• 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

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

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

• We declare an interval: [x_min, x_max]
  • 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%

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

• 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

• 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.

• 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

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

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 \((x ± 𝚫x) \times (y ± 𝚫y)\) 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 ±

First guess, then measure

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

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.”

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