A measurement tells us about a property of something

- It gives a number to that property

Measurements are always made using an instrument

- Rulers, stopwatches, scales, thermometers, etc.

The result of a measurement has two parts:

- A number and a unit of measurement

Measurement Good Practice Guide No. 11 (Issue 2).

A Beginner’s Guide to Uncertainty of Measurement. Stephanie Bell.

Centre for Basic, Thermal and Length Metrology National Physical
Laboratory. UK

There are some processes that might seem to be measurements, but are not. For example

- Counting is not normally viewed as a measurement
- Tests that lead to a ‘yes/no’ answer or a ‘pass/fail’ result
- Comparing two pieces of string to see which one is longer

However, measurements may be part of the process of a test

Uncertainty of measurement is the doubt about the result of a measurement, due to

- resolution
- random errors
- systematic errors

How big is the margin? How bad is the doubt?

We declare an interval: \([x_\text{min}, x_\text{max}]\)

Most of the time we write x ± 𝚫x

Example: 20cm ± 1cm

Do not to confuse *error* and *uncertainty*

*Error* is the difference between the measured and the “true”
value

*Uncertainty* is a quantification of the doubt about the
result

Whenever possible we try to correct for any known errors

But any error whose value we do not know is a source of uncertainty

The measuring instrument

The item being measured

The measurement process

‘Imported’ uncertainties

Operator skill

Sampling issues

The environment

instruments can suffer from errors including

wear,

drift,

poor readability,

noise,

etc.

The thing we measure may not be stable

For example, when we measure the size of an ice cube in a warm room

The measurement itself may be difficult to make

Measuring the weight of small animals presents particular difficulties

Calibration of your instrument has an uncertainty

One person may be better than another at reading fine detail by eye

The use of an a stopwatch depends on the reaction time of the operator

The measurements you make must be representative

If you are choosing samples from a production line, don’t always take the first ten made on a Monday morning

Temperature

Air pressure

Humidity

and many other conditions can affect the measuring instrument or the item being measured

A reading is one observation of the instrument

A measurement may require several reads

For example, to measure a length, we make two reads, and we calculate the difference

The measurement will accumulate the uncertainty

For a single read, the uncertainty depends *at least* on the
instrument resolution

For example, my old water heater showed temperature with 5°C resolution: 50, 55, 60,…

If it shows 55°C, the real temperature is somewhere between 53°C and 57°C

We write 55°C ± 2.5°C

For a single read, 𝚫x = half of the resolution

This is a “one time” error

- We notice it immediately
- It does not change if we measure again

Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results

*Systematic error*which always occurs, with the same value, when we use the instrument in the same way and in the same case*Random error*which may vary from observation to another

**Type A**- uncertainty estimates using statistics- (usually from repeated readings)

**Type B**- uncertainty estimates from any other information.- past experience of the measurements, calibration certificates, manufacturer’s specifications, calculations, published information, common sense

In most measurement situations, uncertainty evaluations of both types are needed

Stephanie Bell. Centre for Basic, Thermal and Length Metrology National
Physical Laboratory. UK

There are other sources of uncertainty: **noise**

When the instrument resolution is good, we observe that the measured values change on every read

In many cases this is due to thermal effects, or other sources of noise

Usually the variability follows a Normal distribution

This would be the case when resolution is very high

Sum of two measurements:

\[(x ± 𝚫x) + (y ± 𝚫y) = (x+y) ±
(𝚫x+𝚫y)\]

Difference between measurements:

\[(x ± 𝚫x) - (y ± 𝚫y) = (x-y) ±
(𝚫x+𝚫y)\]

To calculate \((x ± 𝚫x) × (y ± 𝚫y)\) we first write the uncertainty as percentage

\[(x ± 𝚫x/x\%) × (y ± 𝚫y/y\%)\]

Then we sum the percentages:

\[xy ± (𝚫x/x + 𝚫y/y)\%\]

Finally we convert back to the original units:

\[xy ± xy(𝚫x/x + 𝚫y/y)\]

Assuming that the errors are small compared to the main value, we can find the error for any “reasonable” function

Taylor’s Theorem says that, for any derivable function \(f,\) we have

\[ f(x±𝚫x) = f(x) ± \frac{df}{dx}(x)\cdot 𝚫x + \frac{d^2f}{dx^2}(x+\varepsilon)\cdot \frac{𝚫x^2}{2} \]

When \(𝚫x\) is small, we can ignore the last part.

\[ \begin{aligned} (x ±𝚫x)^2& ≈ x^2 ± 2x\cdot𝚫x\\ & = x^2 ± 2x^2\cdot\frac{𝚫x}{x} \\ & = x^2 ± 2𝚫x\% \end{aligned} \]

because \[\frac{dx^2}{dx}=2x\]

\[ \begin{align} \sqrt{x ±𝚫x}& ≈ \sqrt x ± \frac{1}{2\sqrt x}\cdot 𝚫x\\ & = \sqrt x ± \frac{1}{2}\sqrt x\cdot \frac{𝚫x}{x}\\ & = \sqrt x ± \frac{1}{2}𝚫x\% \end{align} \]

because \[\frac{d\sqrt x}{dx}=\frac 1 {2\sqrt x}\]

These rules are “pessimistic”. They give the *worst case*

In general the “errors” can be positive or negative, and they tend to compensate

(This is valid *only* if the errors are independent)

In this case we can analyze uncertainty using the rules of probabilities

In this case, the value \(Δx\) will
represent the *standard deviation* of the measurement

The standard deviation is the square root of the variance

Then, we combine variances using the rule

“The variance of a sum is the sum of the variances”

(Again, this is valid *only* if the errors are
independent)

\[ \begin{align} (x ± Δx) + (y ± Δy) & = (x+y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx) - (y ± Δy) & = (x-y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx\%) \times (y ± Δy\%)& =x y ± \sqrt{Δx\%^2+Δy\%^2}\\ \frac{x ± Δx\%}{y ± Δy\%} & =\frac{x}{y} ± \sqrt{Δx\%^2+Δy\%^2} \end{align} \]

When using probabilistic rules we need to multiply the standard
deviation by a constant *k*, associated with the *confidence
level*

In most cases (but not all), the uncertainty follows a Normal distribution. In that case

- \(k=1.96\) corresponds to 95% confidence
- \(k=2.00\) corresponds to 98.9% confidence
- \(k=2.57\) corresponds to 99% confidence
- \(k=3.00\) corresponds to 99.9% confidence

Standard deviation of *noise* can be estimated from the data:
\[s=\sqrt{\frac{1}{n-1}\sum_i (x_i - \bar
x)^2}\]

If the measures are random, their average is also random

It has the same *mean* but less *variance*

Standard error of the *average* of samples is \[\frac{s}{\sqrt{n}}\]

Standard deviation of *rectangular* distribution is \[u=\frac{a}{\sqrt{3}}\] when the width of
the rectangle is \(2a\)

The exact distribution is hard to calculate

International standards suggest using computer simulation

They recommend Montecarlo methods

to open the door to infinite wisdom,

but to set a limit to infinite error

Bertolt Brecht, Life of Galileo (1939)