  Confidence intervals are a popular way to visualize and analyze data
distributions. Unlike p-values, they can convey information both about
statistical significance as well as effect size. However, very little work
exists on applying confidence intervals to multivariate data. In this paper we
define confidence intervals for multivariate data that extend the
one-dimensional definition in a natural way. In our definition every variable
is associated with its own confidence interval as usual, but a data vector can
be outside of a few of these, and still be considered to be within the
confidence area. We analyze the problem and show that the resulting confidence
areas retain the good qualities of their one-dimensional counterparts: they are
informative and easy to interpret. Furthermore, we show that the problem of
finding multivariate confidence intervals is hard, but provide efficient
approximate algorithms to solve the problem.
