Writing Numbers with Uncertainties

Question

I am having a little trouble determining the right way to write a number with an uncertainty:

I did a calculation in which the uncertainty was 150, while the value with physics meaning was about to 1427.

I am trying to figure out what is the right way to write it. Now, since the uncertainty cannot have more than two significant figures, my guess would be:

$$1.4*10^3 \pm 1.5*10^2$$

My doubt is about the first number. What are the rules about it? Should it be the same number of significant figures that the uncertainty has? Normally, if the uncertainty is, let's say, "0.003", and the value measured "154.3464", I would write $$154.346 \pm 0.003$$ and I am almost sure this is right, but it does not have the same quantity of significant figures. I am confused

Answer 1

The term "uncertainty" is always vague, an umbrella term for cases that can be extremely different and therefore need to be approached differently. The first rule is always to clearly specify what kind of uncertainty we're speaking about.

For example, it could be that you aren't able to measure some length $a$, but you know that's physically impossible for it to be less than 1276 m and more than 1578 m. Then you can express this with $a=(1427\pm 151)\ \textrm{m}$. In this case it's important to keep all figures, otherwise one couldn't recover from them the physical bounds of 1276 m and 1578 m. But in this case it'd be better to write $a \in [1276, 1578]\ \textrm{m}$, or $1276\ \textrm{m} \le a \le 1578\ \textrm{m}$.

The most common case is that we have an uncertainty about a quantity $a$ expressed by a probability distribution $\mathrm{p}(a)$, for example a normal distribution. This means that there are no hard bounds: $a$ could be between two specific values with a given probability. In this case the probability distribution is summarized giving its mean, say 1427 m, and standard deviation, say 151 m, in the form $a = (1427\pm 151)\ \textrm{m}$. This means that there's roughly a 68% probability that $a$'s true value is between 1276 m and 1578 m (and a 32% probability that it's outside that range!), and rougly a 95% probability that it's between 1125 m and 1729 m (see the summary in the Wikipedia article). Also in this case it's correct to give "151 m" with three significant digits, otherwise we'd be reporting the wrong information about the standard deviation. This precise information can be important when one needs to reverse-calculate the probability distribution, for example to check whether statistics on defective batches in a production plant are within the norm.

Other cases can be more difficult, for example if the uncertainty is expressed by a probability distribution $\mathrm{p}(a)$ that's skewed and has no standard expression. In this case the only precise way to express the uncertainty would be to plot the whole distribution; but usually one gives some quantiles of interest, for example the 2.5%, 50%, and 97.5% quantiles. This uncertainty is usually denoted in a different way, see for example this question.

The figures given in uncertainty expressions are usually rounded for practicality, when the precise value of the uncertainty is unimportant. Its importance depends on the context.

Most important, as you see from the examples above, is to explain clearly what your uncertainty refers to and where it comes from. This is emphasized in the guide below.

The official guide for the expression of uncertainty in measurement, by the Joint Committee for Guides in Metrology, can be found at http://www.bipm.org/en/publications/guides/gum.html

Answer 2

So typically you would write $1.427 \pm 0.150 \cdot 10^3 \text{(units)}$ or so; technically you would put in parentheses but of course that’s up to you. Actually you would probably want to round to $1.43 \pm 0.15 \cdot 10^3 \text{(units)}$, since that difference of 0.02 standard deviations is not going to help anyone in being pedantic.

Atomic/particle physicists in particular got a little tired of typing the $\pm$ symbol and started writing this instead as $1.427(150)\cdot 10^3$ or $1.43(15)\cdot 10^3$, the idea being that you write one digit for every digit that the error applies to. So for example if I go to the Wikipedia page for protons, I find out that their mass is listed as, $$ m_\text{proton} = 938.27208816(29) \text{ MeV/c}^2 $$ This unit, "mega-electron-volts per $c^2$," refers to the amount of mass which if it were present half as matter, half as antimatter, and those were allowed to annihilate, it would generate the amount of energy needed to accelerate an electron through one million volts. The $c^2$ refers to this $E = m c^2$ mass-energy equivalence, the electron-volt is just a common unit of energy in particle physics (where people “just know” that the standard state temperature, 25°C, is 25.7 meV, electrons have a mass of 511 keV, and protons and neutrons have masses of 0.94 MeV).

The above expression is actually shorthand for saying that there is a 68% chance that the mass is between $$938.27208787 \text{ MeV/c}^2 < m_\text{proton} < 938.27208845 \text{ MeV/c}^2.$$So we subtract $29$ from the last two digits of the number, and we add it to the last two digits of the number, to get the actual $1\sigma$-confidence interval, or if we want more confidence we can double that number to get the $2\sigma$-confidence interval with 95% confidence, or triple it to get the $3\sigma$-confidence interval with 99.7% confidence.

The only other thing I will say here is that I find a lot of questionable treatments of uncertainty for undergraduates and, while there are exact mathematical formulas for a lot of operations to combine standard deviations, you can usually much-more-easily program your formula into an Excel spreadsheet, put your measured values into one column, use the Box-Muller formula to generate some randomized values for all the parameters that you know, and produce approximate results. Here is a Google Sheet showing the technique, if you wish to use it.

Funny anecdote: at one point my wife wanted to know how weight correlated to jean sizes at various companies, “what would I need to weigh to fit into size-X-jeans at such-and-so-place?” and I asked for some historical data from her about how much she weighed when she fit into various sizes, how much those sizes were listed as measured on the web sites, and so forth. Then I built a physical model and did some polynomial interpolation to solve this. So, the result of my calculation was some reasonable answer, but I dug a little deeper and used this technique to see how sensitive my technique was to the exact numbers that I was assuming, “maybe these retailers misreport their jean sizes by 0.25 inches or so, maybe my wife misremembers her weight by a kilo or two”, etc. etc.—and the result was that something like 35% of my simulations generated a result that was “obvious garbage.” (Like, she would have to weigh half of what a supermodel weighs for a reasonably-achievable waist size, or she would have to weigh some amount she already had weighed or so.) So I could tell my wife confidently, “Here is the number that it says, but I have to warn you that apparently your retailer sizing is so inconsistent that given the parameters that you have given me, I do not have a consistent way to model this, tiny little variations in the assumptions result in huge changes in this polynomial fit and huge changes in how it estimates weight. So this is a guess but I would say that I have absolutely no confidence in it, because we are comparing all of these different brands.”