# Should the implied uncertainty of a measurement be ±½ or ±1 of its least significant figure?

I've seen conflicting statements on whether the implied uncertainty of a measurement should be interpreted to be ±½ or ±1 of its least significant figure. Wikipedia goes with the ±½ convention:

Applying 10½ meters in a scientific or engineering application, it could be written 10.5 m or 10.50 m, by convention meaning accurate to within one tenth of a meter, or one hundredth. The precision is symmetric around the last digit. In this case it's half a tenth up and half a tenth down, so 10.5 means between 10.45 and 10.55. Thus it is understood that 10.5 means 10.5±0.05, and 10.50 means 10.50±0.005.

However, this definition feels unintuitive, since it allows for no overlap between implied ranges, making it impossible to confidently express measurements whose perceived value lies midway between two units. For example, suppose that I have a ruler with millimeter marks. I measure an object whose length appears to be midway between the 77 mm and 78 mm marks. If the implied uncertainty is ±1, then I could give the measured length as either 77 mm or 78 mm, since both implied ranges would be correct (76–78 mm and 77–79 mm respectively). However, if the implied uncertainty is ±½, then it becomes impossible for me to express the measured length with confidence. If I give it as 78 mm, then my implied range would be 77.5–78.5 mm, making my measurement wrong if the object is actually 77.4999 mm – yet I have no way of detecting this 0.0001 mm discrepancy.

Given the above, why do sources promote the ±½ convention? Are they suggesting that it's acceptable for a measurement to have a confidence level as low as 50% for these midway values, or am I missing something?

Edit: I agree with the feedback that it's better to express the uncertainty explicitly in scientific contexts where it matters. However, I'm more interested in the everyday interpretation of expressed measurements, where significant figures are only intended to convey a rough indication of precision. My argument is that, even in these situations, the ±½ convention is fundamentally flawed, since it results in confidence levels as low as 50%, and any sources that promote it would be better off switching to the ±1 convention.

• General note: significant figures are inherently flawed anyway, as they don't capture the way that uncertainty works in all but the most obviously straightforward situations; as such, you won't see significant figures being used in almost any modern research paper in physics. Instead, you'll see an actual, formal characterization of uncertainty (and if you don't, that's a red flag). Given that it's mostly relegated to education by now, just follow whatever convention your teachers use. Jan 7 '20 at 19:12
• I agree that one should express uncertainty explicitly in scientific contexts where it matters. However, there are many situations in everyday life where one relies on significant figures to convey uncertainty or tolerance, even if loosely. For example, a device that states it requires a 12 V supply would probably be fine with 11.9 V; however, a device that (for some reason) requires a 11.884 V supply wouldn't. Jan 7 '20 at 19:24
• As with many other situations in real life where there are differing conventions, if the distinction you're asking about is important, specify it explicitly (or, if you're not the one stating the figure, ask explicitly which convention they're using). Jan 7 '20 at 19:27

I think you're grossly over-complicating things here. I've listed the convention below for most high-school and undergraduate level treatment of uncertainty. $$\\$$

When a measurement is made with an analogue device, such as a ruler, or a thermometer, then our error is taken to be half of the smallest division. However, this doesn't mean you can't record your measurement to be in between two of the smallest divisions. For example, if I had a rule that measured in $$cm$$, but the object I was measuring was quite close to the middle of two markings, then I would record the measurement to the nearest half centimetre, with uncertainty as $$\pm\frac{1}{2}cm$$. $$\\$$ However, if you are reading off a digital scale, such as with a phone timer, or a digital scale, you should first look for the manufacturer's advice on uncertainty. If you can't find this, then the convention is to have an uncertainty equal to the smallest division. This can be altered if we have knowledge about the rounding procedure of the device, but in many cases this isn't possible. An example of where you could easily go wrong, is with digital clocks, which typically always round down to the nearest division.$$\\$$ In any case, it should be worth clarifying that this concept of your measurement being wrong isn't quite correct either - an uncertainty doesn't mean that the value has to be within those bounds, looking at dozens of measurements of constants from the 1800s should convince you of that - it just specifies a confidence interval, usually $$95\%$$, that the value measured is within the bounds.

• "However, if you are reading off a digital scale …" I would hope that even a high school level and certainly in undergraduate labs, you look at the specification of the device to find the measurement error, instead of blindly applying some arbitrary rule. The whole "significant figures" thing is just a bit of history left over in the high school curriculum. Just do whatever nonsense your teacher or examiner gives you credit for! Jan 7 '20 at 19:30
• I agree that expressing my measurement as 77.5±0.5 mm would be optimal in an academic context. However, I'm interested in everyday contexts where it's impractical to give the explicit uncertainty. A reading of 78 mm would only have a confidence level of 50% if the implied uncertainty is ±0.5 mm, which is nowhere near the 95% that I would (intuitively) expect. Jan 7 '20 at 19:54
• I think that your understanding of confidence levels isn't great here, and it might be worthwhile reading around to see if you can get a better idea of what is meant by a confidence level, and how we might make an estimate of them. As for an everyday measurement, I still don't see what stops you from saying 'it's somewhere in between 77 and 78 mm, let's call it 77.5'.
– MC2k
Jan 9 '20 at 1:36

I think that for meter rule, giving uncertainty to smallest division is more feasible as length of object has two ends and if uncertainty of each end ($$1/2$$ of smallest division) is added - it appears to be smallest division $$0.1 \;\text{cm}$$. Furthermore, this rule applies to static measurements.

For dynamic measurements, the uncertainty increases. For example, if you are measuring rebound height of ball using meter rule, thee uncertainty may be $$1$$ or $$2 \;\text{cm}$$.