Unit 1: Scientific Notation, Data Analysis, and Experimental Error This unit consists of a review of some basic concepts you may remember from courses in mathematics and experimental science. You may skim through the material, but you will need to be precise in later work about the nomenclature used for reporting errors and statistics. The reference immediately below is a handbook for engineering statistics; it is not meant to be read from start to finish but rather to be used as a reference for specific problems at hand. You may wish to familiarize yourself with the nomenclature and organization of the handbook.
Unit 1 Time Advisory
This unit should take you 19 hours to complete.
☐ Subunit 1.1: 4 hours
☐ Subunit 1.2: 6 hours
☐ Subunit 1.3: 1 hour
☐ Subunit 1.4: 2 hours
☐ Subunit 1.5: 2 hours
☐ Subunit 1.6: 1 hour
☐ Subunit 1.7: 2 hours
☐ End of Unit Self-Assessment: 1 hour
Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Use scientific notation for engineering calculations.
- Compute mean and standard deviation and understand their significance.
- Compute how uncertainty in parameters propagates through simple calculations.
Perform elementary regression for parameter estimation (e.g. linear least-squares analysis).
Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010) Link: NIST/SEMATECH’s e-Handbook of Statistical Methods(2010) (PDF)
Also available in:Instructions: Use this handbook as a resource throughout the course. Peruse the introductory material (“How To Use this Handbook” and “Tools and Aids”) at this stage in order to facilitate later use.
Terms of Use: This material is in the public domain.
1.1 Numerical Precision
1.1.1 Scientific Notation
- Reading: All About Circuits’ “Scientific Notation”
Link: All About Circuits’ “Scientific
Notation”
(PDF)
Instructions: Read the page linked above. You may skim through the
page if it is review for you. To view as a PDF, click the PDF link
in the top right corner.
Terms of Use: This material has been released under the terms of
the Design Science
License.
- Web Media: Khan Academy’s Scientific Notation 1; Scientific
Notation 3 (new); and Scientific Notation 3
Link: Khan Academy’s Scientific Notation
1 (YouTube); Scientific
Notation 3
(new) (YouTube);
and Scientific Notation
3
(YouTube)
Instructions: View these three videos to review the details of scientific notation and arithmetical operations. You may skim through these topics if they are review for you.
Terms of Use: These videos are licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. They are attributed to the Khan Academy.
1.1.2 Significant Figures
- Reading: Connexions: Sunil Kumar Singh’s “Significant Figures”
Link: Connexions: Sunil Kumar Singh’s “Significant
Figures”
(PDF)
Also available in:
[PDF](http://cnx.org/content/m15032/1.3/content_info#cnx_downloads_header)
[EPub Format](http://cnx.org/content/m15032/1.3/?format=epub)
Instructions: Read these notes and pay particular attention to the
effect of mathematical operations upon significant figures. For
example, consider how many significant figures might be in the
result of the operation 1.23(4.4/6,873 +2.0).
Terms of Use: This work is licensed under a [Creative Commons
Attribution 2.0 Generic
License](http://creativecommons.org/licenses/by/2.0/). It is
attributed to Sunil Kumar Singh and can be found in its original
form [here](http://cnx.org/content/m15032/latest/).
1.2 Statistics
1.2.1 Introduction to Statistics
- Reading: National Institute of Standards and Technology (NIST):
Dr. W. J. Youden’s Experimentation and Measurement
Link: National Institute of Standards and Technology (NIST): Dr. W.
J. Youden’s “Experimentation and
Measurement”(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-0-W.J.-Youden.epub)
Instructions: Click the link for the PDF *Experimentation and
Measurement* under “Calibration Related Publications.” Read
Chapters 4 (*Typical Collections of Measurements*) and 5
(*Mathematics of Measurement*).
Terms of Use: This material is in the public domain.
1.2.2 Mean
- Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of Location”
Link: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of
Location”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-1.1-National-Institute-of-Standards.epub)
Instructions: Read the linked section above on descriptors of
averages. Note that there are other descriptors (besides the mean
and median) that may be useful. As you read, you may wish to
consider hypothetical cases in which the median might be more useful
than the mean.
Terms of Use: This material is in the public domain.
1.2.3 Variance and Standard Deviation
- Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of Scale”
Link: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of
Scale”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-1.1-National-Institute-of-Standards.epub)
Instructions: Read the linked section above, which presents
descriptors of width. The most commonly used descriptors are
variance and standard deviation.
Terms of Use: This material is in the public domain.
1.2.4 Skewness and Higher Moments
- Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of Skewness and Kurtosis”
Link:NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Measures of Skewness and
Kurtosis”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-1.1-National-Institute-of-Standards.epub)
Instructions: Read this section on the higher moments of a
distribution. Under what circumstances might the higher moments be
significant?
Terms of Use: This material is in the public domain.
1.3 The Normal or Gaussian Distribution
- Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Normal Distribution”
Link: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“Normal
Distribution”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-1.1-National-Institute-of-Standards.epub)
Instructions: Read the linked section on the properties of the
Normal or Gaussian distribution. Pay attention to the significance
of the mean and standard deviation to the position and width of the
distribution.
Terms of Use: This material is in the public domain.
1.4 Sources of Error
- Reading: Simon-Fraser University: Stephen Lower’s Chem1 General
Chemistry Virtual Textbook: “The Meaning of Measure: Dealing with
Error and Uncertainty in Measured Values”
Link: Simon-Fraser University: Stephen Lower’s Chem1 General
Chemistry Virtual Textbook: “The Meaning of Measure: Dealing with
Error and Uncertainty in Measured
Values”(PDF)
Instructions: Review this section, ensuring that you are able to
distinguish between random error, systematic error, accuracy, and
precision. This reading addresses sections 1.4.1 and 1.4.2. To
view as a PDF, click the “download” link in the bottom right
corner.
Terms of Use: This work is licensed under a Creative Commons
Attribution-NonCommercial 2.5 Generic
License. It is
attributed to Stephen Lower and can be found in its original
form here.
1.4.1 Systematic Error and Accuracy
1.4.2 Random Error and Precision
1.5 Error Propagation
- Reading: University of Toronto: David Harrison’s “Error Analysis
in Experimental Physical Science”
Link: University of Toronto: David Harrison’s “Error Analysis in
Experimental Physical
Science”
(PDF)
Instructions: Read the linked section above. Answer questions
9.1-9.4; ignore Exercise 9.1. This section is concise; you may wish
to practice the exercises a few times.
Terms of Use: This work is licensed under a Creative Commons
Attribution-NonCommercial-ShareAlike 2.0 Generic
License. It is
attributed to David Harrison and can be viewed in its original
form here.
1.6 Parameter Estimates and Confidence Intervals
- Reading: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“How Are Estimates of the Unknown Parameters Obtained?”
Link: NIST/SEMATECH’s e-Handbook of Statistical Methods (2010):
“How Are Estimates of the Unknown Parameters
Obtained?”
(PDF)
Also available in:
[EPUB](http://www.saylor.org/site/wp-content/uploads/2011/07/ME301-1.1-National-Institute-of-Standards.epub)
Instructions: Read this section and the links titled “Least
Squares” and “Weighted Least Squares.” This information may seem
rather technical at the moment, but as you gain experience with
using the methods, it will make more sense to you.
Terms of Use: This material is in the public domain.
1.7 Exercises
1.7.1 Calculation of Mean, Standard Deviation, and Variance
- Reading: College of St. Benedict and Saint John’s University:
“Descriptive Statistics”
Link: College of St. Benedict and Saint John’s University:
“Descriptive
Statistics”
(PDF)
Instructions: Read the linked section above. Perform the
calculations of statistics as prompted. You may wish to experiment
with peculiar distributions exhibiting distinctive shapes (such as
bimodal, triangular, or square distributions) and compare them with
other distributions.
Terms of Use: The material above has been reposted with permission
for educational use by Thomas W Kirkman. It can be viewed in its
original
form here.
1.7.2 Linear Least-Squares Estimates of Slope and Intercept
- Reading: Yale University Department of Statistics’ “Inference in
Linear Regression”
Link: Yale University Department of Statistics’ “Inference in
Linear
Regression”
(HTML)
Instructions: This section caters to those with an abstract,
mathematical bent. You can develop some appreciation for the
concepts involved by skimming the text and studying the graphs as
examples rather than by trying to understand the details of the
mathematical analysis. Note that this material may be more
appealing after you have some experience using the methods.
Terms of Use: Please respect any copyright and terms of use
displayed on the webpage above.
1.7.3 Nonlinear Regression
- Web Media: University of Colorado’s PHET Curve-Fitting
Demonstration Package: “Curve Fitting”
Link: University of Colorado’s PHET Curve-Fitting Demonstration
Package: “Curve
Fitting”
(Adobe Flash)
Instructions: Drag data points from the data-point-bin onto the
graph. Adjust the error bars. Examine the best-fit curves for
different situations. You should play with this exercise to get an
intuitive feel for how data and error bars affect the best fit
curves. Please also try to fit data with different types of curves
(for example, lines and higher order polynomials). By playing
around with this online demo, you can quickly obtain experience that
can be easily applied to concrete situations.
Terms of Use: This material is licensed under the GNU General
Public License v2.0. It
is attributed to PhET Interactive
Simulations, University of Colorado and
the original version can be found here.
Unit 1 Assessment
- Assessment: The Saylor Foundation's "ME301: Unit 1 Quiz"
Link: The Saylor Foundation’s “ME301: Unit 1
Quiz”
Instructions: Please complete the linked assessment.
You must be logged into your Saylor Foundation School account in
order to access this exam. If you do not yet have an account, you
will be able to create one, free of charge, after clicking the link.
This quiz should require less than one hour to complete.