, the median is the average of the two elements in the middle.

Max, which is the largest value in the data set.

Min, which is the smallest value in the data set.

Sample mean (a. k. a., average): 1

1 N

ii

x xN

�

� � ; 1

1

1 classes

classes

N

j jj

N

jj

x NN

x

N

�

�

�

�

�

(1)

Sample standard deviation � �

2

1

1

N

ii

x

x xS

N�

�

�

�

� ;

2

1

1

( )

( ) 1

classes

classes

N

j jj

x N

jj

x x N

S

N

�

�

�

�

�

�

�

(2)

6.3 Probability

By definition, the probability that a certain member of the population, X, lies in the range (x1, x2] is given by

� �2

1

1 2 ( )x

x

P x X x = f x dx� � � (3)

Hence, the probability that a certain member of the population lies below a given value, x, is given by

� � � �( )x

F x P X x = f d� �

��

� � � (4)

The function F(x) is called the cumulative distribution. Note that, by definition of probability, ( ) 1F � � .



By substituting Equation (3) into (4), one can find the expression of F(x) for a normal (Gaussian) distribution in the form of the integral:

� �

2

221( )2

x

F x e d� �

� ��

�

�

��

� � (5)

However, calculating this integral every time is unnecessary. The textbook gives in Table 3.2 on page 62, numerical values for the function z-distribution which equals F(x) - 0.5. Table 3.2 is based on the standard normal distribution (Figure 1), which is obtained from Equation (3) by making the substitution

, and xz x z��

�

�

� � � � (6)

Figure 1 Standard normal distribution curve (Beckwith et al., 1995, pp. 61).

This give the expression

2

21( )2

z

f z e� �

�

� (7)

which is the same with Equation (3) with � = 0 and � = 1. The standard normal distribution has zero mean (� = 0) and unity standard deviation (� = 1).

For a given value z, the value of the z-distribution is the given by the area shown in gray in Figure 1. Its numerical values are listed in Table 3.2 (Beckwith et al., 1995, pp. 60). Because the curve is symmetric, the same table can be used to find values for negative z’s.

6.3.1 Example 1: Given: Assume that the population has a normal distribution with mean � = -4 and standard deviation � = 5.2, as shown in Figure 2.



Find the probability of a member lying between x1 = -15 and x2 = 6.

Solution: See table below. Use formula (7) to get z1 = -2.12 and z2 = 1.92. Look up the z-distribution in Table 3.2 and find corresponding values 0.4830 and 0.4726. These are the areas corresponding to z1 and z2, as measured from the z=0 axis. Since the areas are on opposite sides of the z=0 axis, they add. The total area is 0.9556. This means that the probability of a member of this population lying between x1 = -15 and x2 = 6 is 0.9556. In other words, 95.56% of the members are lying in the interval (-15, 6].

x1 x2 Total

x -15 6

z -2.12 1.92

z-distr 0.4830 0.4726 0.9556

6.3.2 Example 2: Given: Assume that the population has a normal distribution with mean � = -4 and standard deviation � = 5.2, as shown in Figure 2.

Find the probability of a member lying between x1 = 15 and x2 = 6.

Solution: See table below. Use formula (7) to get z1 = 1.92 and z2 = 3.65. Look up the z-distribution in Table 3.2 and find corresponding values 0.4726 and 0.4999. These are the areas corresponding to z1 and z2, as measured from the z=0 axis. Since the areas are both on the positive z side, they subtract. The resulting total area is 0.0273. This means that the probability of a member of this population lying between x1 = 6 and x2 = 15 is 0.0273. In other words, only 2.73% of the members are lying in the interval (6, 15].

x1 x2 Total

x 6 15

z 1.92 3.65

z-distr 0.4726 0.4999 0.0273

0.0000

0.0200

0.0400

0.0600

0.0800

-40 -30 -20 -10 0 10 20 30 40

Figure 2 Normal distribution with mean � = -4 and standard deviation � = 5.2 used in the examples.



6.3.3 Example 3: Given: Assume that the population has a normal distribution with mean � = -4 and standard deviation � = 5.2.

Find the interval in which 90% of the population lies.

Solution: See table below. Convert from % to decimal, and get that the required probability is 0.9000. We are looking for an area on the standard normal distribution that equal 0.9000. Assume that the interval is symmetric about the mean, and split the total value 0.9000 into two 0.4500 values. We need to find the value of z that gives 0.4500 in Table 3.2. By inspection of Table 3.2, we identify that this value lies between 0.4595 corresponding to 1.64z � and 0.4505, corresponding to 1.65z � . Hence, conclude that the required value of z is 1.645z � . This means that the interval is (-1.645; 1.645). Use Equation (7) to obtain the values x- and x+ corresponding to the negative and positive values of z. i.e., x- = -12.544 and x+ = 4.554. We conclude that 90% of the population lies in the interval (-12.554, 4.554). This fact can also be expressed as: 4.000 8.544x � � � for 90% of the population.

On the other hand, most mathematical software packages have the functions f(x) and F(x) built in as functions. For example, MS Excel has the function NORMDIST, which returns either f(x), or F(x), depending on the value of a selection (logical) parameter. (Explore and enjoy the MS Excel Help !!! You will find this much more fun and easier to use).

6.4 Confidence Intervals

In most cases, we know the mean, x , and standard deviation, xS , for certain samples, but we do not have the means to measure the entire population. However, we choose to approximate the population mean, �, and population standard deviation, �, with the values determined for the sample. This means that we make the approximations:

x

xS

�

�

�

� (8)

How good are these approximations? How much confidence shall we have in these numbers being close to the true values. The answer to these questions is given by the confidence intervals.

6.4.1 Confidence intervals for large samples If the sample is large (N>30), the confidence intervals are calculated as follows:

Confidence interval for the mean:

/ 2 / 2x x

c cS Sx z x zN N

� � � �� (9)

where c is the confidence with which we want to determine the interval. The quantity /xS N is called standard error.

Confidence interval for the standard deviation:

� � � �2 2

22 2

/ 2 1 / 2

1 1x xN S N S

�

� �

� �

� �

��

(10)

+ -

Area 0.4500 0.4500

z 1.645 -1.645

x 4.554 -12.554



where � = 1-c. The functions 2/ 2�

� and 21 / 2��

� , called chi squared, are given in Table 3.5, pp. 69 in the textbook, as functions of �� and of sample degrees of freedom, � = n-1.

6.4.2 Example 4 Given: Assume that a sample of N = 20 measurements has produced a sample mean x = -3.95 and a sample standard deviation Sx = 5.02.

Find: Determine with confidence of 95% the actual mean and standard deviation of the population.

Solution:

(a) Confidence interval for the mean value: The value of c is c = 0.95. Hence c/2 = 0.475. Table 3.2 give the corresponding z value as zc/2 = 1.96. Hence, using Equation (10),

5.02 5.023.95 1.96 3.95 1.9620 20

�� (11)

After performing the arithmetic, one gets:

6.15 1.75�� (12)

This means that we can say, with 95% confidence, that the population mean lies in the interval (-6.15, -1.75). Another way of saying this is that, with 95% confidence, the population mean is �95% = -3.95� 2.2.

(b) Confidence interval for the standard deviation: Calculate � = 1 - 0.95 = 0.05. Then, look up in Table 3.5 under � = 20-1 = 19, for the values of 2

0.025X = 32.852 and 20.975X = 8.907. Then, using Equation (11),

� � � �2 2220 1 5.02 20 1 5.02

32.852 8.907�

� �

� � (13)

After performing the arithmetic, one gets:

214.575 53.756�� (14)

After taking the square root,

3.818 7.332�� (15)

This means that we can say, with 95% confidence, that the population standard deviation lies in the interval (3.818, 7.332). Another way of saying this is that, with 95% confidence, the population mean is �95% = 5.575+/- 1.757.

6.4.3 Confidence intervals for small samples If the sample is small (N<30), the confidence intervals are calculated as follows:

Confidence interval for the mean:

/ 2, / 2,x xS Sx t x tN N

� � � ��

� (16)

where t�/2,� is the Student distribution. This distribution is given in the textbook.

On the other hand, most mathematical software packages have built functions for this analysis. Explore and enjoy the MS Excel Help !!! You will find this much more fun and easier to use.



6.5 Excel statistical functions

AVEDEV Returns the average of the absolute deviations of data points from their meanAVERAGE Returns the average of its arguments

BETADIST Returns the cumulative beta probability density function

BETAINV Returns the inverse of the cumulative beta probability density function

BINOMDIST Returns the individual term binomial distribution probability

CHIDIST Returns the one-tailed probability of the chi-squared distribution

CHIINV Returns the inverse of the one-tailed probability of the chi-squared distribution

CHITEST Returns the test for independence

CONFIDENCE Returns the confidence interval for a population mean

CORREL Returns the correlation coefficient between two data sets

COUNT Counts how many numbers are in the list of arguments

COUNTA Counts how many values are in the list of arguments

COVAR Returns covariance, the average of the products of paired deviations

CRITBINOM Returns the smallest value for which the cumulative binomial distribution is less than

or equal to a criterion valueDEVSQ Returns the sum of squares of deviations

EXPONDIST Returns the exponential distribution

FDIST Returns the F probability distribution

FINV Returns the inverse of the F probability distribution

FISHER Returns the Fisher transformation

FISHERINV Returns the inverse of the Fisher transformation

FORECAST Returns a value along a linear trend

FREQUENCY Returns a frequency distribution as a vertical array

FTEST Returns the result of an F-test

GAMMADIST Returns the gamma distribution

GAMMAINV Returns the inverse of the gamma cumulative distribution

GAMMALN Returns the natural logarithm of the gamma function, G(x)

GEOMEAN Returns the geometric mean

GROWTH Returns values along an exponential trend

HARMEAN Returns the harmonic mean

HYPGEOMDIST Returns the hypergeometric distribution

INTERCEPT Returns the intercept of the linear regression line

KURT Returns the kurtosis of a data set

LARGE Returns the k-th largest value in a data set

LINEST Returns the parameters of a linear trend

LOGEST Returns the parameters of an exponential trend

LOGINV Returns the inverse of the lognormal distribution



LOGNORMDIST Returns the cumulative lognormal distributionMAX Returns the maximum value in a list of arguments

MEDIAN Returns the median of the given numbers

MIN Returns the minimum value in a list of arguments

MODE Returns the most common value in a data set

NEGBINOMDIST Returns the negative binomial distribution

NORMDIST Returns the normal cumulative distribution

NORMINV Returns the inverse of the normal cumulative distribution

NORMSDIST Returns the standard normal cumulative distribution

NORMSINV Returns the inverse of the standard normal cumulative distribution

PEARSON Returns the Pearson product moment correlation coefficient

PERCENTILE Returns the k-th percentile of values in a range

PERCENTRANK Returns the percentage rank of a value in a data set

PERMUT Returns the number of permutations for a given number of objects

POISSON Returns the Poisson distribution

PROB Returns the probability that values in a range are between two limits

QUARTILE Returns the quartile of a data set

RANK Returns the rank of a number in a list of numbers

RSQ Returns the square of the Pearson product moment correlation coefficient

SKEW Returns the skewness of a distribution

SLOPE Returns the slope of the linear regression line

SMALL Returns the k-th smallest value in a data set

STANDARDIZE Returns a normalized value

STDEV Estimates standard deviation based on a sample

STDEVP Calculates standard deviation based on the entire population

STEYX Returns the standard error of the predicted y-value for each x in the regression

TDIST Returns the Student's t-distribution

TINV Returns the inverse of the Student's t-distribution

TREND Returns values along a linear trend

TRIMMEAN Returns the mean of the interior of a data set

TTEST Returns the probability associated with a Student's t-Test

VAR Estimates variance based on a sample

VARP Calculates variance based on the entire population

WEIBULL Returns the Weibull distribution

ZTEST Returns the two-tailed P-value of a z-test

emch 361 – fall 2001 measurements and instrumentation 361 handout gen … · emch 361 – fall...

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