Chapter V. Uncertainty Analysis in Experiments

Experimental Approach

1. What is the problem ? (What questions are we trying to answer ?)

2. How accurately do we need to know the answer ? (where to use)

3. What physical principles are involved ? (governing phsical laws)

4. What experiment or set of experiments might provide the answer ?

5. What variables must be controlled ?   How well ?

6. What quantities must be measured ?  How accurately ?

7. What instrumentataions is to be used ?

8. How are the data to be acquired, conditions, and stored ?

9. How many data points must be taken ?  In what order ?

10. Can the requirements be satisfied within the budget and time

constraints ?

11. What techniques of data analysis should be used ?

12. What is the most effective and revealing way to present the data ?

13. What unanticipated questions are raised by the data ?

14. In what manner should the data and results be reported ?

1. Importance of Uncertainty Analysis

☞ hardly present an experimental result without clear explanation of uncertainty

☞ The uncertainty analysis is a rational tool for an experimenter to assess the

importance of his scatter data in repeated experiment.

Also, it can support the data user to decide the coincidence of the data with

other experiments in the experimental range.

◦ what the uncertainty analysis is

◦ what it can do

◦ how to interpret its results

(Ex) Archimedes decision for 18-karat gold crown

[eqn: rm &rho_gold ~=~ 15.5 ~gm/cm^3#rm &rho_alloy ~=~ 13.8 ~gm/cm^3]

Two measurements

	Expert A (quick measurement)	Expert B (precise measurement)
Best estimate Probable range	15 13.5 ～ 16.5	13.9 13.7 ～ 14.1

- Since the range where both experimenter confide is overlapped, possible to say that

both are “correct".

- A has a wide range uncertainty ∴ maybe "useless"

- B shows the crown is not made of pure gold.

2. Definitions

(1) accuracy [eqn: ==] closeness of agreement between a measured and the true value

(2) error [eqn: ==] difference between its true value and the measured value

- degree of inaccuracy

- calibrate experiments against baseline data or against one of the basic conservation         laws.

▶ clear but not very useful

total error(δ) = bias error(β) + precision error(ε)

① bias error ; fixed, systematic, or constant component of total error

② precision error ; random component of total error (repeatability error)

For each measurement,

[eqn: rm delta_i ~=~ beta + epsilon_i]

(3) uncertainty [eqn: ==]  a possible value that an error may have

"What we think the error would be if we could and did measure it by calibation."

⇨ an estimate of experimental error

(4) uncertainty analysis [eqn: ==] Process to estimate the effect of uncertainties in each

experiment on the final result

◦ drawing error bars

◦ calculate rms deviation of data from a curve fit

- statistical interpretation of the errors in well-replicated experimental results.

Since, it is not possible to specify the exact bias and precision errors, we are C %          confident that the true value of X lies within the interval

[eqn: rm X_best ~under+ ~U_X]

where [eqn: rm X_best] is usually assumed to be the mean value of N reading.

Consider

◦ single-sample data  ;  1 pressure gage no matter how many readings.

◦ multi-sample data  ;  more than one gage

① number of independent data or observation at each test point

② data-sampling rate

③ the spectrum of significant frequency

④ time of data-averaging for processing

1) Single sample uncertainty

☞ the zeroth-order uncertainty : evaluates the contribution to the total uncertainty                                        introduced by the measuring system

☞ the first-order uncertainty : predicts the scatter that should be observed on                                           repeated trials with the same equipment and the                                        same instruments.

☞ the N-th order uncertainty : the overall uncertainty in the experiments as it                                           was run.

2) Multiple sample uncertainty

; produces estimates of the total fixed error and the total random error in the result.