The condition number of a sample.

condition-number floating-point numerical-analysis

Today I learned about the condition number, \(\kappa\), of a sample \(\{ x_i \}\) of \(N\) data points, with respect to computing the variance. Previously, I had only know about condition numbers of matrices Nick Higham, “What is a condition number?”.

First, some references:

  • T. F. Chan and J.G. Lewis. “Rounding error Analysis of algorithms for computing means and standard deviations.” Tech. Rep. No. 284, Dept. of Mathematical Sciences, The Johns Hopkins University, April 1978. (I couldn’t find this paper online.)
  • T. F. Chan, G. H. Golub, and R. J. Leveque, “Algorithms for Computing the Sample Variance: Analysis and Recommendations,” The American Statistician, vol. 37, no. 3, pp. 242–247, Aug. 1983, doi: 10.1080/00031305.1983.10483115. Published paper link, Open access technical report link.

The condition number \(\kappa\) in the context of computing the variance is

\[\kappa = \sqrt{1 + \frac{\bar{x}^2 N}{S}}\]

where \(\bar{x}\) is the mean.

From Chan et al., 1983:

This condition number measures the sensitivity of \(S\) [the variance] for the given data set.

And Chan et al. 1983 references Chan and Lewis, 1978 for the condition number. However, I could not find a copy of Chan and Lewis, 1978.

See Chan et al. 1983 for more discussion about condition numbers and approximations of the condition number.

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