What is a Measurement?

A measurement is a quantitative comparison between a predefined standard and a measurand.

What are some examples of different types of measurements we can make?

• Temperature

• Length

• Time

• Stress

• Strain

• Viscosity

• Force

• Torque

• ....

The process or act of measurement is a quantitative comparison between a predefined standard and a measurand.

Measurand: the particular physical parameter being observed and quantified, i.e., the input quantity of the measuring process

Standard: Must be of the same character as the measurand and usually recognized by the NIST (National Institute of Standards and Technology)

What are some standards?

• Tape Measure
• Balance
• Speed of Light
• ...

The act of measurement produces a result.

ADD FIGURE

The significance of measurements:

• Obtain quantitative information on the actual state of physical variables and processes (i.e. T, P, V, etc.)

• 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. central power station)

• Establishing costs

Measurements are only useful if they are reliable. This brings up the concept of the accuracy or uncertainty of a measurement. No measurement is perfect and therefore engineers must place tolerances on measured values.

There are two basic methods of measurement: i) direct and ii) indirect

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.

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.

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.

Gaussian and Normal Distribution

One of the most common distributions encountered is the Gaussian distribution, also called the normal distribution or bell curve.

The Gaussian distribution describes an infinite population of data where each datum represents a measurement of a single quantity. Each value x differs due to precision error.

The probability of obtaining a specific value x is described by the probability density function (PDF), f(x).

The total area under the curve equals 1, which represents 100% probability:

\[P = \int_{-\infty}^{\infty} f(x)\, dx = 1\]

Gaussian PDF:

\[f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)\]

Where \(\mu\) = population mean and \(\sigma\) = population standard deviation.

To simplify, define \[z = \frac{x - \mu}{\sigma)}\], so:

\[f(z) = \frac{1}{\sqrt{2 \pi)}} e^{-z^2/2}\]

You can use Z-tables to find the area under the curve for a given z. For example, the 95% confidence interval contains most values within ±1.96\(\sigma\).

Outliers

Outliers are data points that are much higher or lower than the mean.

You can disregard a data point if it exceeds 3\(\sigma\), but must report it and justify its exclusion.

Confidence Intervals

The area between ±z under the standard curve defines a confidence interval.

Example: A 95% confidence interval covers ~95% of the population's measurements.

Example Problems - Confidence Intervals

Working with Z Tables

• What is the area under the standard normal distribution curve between z = -1.43 and z = 1.43?

• What percent of the population lies within this range?

• What range of z contains 90% of the data?

Running and Tumbling E. coli

Given: mean = 3 \(\mu\)m/s, \(\sigma\) = 0.1 \(\mu\)m/s, range = [2.857, 3.143] \(\mu\)m/s

• Assume normal distribution

• Compute z for bounds

• Use standard normal table to compute area

• Estimate number of samples within range

• Determine 95% confidence bounds

Marble Diameters

Given: N = 765 marbles, \(\mu\) = 200 mm, \(\sigma\) = 20 mm

• Determine diameter range for 60% of marbles

Ultimate Tensile Strength Measurements

Given: \(\mu\) = 303 MPa, \(\sigma\) = 33 MPa

• What is the probability a sample exceeds 350 MPa?

Voltage Measurements

Given: \(\mu\) = 10 V, \(\sigma\) = 3.4 V

• How many of 150 measurements fall between 10–15 V?

Under Pressure Measurements

Given: \(\mu\) = 39 Pa, \(\sigma\) = 4 Pa, N = 120

• How many measurements fall between 35–45 Pa?

• How many below 35 Pa?

• How many below 45 Pa?