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Inspired by these two question on tex.SX

I'd like to ask for a nice explanation for these kind of uncertainties, like $10_{-2}^{+6}$. I never saw them in real life. I'd like to give some question that may be stupid or redundant to each other but should show my problems that I currently have with them:

  • What exactly do they mean? Example: $10_{-2}^{+6}$ means that the real value is much more likely to be bigger than $10$. So publishing $10_{-2}^{+6}$ as a value will much more likely be "more off" than someone publishing $11 \pm 3$ for the same quantity. What I mean: repeating the experiment often and always finding more possible "+" than possible "-" means in a statistical sense "value to low" for sure.
  • How to calculate them? They seem not to be based on some gaussian distribution or statistical assumptions.
  • Can they be "made" symmetric in some way?
  • Why and when is it useful to use them?
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2 Answers

up vote 2 down vote accepted

As you probably know, whenever you make a measurement, there is a distribution of values you might get, due to noise, systematic errors, etc. Only in very contrived circumstances is this distribution symmetric about the mean, and only in even more restrictive cases is it Gaussian, but nonetheless we often make those assumptions for simplicity, waving our hands and chanting "Central Limit Theorem."

Sometimes, though, we want to convey a bit more about the shape of this distribution than just its mean and some proxy for its "width." The next-simplest thing to do is report an average and a confidence interval. Instead of $10\pm3$, which means "mean of $10$, and a $68\%$1 chance the value is within $3$ of $10$," we might say $10^{+6}_{-2}$, which means "mean of $10$, a $16\%$ chance the value is below $8$, and a $16\%$ chance the value is above $16$." The percentiles are chosen so as to encompass the same area under the PDF of the distribution as in the symmetric case, but we allow for the distribution to be asymmetric. Whatever the distribution is (and we have to have some way of getting at it, either analytically or through enough repeated trials), we choose a lower percentile such that the probability of being below this is the same as the probability of being less than $1\sigma$ below the mean in a Gaussian distribution, and ditto for the upper value.

So to address the points explicitly:

  • $10^{+6}_{-2}$ means you are as likely to be above $16$ as below $8$, despite the average being $10$.
  • In simple cases you can calculate these by finding the appropriate percentiles from a known PDF/CDF. Probably the most illustrative case is with a continuous Poisson variable with small mean, say $1.10$. In a unit of time, you expect $1.10$ events. The standard deviation is about $1.05$, but saying $1.10\pm1.05$ signals that negative counts are not too unlikely. Instead, you might say you expect $1.10^{+3.01}_{-0.67}$. In more complicated cases, one often runs the data through the pipeline many times, varying the parameters slightly, to estimate how the final measurement is likely to vary as the unknown parameters vary over reasonable values.
  • You can't really convert back to something symmetric without knowing the full underlying distribution. For some purposes it may be acceptable to keep the same width, so $+6, -2$ becomes $\pm4$.
  • It's useful whenever there is an asymmetric underlying distribution of measurements, and further use of the reported values should take that into account. If your experiment crucially depends on the voltage not dropping below $120\ \mathrm{V}$, then a $4$-$\sigma$ voltage of $130^{+40}_{-4}\ \mathrm{V}$ is acceptable, whereas if someone just told you they could supply $130\pm22\ \mathrm{V}$, you could not trust the source.

1 This comes from $\int_{-\sigma}^\sigma (1/\sqrt{2\pi\sigma^2}) \mathrm{e}^{-x^2/(2\sigma^2)} \ \mathrm{d}x \approx 0.68$. If we had implicitly discussing $2$-$\sigma$ uncertainties rather than $1$-$\sigma$, the limits of integration would have doubled and we would say $95\%$.

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'waving our hands and chanting "Central Limit Theorem."' Nice. I believe you win the internet for an hour or so. –  dmckee Apr 15 '13 at 23:30
    
Thanks, I think this explains most of my problems. I guess these are then, for example, used on single photon experiments where you often have Poissonian distributions in the count rate? –  Foo Bar Apr 16 '13 at 11:08
    
@FooBar That's one example. More often, they're used in very complicated experiments (think CMB mapping, LHC, LIGO, etc.) where the output at the end of the data reduction depends in a rather complicated, often not completely known, way on the inputs. –  Chris White Apr 16 '13 at 15:54
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The first google item looking for asymmetric errors, and same as the fourth, seems quite good. I haven't done a exhaustive lecture, but it seems that apoints your problem.

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