EXPERIMENTAL DESIGN 



219 



Fig. 16-3. A Latin square. 

 Any treatment is present in 

 each row and in each col- 

 umn. 



simplicity, let us reduce the number of plants in each treatment to three. 

 We then have three replications of each of three treatments, which can 

 be arranged in a Latin square as shown in Fig. 16-3. This is only one 

 of twelve possible 3X3 Latin squares, and 

 the actual placing of the plants was done at 

 random. 



Other Variations: Agricultural statisticians 

 have developed a vast list of modified experi- 

 mental designs, such as Partially Balanced 

 Incomplete Block Designs and Lattice De- 

 signs. Most of these are highly specialized, 

 however, and it is unlikely that they would 

 be of utility to the ordinary laboratory biolo- 

 gist. The experiment on the growth of sugar 

 beets in a plant growth room is very similar 

 to an agricultural experiment, of course. Simi- 

 lar designs can be adapted to strictly labora- 

 tory experiments, such as the effects of several treatments on muscle con- 

 traction or the metabolism of cells, but the analogies to blocks, replicates, 

 rows, and columns are often abstruse. Frequently single simply designed 

 experiments go so rapidly that it is not worth the effort to use one of the 

 abstract designs. 



Analysis of Variance: In the experiment described above, more than 

 two populations are being described simultaneously. Differences between 

 sample means can arise because the populations are different but also 

 from chance alone. Variances are more difficult to estimate. Our experi- 

 ment can be tested most effectively by means of analysis of variance. The 

 following assumptions will be made: the samples are random, the varia- 

 bility is distributed according to the normal curve, and the variances of 

 the different populations are equal. Again we shall use the hypothesis 

 that the three treatments produce equal results, or that all the plants 

 belong to the same population, and then we shall find the probability 

 that the observed differences could arise from chance alone. Since the 

 randomized block is used more frequently than the other designs, that 

 one has been chosen for analysis. 



After a period of growth, the plants of Fig. 16-2 were harvested, and 

 the leaves were weighed. The weights are recorded in Table 16-2. There 

 are three sample means, x«, and these are calculated at the bottom of the 

 table. The working formula, 



Xx^-(Xxy/n 



^ = 



n — 1 



