Comments on the Calculation of Mean Harvest Indices

The harvest index Z is defined, for example for cereals, as the ratio between grain yield X and total yield Y: Z = X/Y. For individual measurements x_i of X and y_i of Y the individual harvest index is calculated by z_i= x_i/y_i. These individual measurements i are based on N different experimental units (single plants, random samples from plots or total plots).
Mean harvest indices are commonly calculated either by     or by     where the means of X, Y and Z are denoted by , and respectively.
Both approaches     are, in general, different. But, in this paper it is shown that for most situations which may occur in the field of practical applications in agronomy and plant breeding both procedures will lead to nearly identical results. Only in some special situations the two results differ significantly. These conditions are investigated and explicitly given.
Some explicit formulae (approximations) for the calculation of mean harvest indices are derived and discussed.
Finally, all the theoretical investigations and results are demonstrated and applied to a numerical example of winter-rapeseed data.