ME 322: Instrumentation Lecture 2 January 23, 2015 Professor Miles Greiner Quad measurement calculations and results for Lab 2, Probability Distribution Functions, Examples (symmetric, one-sided)
ME 322: InstrumentationLecture 2
January 23, 2015
Professor Miles Greiner
Quad measurement calculations and results for Lab 2, Probability Distribution Functions,
Examples (symmetric, one-sided)
Announcements• Lab 1 work sheet due now• HW 1 due Monday– Use ME 322 ID number (from WebCampus), not name
• If you have requested to move to a new section – we will send an email to let you know if we were able to
fulfil you request• Extra-Credit Opportunity– Help out at Science Olympiad• Saturday, February 7, 2015, 9:00am to 3:00pm• ~1% course grade extra credit
– ~One homework or lab assignment– ~4 points on a test
• Sign up by Wednesday, January 28, 2015
Results of the Quad Measurements• Data is on Lab 2 website• How to process that data,
– You will repeat and present this in Lab 2, Analysis of Quad Measurement• Spreadsheet has Measured Data and room for Calculations
– H, DC , NCi, NCA, F = DC/NCA
– NSi, NSA, S=F*NSA
– NLi, NLA, L=F*NLA – A = LS– C = A*($3.49/200 ft2)
• How to Plot– F versus H; L vs S (scatter plots)– Cost Estimate Histogram (install analysis toolpack; file, options, add-in)
• Questions– Is stride length F highly correlated with height, H?– What are the distributions, sample mean and standard deviation of the cost estimates?
• Are any of them “out of place?”– Are the measured values of L and S correlated? Should they be?– If you budget the amount of your cost estimate, you are only 50% sure to have enough to cover
quad (be above the average value, which we assume is the most accurate estimate)• How much money should you budget to be 90% sure to have enough
Randomly Varying Processes• The output of a measurement instrument is affected by the
measurand (the quantity being measured) and many uncontrolled (undesired) factors
• Consider a process (such as a measurement) that has a very large number of factors that can, independently, increase or decrease the value of the outcome
• Its not likely that all or a large majority of the factors will push the outcome in the same direction.
• Its more likely that “most” of the factors will cancel each other, and push the outcome only “slightly” in one direction or the other.
• This describes how uncontrolled factors affect the output of measurement systems (instruments)
Gaussian (Normal) Probability Distribution Function
• Describes Randomly Varying Processes• Looks like the pattern observed from the cost estimate
histogram in Labs 1 and 2– We were able to estimate m ~ and s ~ for that data
𝑓 (𝑥 ;𝜎 ,𝜇 )= 1𝜎 √2
𝑒(− 1
2𝑥−𝜇𝜎
)2
How can we use this?• If a sample is very large, and if the process variations
are “normally” distributed, – Then expect sample histogram to take a bell shape, – And, if we know s and m, the probability that the next
measurement x will be in the range x1 < x < x2 is
• Note, for any s and m :∫−∞
∞
𝑓 (𝑥 ;𝜎 ,𝜇)𝑑𝑥=1=100 %
Non-Dimensionalization
• Define– Number of standard deviations x is above the mean
• We can show that the probability that the next measurement is between z1 and z2 is:– – Where
– This integral is tabulated on page 146, for z > 0
Graphical RepresentationArea from center (z = 0) to z
For z > 0 :
Note:
This integral is tabulated on page 146, for z > 0
For negative values of z
For z1 < 0 :
¿ 𝐼 (𝑧2 )−[− 𝐼 (−𝑧1 ) ]𝑃 (𝑧1<𝑧<𝑧2 )=𝐼 ( 𝑧2 )− 𝐼 (𝑧 1)
𝐼 ( 𝑧1 )=− 𝐼 (−𝑧 1)
Symmetric Example
Find the Probability a measurement is within one standard deviation (s) of the mean (m).
= = -1
= = 1
= =
Page 146
𝑃 (−1<𝑧<1 )=2 𝐼 (1 )=2∗0.3413=0.6826=68. 26 %
Next measurement is within 2s and 3s of the mean
𝑃 (−2<𝑧<2 )=2 𝐼 (2 )=2 ( .4772 )=95.44 %
One-sided example• From Lab 2, what seed cost will cover (be greater than) 90%
of all future estimates?
• One-sided example• P=0.9= I(z2)-I(z1)
• z2 = ? But z1-∞, so I(-∞)= -I(∞)=-0.5
• So P = 0.9 = I(z2) – [-0.5]
• 0.4 = I(z2)
• Interpolate between z2 = 1.28 and 1.29
• Get z2 = 1.2817
Lab 2
• If you make a measurement, there is a 50% likelihood it is below the mean (best) value.
• How much should you add to your best estimate to be 90% you are above the mean?
• Answer: 1.282 standard deviations
Extra Slides
Area of UNR Quad
• Find Short Side (S)– NSi
– NSA
• Find Long Side (L)– NLi
– NLA