I've found I rarely use "average" to describe any discrete one-dimensional data sets anymore, and tend to use "minimum", "maximum", and some set of
k-th percentiles, usually including "median" (
k=50 if the number of data points is odd).
(If you sort
N measurements into entries 1 to
N, inclusive, you can find the approximate
k-th percentile measurement at entry
round(N·
k/100). If
N is odd, then
k=50 is median, exactly.)
k-th percentiles can be expanded into error bars around median. One possible notation is \$\left.m\right.^{+a}_{-b} (\text{p}\%)\$, where \$m\$ is the median value (
k=50), \$a\$ is the difference between value at
k=(100+p)/2 and median, \$b\$ the difference between median and value at
k=(100-p)/2, and p is the percentage of data points the entire range covers, centered on median. Most often the parenthesized range of included results is omitted, and instead the error bars shown are described generally as 1σ ("one sigma error bars", corresponding to p=68.3), 2σ ("two-sigma error bars", p=95.4), or 3σ ("three-sigma error bars", p=99.7), derived from the fact that in a standard/Gaussian distribution 68.3% of values are within one standard deviation of the median, 95.4% of values are within two standard deviations of the median, and 99.7% of values are within three standard deviations of the median. If p is not explicitly or implicitly defined, the error bars are 1σ by convention, and the range bracketed by the error bars cover 68.3% of all samples. If \$a ≃ b\$, it is common to use \$m \pm a\$.
Note that such error bars do not assume that the shape of the distribution is Gaussian, though: they can be used with any distribution, and even with discrete sets of samples where you do not know (or examine) the shape at all. In fact, I've found error bars the most useful measure of
statistical dispersion (=variability,scatter,spread) for discrete data sets both in physics and in numerical computation in general.
If you look at your results as a histogram or graph, it can be useful to pick p by hand so that it optimally includes valid samples but excludes invalid/erroneous/suspicious samples. The larger p is, the more reproducible your test/measurement setup is –– but the larger the error bars, the less specific the result is. By picking p by hand, you choose the balance between the two, and readers can assume you did so to the best of your ability, balancing the usefulness of the results with your trust in the measurement setup/run/iteration count.
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