Sensitivity and Uncertainty in measurements

[tyneoh]

Greetings fellow members, I have some queries on a laughably rudimental topic regarding measurements.

Say you have a metre rule with sensitivity of 0.1cm, and you are measuring a wire which stretches from the 0.0cm starting point to the middle point between 7.3 cm and 7.4cm. My confusion arises here, do you measure the wire as 7.35cm or round it up to 7.4cm?

For me I would normally choose 7.35 cm but after learning about uncertainty and sensitivity, I am starting to doubt my original comprehension about measurements. When you measure it as 7.35cm, are you exceeding the sensitivity(0.1 cm) of the metre rule thereby "creating" an arbitrary reading? My teacher said that when you round it to 7.4cm, your reading includes 7.35 cm as the metre rule has an absolute uncertainty of 0.05cm, thus your reading would be (7.40+/-0.05)cm, which includes the 7.35cm.

To round up my question(no pun intended), when you measure an object do you measure it to the sensitivity or the uncertainty of the measuring equipment, 7.35cm or (7.40+/-0.05)cm?

 

[Simon Bridge]

Say you have a metre rule with sensitivity of 0.1cm, and you are measuring a wire which stretches from the 0.0cm starting point to the middle point between 7.3 cm and 7.4cm. My confusion arises here, do you measure the wire as 7.35cm or round it up to 7.4cm?

In practice, you can use a standard meter ruler to 0.5mm accuracy for ±0.25mm uncertainty. If you rounded the reading to the nearest actual marking on the scale, then the uncertainty you are introducing is bigger ±0.5mm would be a common estimation.

What you want to do is figure what the distribution of a large number of measurements would be like.

 

[tyneoh]

By rounding to the nearest actual marking, the systematic uncertainty increases from 0.25mm to 0.50mm? Is that what you mean?

 

[Simon Bridge]

By rounding to the nearest actual marking, the systematic uncertainty increases from 0.25mm to 0.50mm? Is that what you mean?

It's a statistical or random uncertainty rather than a systematic one.
http://www.physics.umd.edu/courses/Phys276/Hill/Information/Notes/ErrorAnalysis.html

But that's the idea.
A common strategy is to measure to the nearest graduation on the scale, and estimate the error to plus-or-minus half the resolution of the instrument.
The estimate assumes something about the distribution of many measurements - so you need to select a method for estimation that takes into account what you now about this.

Estimating uncertainties can be something of an art-form.
In this case, the uncertainty introduced by the rounding off is probably bigger than any other random variation. When you say you got a reading of, say, 100mm, that means the length is somewhere between 99.5mm and 100.4mm. If the distribution of many measurements is Gaussian then there is a non-zero probability that a length a little outside that range would still get measured as somewhere inside that range. Estimating the standard deviation to 0.5mm would be an over-estimate.

Of course, just because the ruler is marked in millimeters does not mean you have to measure in millimeters. You could round to the nearest cm, for example. That does that do the the possible variation in repeated measurements?

The bottom line is that the strategy used depends on what you hope to do with the measurement.