Given: The smallest increment on thermometer is 5°C.

When reading a temperature, should I note 15°C or 15.0°C?

I learnt from 2.2 Accuracy, Precision and Resolution, that the last sig fig in a number is always guessed and therefore uncertain.

They say that $resolution=frac{smallest increment}2$.

Here we would get $resolution = 2.5°$.

Therefore I would be able to write $15.0°pm2.5°$, wouldn’t I?

Would it be also correct to write $18.0°pm2.5°$? I guess no.

Because here Uncertainties – Physics A-level & GCSE they say, that it is not possible to use any value which is not an increment of the measuring scale.

So is the notation of $xpm resolution$ only valid for multiples of $Delta x=2times resolution$?

From the logical point of view I would interpret a notation of

A)

$$x pm y$$

as

$$x_{true} in (x – y; x + y)

$$

But what about rounding.

Let’s say, that we want to round to the $10^{-3}$.

B)

$$x_{true}=1.23456$$

$$y=round(x_{true},3)=1.235$$

If we define

$$f(x)=round(x,3)=y$$

then $$f^{-1}(y)={x|f(x)=y}$$

In this case the notation of $pm$ would just mean:

if y is positive (the minus case):

$$1.235 pm 0.001$$

$$equiv$$

$$1.235 – 0.001 = 1.234$$

if y is negative (the plus case):

$$-1.235 pm 0.001$$

$$equiv$$

$$-1.235 + 0.001 = -1.234$$

Because in the set of $f^{-1}(1.235)=(1.2345;1.2355($ some of the elements are $=….5$ and some are $=….4$.

For the latter case, that is the reason for writing $pm 0.001$?

So which interpretation of $x pm y$ is correct?

A) or B)

What’s the relation of rounding and sigfigs.

If a number is rounded, does it change the certainty quality of the last digit into uncertain?

Please also have a short look at these video clips, to understand my confusion:

Which numbers are certain in each measurement:

In spite of 1 and 3 are certain in 130 pounds, it’s still possible that these sigfigs are affected by rounding ?

If a digital scale reads 133.6, does that mean that the 0.6 is estimated or is it the rounding result of the estimated digit in the $10^{-2}$ position?

Measuring with a metric ruler:

Possible readings 3.38 … 3.41 by having a scale with $Delta = 0.01$

Measuring with a graduated cylinder:

Is it here also possible to measure it as 37.1 ml?

Measure pencil with a ruler $Delta 1mm$

Is 1mm the resolution or is it $0.5mm$?

If the ruler increment is $Delta 10 cm$, are we free to claim the length any value in the range $(0;10)$?

Readings vs Measurements

Uncertainty $frac 1 2 times resolution$ vs $uncertainty=resolution$

Thx in advance.