2.1.1 What is the significance of measurements, why do we experimentally measure parameters?

• Obtain quantitative information on the actual state of physical variables and processes (i.e. T, P, V, etc.), this is critical especially for thermodynamics and finding thermodynamic equilibrium

• Vital for control processes which require measured discrepancy between actual and desired output (i.e. temperature control on sheet annealing line)

• Daily operations (i.e. temperature controls, heating and cooling, tire gauge, oil gauge, etc.)

• Establishing costs, Ti $140/lb precision is important when dealing with tons of Ti.

• Temperature in bioreactor can influence cell growth

• Thermal conductivity, acceleration, material properties

2.4 Error and Uncertainty:

Error is the difference between the measured and true value of the quantity being measured. Unfortunately, the true value can never really be known so instead we quantify the uncertainty of a measurement at specific levels of confidence. Error can be categorized into two basic categories either bias or precision error/uncertainty. We will cover uncertainty and error analysis much more in-depth in the next several lectures...but never use percent error in this course.

Error

We can begin with our discussion of error by defining the following equation

\begin{equation}
Error = x_{m} - x_{true}
\end{equation}

where \(x_m\) is the measured value and \(x_{true}\) is the true value.

When designing an experimental system or executing an experiment the experimentalist’s primary objective should be to minimize the error. After the experiment is completed we need to estimate a bound on the error with a degree of certainty or a confidence interval (i.e. 90%, 95%, 99%), hopefully you have encountered confidence intervals before but if not no worries we will discuss this in depth.

There are many different sources of inaccuracy or error however most errors can be classified as:

(1) Precision

(2) Bias

(3) Illegitimate

(4) Errors that are sometimes bias or sometimes precision errors

In this course we will spend a considerable amount of time discussing 1. Precision and 2. Bias error in great detail. But we will discuss all these sources of error and how to account for them when designing experiments or presenting your results. Let’s start with Precision Error.

3. Illegitimate Error

Illegitimate error can and most likely will occur to you at some point as you perform experiments as an engineer. Illegitimate errors include

• Blunders and mistakes

• Computational or calculation errors

• Analysis errors

Figure \(\PageIndex{2}\): Bias Error Distribution

While these things can occur in lab you must always perform the experiment again. You cannot simply present your results in either a presentation or a form of technical writing and attribute error to illegitimate error. You must perform the experiment again.

4. Errors That Are Sometimes Bias and Sometimes Precision Error:

These are the most difficult sources of error that we must analyze as it is difficult to determine what type of error analysis must be done and which error has a distribution and what does not.

Quantifying Total Uncertainty, Comprehensive Error Analysis

Again typically we are concerned with bias and precision error when evaluating experimental data. And this data comes from two classes of experiments: i) single- sample and ii) repeated-sample experiments.

A sample refers to an individual measurement of a specific quantity. Ex. Measuring strain in a material several times under identical loading is a repeated-sample experiment. A single measurement of strain is a single-sample experiment and we cannot extract the distribution of precision error in this case. In this course and in your career as an engineer you will most likely be taking multiple measurements,

i.e. repeated sample experiments. Single sample experiments are more uncommon and typically occur when the measurement you are making is extremely difficult and multiple experiments are non realistic to conduct, typically due to economic or practicality reasons.

The total uncertainty \(U_x\) in a measurement of x is

\begin{equation}
U_{x} = \sqrt{B_{x}^2 + P_{x}^2}
\end{equation}

where \(B_x\) if the bias uncertainty in a measurement of x and \(P_x\) is the precision uncertainty. Note that x is just a variable and we can talk about uncertainty in measuring volume (V), pressure (P), stress (σ), etc.

This is a largely empirically derived formula however it does depend on the assump-tion that the bias and precision uncertainty are independent sources of error. Also the bias and precision uncertainty should have the same confidence level when calculating the total uncertainty. We will come back to this a bit later in excruciating detail for this calculation so please be patient.

Minimizing Error in Designing Experiments

Best time to minimize experimental error is in the design stage. You should perform a single-sample uncertainty analysis of the proposed experimental arrangement prior to building the apparatus. This allows one to make a decision on whether the expected uncertainty is acceptable and to identify the sources of uncertainty. So when designing an experiment:

• Avoid approaches that require two large numbers to be measured to determine the small difference between them

• Design experiments or sensors that amplify the signal strength to improve sensitivity

• Build null designs in which the output is measured as a change from zero rather than as a change in nonzero value

• Avoid experiments with large correction factors must be applied as part of the data-reduction procedure

• Attempt to minimize the influence of the measuring system on the measured variable

• Calibrate entire system to minimize calibration related bias errors.

Once we have established the error and uncertainty with a measurement we also have to report these results and explain our analysis. Technical writing is a very distinct class of writing with many different formats and audiences. It can be difficult to strike the right tone and level of technical information that must be provided at times. In this class we will focus on Laboratory Reports but as you can see below there are many different form of technical writing.

Forms of Technical Writing:

• Executive summary

– Audience is a supervisor. Summary should be concise, usually only a few paragraphs, and highlight key aspects of work in first paragraph.

– Consequences can be drastic as in the case of the Challenger Shuttle tragedy.

• Laboratory note or technical memo

– Limited audience, immediate supervisor

– Sufficient information should be included so the experimentalist can re- construct the experiment

• Progress report

– Audience is supervisor of sponsoring agency

– Can vary in terms of length, structure, timelines, etc.

• Full technical report

– Wider audience typically not people closely associated with the work

– Relates all facts pertinent to project

– Contains enough detail so another professional can repeat experiments

• Full technical report

– Wider audience typically not people closely associated with the work

– Relates all facts pertinent to project

– Contains enough detail so another professional can repeat experiments

Line Fitting: Identify Outliers and Physical Interpretation

When you are fitting a curve or a line to a data set one must be careful to select a fit that is appropriate for the experiment being conducted. Often researchers are too consumed with obtaining a high R² value. While the goodness of a fit is important you must make sure that the equation that you are fitting matches with the physical experiment that you are conducting. For example if you are measuring how an electrochemical cell voltage changes as a function of temperature you must use the Nernst Equation. Looking at the data perhaps a 12th degree polynomial fit will give you a better R² value but that equation has no physical meaning in terms of your analysis. So always keep that in mind.

When identifying outliers and a particular point is in doubt, temporarily exclude the data point and fit a new line through the remaining data. If the fit is better the point can possibly be dropped however it could indicate that the relation- ship is not linear it may indicate another scaling regime or a transition in behavior.

We can also apply this same error analysis procedure and to optimize your experimental apparatus and minimize error/total uncertainty. One can identify what component or part of the measuring process is generating the most error and then one can re-design the apparatus to minimize that error.