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.

 

[pmacias]

I can understand your confusion about sensitivity and uncertainty in measurements. These are important concepts to consider when conducting any type of scientific research.

First, let's define sensitivity and uncertainty. Sensitivity refers to the smallest change that can be detected by a measuring instrument. In this case, the sensitivity of your meter rule is 0.1cm, which means that it can detect changes as small as 0.1cm. Uncertainty, on the other hand, refers to the range of values within which the true value of a measurement is likely to fall. It takes into account the limitations and potential errors of the measuring instrument.

In the scenario you described, it is important to consider both sensitivity and uncertainty when measuring the wire. The sensitivity of your meter rule is 0.1cm, so you can measure the wire to the nearest 0.1cm, which would be 7.3cm or 7.4cm. However, the uncertainty of your meter rule is 0.05cm, which means that the true value of the measurement could be anywhere within a range of +/- 0.05cm. This means that you could report the measurement as 7.35cm or (7.40+/-0.05)cm, both of which would be considered accurate and within the uncertainty range of the measuring instrument.

In general, when measuring an object, it is important to consider both sensitivity and uncertainty. You should measure to the nearest sensitivity of the instrument, but also report the measurement with the uncertainty range to account for any potential errors. This ensures that your measurement is as accurate and precise as possible.

I hope this helps to clarify the concept of sensitivity and uncertainty in measurements. It is always important to carefully consider these factors when conducting scientific research to ensure accurate and reliable results.