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