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I'm curious as to why this correction factor is needed.

The repeatability standard deviation was calculated by dividing the range by d2*, which is fair enough. The bias standard deviation is then the standard error of the repeatability, fair enough.

The manual then uses the bias divided by this bias standard deviation as the t statistic, yet again no problem.

But then it introduces this correction factor into the calculation of the confidence bounds, and I'm stumped as to why.

It won't make much difference because d2 and d2* are almost equal for the fairly large (>10) sample sizes involved, but why do it?