Consider an example where we expect a mean experimental yield of 3000 kg/ha, a CV of 20% which
implies s = 600 kg/ha and where we would hope to detect a true difference, if it exists, of 750 kg/ha. The
SE should therefore be no bigger than 250 kg/ha and we thus require that n shall be at least big enough to
make

 250 = 600 4(2/n)

 This implies that

 n/2 = (600/250)2 = 5.76

 and n should be at least 12.

 We can approach the problem from the other end, and consider what precision would be achieved and what
 differences detected for various possible n.

 If we can afford 8 observations per treatment our standard error will be

 SE (for n=8) = 600 4(2/7) = 300.

 This would give us a 5/6 chance of detecting a difference of 900 kg/ha using a 5% sig. level.

 If we could use 20 observations.

 SE (for n = 20) = 6004(2/20) = 186,

 and we would now have a 5/6 chance of detecting a difference of 560 kg/ha.

 There are a few additional comments. We can work with CV and A rather than s and d and everything is
 expressed in percentages but gives the same answer:

 CV = 20%, A = 25%

 25/3 = 204(2/n)

which gives n/2 = (20x3/25)2 = 5.76.

These calculations assume that the experimenter can assess the appropriate value for d. More crucially the
calculations assume that s is known. In practice previous experience with the same crop for similar plots in
similar conditions will often provide, through the "EMS. a credible value for s; sometimes less closely
related information will have to be used to guess the likely value of s. If the actual experimental value of s
is smaller than the assumed value, then our detection chances improve.

Note that although the SE decreases as n is increased, the reduction of SE for an increase of 1 to n gets
smaller as n increases (the law of diminishing returns). We can make the statements about detection
probabilities and significance probabilities more precise by expressing

 d as Za + Zp

where Zp is the standardized normal deviate corresponding to an *a % significance level (Za = 1.96 for a
= 0.05) and Z is the standardized normal deviate for a risk of non-detection (Zp = 1.0 for P = 1/6).