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The condition number of a sample.

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:

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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