Salter, M J; Ridler, N M; Cox, M G
(2000)
*Distribution of correlation coefficient for samples taken from a bivariate normal distribution.*
NPL Report.
CETM 22

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## Abstract

When measuring a complex S-parameter several repeat measurements are made. The best estimate of the S-parameter and an elliptical uncertainty region are calculated from statistics of the repeat measurements. In particular the correlation coefficient between the real and imaginary parts r(x,y) is used in calculating the uncertainty region. The set of repeat measurements can be considered as a sample from a bivariate normal distribution. In this report the distribution of correlation coefficient r(x,y) calculated for samples from a bivariate normal distribution is investigated by generating a large number of samples using the multivariate normal distribution simulator MULTNORM. The effect of the population correlation coefficient p(x,y) and of the sample size n on the distribution is examined. For small samples it is found that the distributions are non-normal, broad and sometimes skew. This has implications for the reliability of confidence regions arrived at based on a small number of repeat measurements. The distribution of Fisher's z (a statistic defmed in terms of r) is also investigated and is found to be more normal than the distribution of r. This statistic is useful for estimating 95% confidence intervals for p.

Item Type: | Report/Guide (NPL Report) |
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NPL Report No.: | CETM 22 |

Subjects: | Electromagnetics |

Last Modified: | 02 Feb 2018 13:17 |

URI: | http://eprintspublications.npl.co.uk/id/eprint/1802 |

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