1. Probability (Ch. 6)
Probability: “…the chance of occurrence of an event in an experiment.” [Wheeler & Ganji]
Chance: “…3. The probability of anything happening; possibility.” [Funk & Wagnalls]
A measure of how certain we are that a particular outcome will occur.

2. Probability Distribution Functions
Descriptors of the distribution of data.
Require some parameters:
_______, _______________.
Degrees of freedom (__________) may be required for small sample sizes.
Called “probability density functions” for continuous data.
Typical distribution functions:
Normal (Gaussian), Student’s t.
average
standard deviation
sample size

3. Probability Density Functions
Suggests integration!
Normal Probability Density Function:
=0
=1

4. Normal Distributions
Let
Transform your data to zero-mean, =1, and evaluate probabilities in that domain!

5. Normal Distribution
Standard table available describing the area under the curve from “0 to z” for a normal distribution. (Table 6.3 from Wheeler and Ganji.) So, if you want X%, look for (0X/2).

6. Student’s t Distribution
Data with n30.
Result we’re looking for:
a/2
a/2
w/ confidence:
ta/2
-ta/2
How do we get ta/2?
Based on calculating the area of the shaded portions.
Total area = a.

7. Student’s t Distribution

8. Chapter 7 Uncertainty Analysis
Student Pages:
1,2,3,4,5,6
8,9,10,12,13,14
15,16,17,18,20,23
In PDF file leave out 6, 10

9. Plot X-Y data with uncertainties
Where do these come from?

10. Significant Digits
In ME 360, we will follow the rules for significant digits
Be especially careful with computer generated output
Tables created with Microsoft Excel are particularly prone to having…
- excessive significant digits!

11. Rules for Significant Digits
In multiplication, division, and other operations, carry the result to the same number of significant digits that are in the quantity used in the equation with the _____ number of significant digits.
least
234^2 =
>
54800
If we expand the limits of uncertainty:
233.5^2 =
>
54520
234.5^2 =
>
54990

12. Rules for Significant Digits
In addition and subtraction, do not carry the result past the ____ column containing a doubtful digit (going left to right).
first
“doubtful” digits
“doubtful” digits
1270.2
383700

13. Rules for Significant Digits
In a lengthy computation, carry extra significant digits throughout the calculation, then apply the significant digit rules at the end.
As a general rule, many engineering values can be assumed to have 3 significant digits when no other information is available.
(Consider: In a decimal system,
three digits implies 1 part in _____.)
1000

14. Sources of Uncertainty
Precision uncertainty
Repeated measurements of same value
Typically use the ____ (±2S) interval
___ uncertainty from instrument
Computed Uncertainty
Technique for determining the uncertainty in a result computed from two or more uncertain values
95%
Bias

15. ux = % of reading + n digits
Instrument Accuracy
Measurement accuracy/uncertainty often depends on scale setting
Typically specified as
ux = % of reading + n digits
Example:
DMM reading is 3.65 V with uncertainty (accuracy) of ±(2% of reading + 1 digit):
ux =± [ ] =
(0.02)*(3.65) +
(0.01)
±[ ] =
±0.083 V
DON’T FORGET!

16. Instrument Accuracy
Data for LG Precision #DM-441B True RMS Digital Multimeter
What is the uncertainty in a measurement of volts (DC)??
pass out multimeter of each - see Barry or Dan for analog type?

17. DMM (digital multimeter)
For DC voltages in the 2-20V range, accuracy =
±0.1% of reading + 4 digits
4 digits in the least significant place
First “doubtful” digit

18. DMM (digital multimeter)
What is the uncertainty in a measurement of volts AC at 60 Hz?
For AC voltages in the 2-20V, 60 Hz range, accuracy =
±0.5% of reading + 20 digits
First “doubtful” digit - ending zeros to the right of decimal points ARE significant!

19. Sources of Uncertainty
Precision uncertainty
Repeated measurements of same value
Typically use the ____ (±2S) interval
___ uncertainty from instrument
Computed Uncertainty
Technique for determining the uncertainty in a result computed from two or more uncertain values
95%
Bias

20. Uncertainty Analysis #1
We want to experimentally determine the uncertainty for a quantity W, which is calculated from 3 measurements (X, Y, Z)

21. Uncertainty Analysis #2
The three measurements (X, Y, Z) have nominal values and bias uncertainty estimates of

22. Uncertainty Analysis #3
The nominal value of the quantity W is easily calculated from the nominal measurements,
What is the uncertainty, uW in this value for W?

23. Blank Page (Notes on board)

24. Uncertainty Analysis #4
To estimate the uncertainty of quantities computed from equations:
Note the assumptions and restrictions given on p. 182! (Independence of variables, identical confidence levels of parameters)

25. Uncertainty Analysis #5
Carrying out the partial derivatives,

26. Uncertainty Analysis #6
Substituting in the nominal values,

27. Uncertainty Analysis #7
Substituting in the nominal values,
Square the terms, sum, and get the square-root:

28. Uncertainty Analysis #12
Simplified approach:

29. Uncertainty Analysis #14
Which of the three measurements X, Y, or Z, contribute the most to the uncertainty in W?
If you wanted to reduce your uncertainty in the measured W, what should you do first?

30. Exercise #1a
Experimental gain from an op-amp circuit is found from the formula
Compute the uncertainty in gain, uG, if both Ein and Eout have uncertainty:

31. Exercise #1c
Equation:

32. Exercise #1d
Answers:

33. Exercise #2
What is the uncertainty in w if E, M, and L are all uncertain?

34. Exercise #2a
Show that

35. Exercise #2b
Base form
Simplified form

36. Exercise #2c
Compute the nominal value for w and the uncertainty with these values:

37. Combining Bias and Precision Uncertainties
Use Eqn (p. 165)
generally compute intermediate uncertainties at the 95% confidence level