Which summary statistics are always nonnegative?

So far, the only ones I can think of are range and standard deviation/variance (which are positive except in trivial cases).

  • 1
    $\begingroup$ Adding to @Kurtosis' list (+1): coefficient of determination; coefficient of variation; estimate of a probability / proportion; standard error; various indexes of similarity; sample mean, median, mode of distribution with non-negative support (binomial, Poisson, geometric, hypergeometric; exponential, gamma, beta, Rayleigh, Weibull, Pareto, etc.). $\endgroup$
    – BruceET
    Jul 28 '20 at 7:18
  • $\begingroup$ The exponential of any summary statistic is a non-negative summary statistic. If we stipulate $\log(0)=-\infty$ and allow $-\infty$ as a possible value, then the log of any non-negative summary statistic is a summary statistic. Thus, your question is tantamount to asking for an enumeration of all summary statistics. That's beyond the capacity of this site. $\endgroup$
    – whuber
    Jul 28 '20 at 14:12
  • Range, IQR, and really any quantile interval lengths;
  • standard deviation/variance, kurtosis (not excess kurtosis), and all other even-power moments; and,
  • likelihood and other chi-squared statistics (e.g. Wilk's lambda).

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