METHODS 185 



is y^. If it is 3-8 or under, and particularly if it is under 2-7, there is 

 no good reason for supposing that the results are anything but chance. 

 If yi^ hcs between these two numbers, it may be worth considering 

 whether to collect more data to make matters more certain. If, on the 

 other hand, y^ is over 3-8, and especially if it is over 6-7, the actual 

 results caimot fairly be ascribed to chance. This would be equivalent to 

 very strong evidence for the view that temperature does affect spawning. 



I have purposely left out two minor precautions. The numbers 

 expected in any cell ought not to be less than five: otherwise there is 

 really not enough data, although the information can nevertheless be 

 useful, for any guidance is sometimes better than none. If the numbers 

 are rather small, it is better to subtract 0-5 from the "expected" number 

 if this is larger tlian the actual number and add 0-5 if it is smaller. If 

 the numbers are all large, it does not matter much whether this cor- 

 rection is applied or not. Now there is surely no one likely to be 

 reading this who could admit to inability to do this kind of arithmetic. 

 If he can do it, then he can "do statistics" and can prevent himself 

 reaching conclusions that have no sound basis, or, what is just as 

 important, failing to see connexions between events that are in the 

 data. The chi-squared test can be used to compare any series of numbers 

 expected on some hypothesis with those actually found, and more 

 comphcated examples can be found in the books, or even in this one. 

 For simple examples of the test, see Appendixes ic, i^, and i/i. 



A more elaborate example is to be found in Appendix i^ in which 

 the number of parasites found is compared with those expected on the 

 hypothesis that the infestation occurs purely by chance encounters 

 between parasite and host, swimming in the same pond. The work of 

 the statisticians has given the proportions expected in such a case, and 

 it was found that the actual results cannot be reconciled with this 

 hypothesis. In Table 6, the total number of frogs killed during 

 migration has been distributed in proportion to the frequency of the 

 winds from various directions. If migration had been random, the last 

 two columns should have agreed. But they did not, and a calculation 

 oiyj' shows that there is no reasonable chance that the disagreement 

 is an accident. Migration is definitely related to wind direction. 



Analysis of Variance 



This more elaborate test has so many variations that it cannot be 

 described here in detail. Appendix kj is an example appHed to size 



