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I am working with 5 groups of measurements, all having a measuring uncertainty of 0.5 mm - I used the one-way ANOVA test to reject the null hypothesis and Fisher's Least Significant Difference to compare individual groups. The requirement of the paper I am writing is to account for all measurement errors. How could I include the measurement error in the F-ratio and LSD? Should I just follow standard error propagation rules in order to determine the absolute uncertainty of both, or are there any easier formulas/methods to this effect?

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The residual mean square from the ANOVA includes all sources of variance among individual observations unaccounted for by group membership. That includes the variance of the measurements themselves. The measurement error is already included in the F and LSD tests.

What you can do is estimate how much of the residual mean square is due to measurement error. It seems reasonable to assume that the measurement variance is independent from other sources of residual variance. In that case the residual variance (estimated by the residual mean square) is the sum of the individual-measurement variance and all other sources of variance. If the 0.5-mm value is the standard deviation of individual measurements on the same object, then the individual measurement variance is $(0.5\text{ mm})^2=0.25 \text{ mm}^2$. You can use that to estimate the magnitude of other sources of variance.

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