March 4, 2020

The aim of science is not to open the door to infinite wisdom, but to set a limit to infinite error

Bertolt Brecht, Life of Galileo (1939)

What is a measurement?

  • A measurement tells us about a property of something
    • It gives a number to that property
  • Measurements are always made using an instrument of some kind
    • Rulers, stopwatches, weighing 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

What is not a measurement?

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

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

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

Measurements have uncertainty

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

  • resolution
  • random errors
  • systematic errors

Uncertainty must be declared

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

  • We declare an interval: [xmin, xmax]
    • Most of the time we write x ± 𝚫x
  • We declare the confidence level: how much we are sure that the real value is in this interval

Example: 20cm ± 1cm, at a level of confidence of 95%

Error versus uncertainty

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

Where do errors and uncertainties come from?

Flaws in the measurement can come from:

  • The measuring instrument – instruments can suffer from errors including bias, wear, drift, poor readability, noise, etc.
  • The item being measured – which may not be stable (measure the size of an ice cube in a warm room)
  • The measurement process – the measurement itself may be difficult to make. Measuring the weight of small animals presents particular difficulties
  • ‘Imported’ uncertainties – calibration of your instrument has an uncertainty

Where do errors and uncertainties come from?

  • Operator skill – 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
  • Sampling issues – 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
  • The environment – temperature, air pressure, humidity and many other conditions can affect the measuring instrument or the item being measured

What is not a measurement uncertainty?

  • Mistakes made by operators. They should not be counted as contributing to uncertainty. They should be avoided by working carefully and by checking work
  • Tolerances. They are acceptance limits which are chosen for a process or a product.
  • Specifications. A specification tells you what you can expect from a product.

Two ways to estimate uncertainties

  • 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

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

Reading v/s measurement

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

Uncertainty of a single read

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

For example, my water heater shows 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, with 100% confidence

For a single read, 𝚫x = half of the resolution

Combining Uncertainty

Sum of two measurements:
\[(x ± 𝚫x) + (y ± 𝚫y) = (x+y) ± (𝚫x+𝚫y)\]

Difference between measurements:
\[(x ± 𝚫x) - (y ± 𝚫y) = (x-y) ± (𝚫x+𝚫y)\]

Multiplying Uncertainty

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

\[(x ± 𝚫x/x\%) \times (y ± 𝚫y/y\%)\]

Then we sum the percentages:
\[xy ± (𝚫x/x + 𝚫y/y)\%\]

Finally we convert back to the original units:
\[xy ± xy(𝚫x/x + 𝚫y/y)\]


\[ \begin{aligned} (x ± 𝚫x) \times (y ± 𝚫y) & = x(1 ± 𝚫x/x) \times y(1 ± 𝚫y/y)\\ & = xy(1 ± 𝚫x/x)(1 ± 𝚫y/y) \\ & = xy(1 ± 𝚫x/x ± 𝚫y/y ± (𝚫x/x)(𝚫y/y)) \\ & = xy(1 ± 𝚫x/x + 𝚫y/y) \\ & = xy ± xy(𝚫x/x + 𝚫y/y)\\ \end{aligned} \]

We discard \((𝚫x/x)(𝚫y/y)\) because it is small

We choose the pessimistic alternative on ±


Measure the table top

First guess, then measure

  • What is the length?
  • What is the perimeter?
  • What is the area?
  • What is the volume?

Significant figures

Dinosaur Bones

Some tourists in the Museum of Natural History are marveling at some dinosaur bones. One of them asks the guard, “Can you tell me how old the dinosaur bones are?”

The guard replies, “They are 3 million, four years, and six months old.”

“That’s an awfully exact number,” says the tourist. “How do you know their age so precisely?”

The guard answers, “Well, the dinosaur bones were three million years old when I started working here, and that was four and a half years ago.”

Significant Figures

Lets be honest about what we know and what we do not know

We write the values that have real meaning

3 million years means 3±0.5 ⨉ 106

Adding 4.5 years is meaningless

It is the mark of an instructed mind to rest assured with that degree of precision that the nature of the subject admits, and not to seek exactness when only an approximation of the truth is possible

Aristotle (384–322 BC), Nicomachean Ethics