l84 ECOLOGY AND LIFE HISTORY OF THE COMMON FROG 



only one that is sufficient, a belief that seems to be supported by the 

 fact that two authors of general books, Tippett (1952) and Kendall 

 (1952), both refer their readers to Ezekiel after giving their own short 

 treatment. Ezekiel' s book is quite a large one, and is a special text 

 dealing only with this subject, but it is lucidly written, and easy to use. 



Some Examples of the Application of Statistical Methods 



This section and the Appendixes are intended to serve a double func- 

 tion. The statistical evidence for many of the statements in the preceding 

 chapters is here set out, and serves also to illustrate the methods. 



In many parts of the book there are examples of the simple 2X2 

 tables. All that has to be done to the data is to arrange it in the form of 

 a table in which all the four possible combinations of the two factors 

 are represented. If it is not at once obvious what classification to adopt 

 it may be necessary to make an arbitrary decision. For example, in 

 Appendix i, (c) (on p. 205 below) I had to decide for the purpose of 

 one of these tables what was to be regarded as a warm day. I chose io°C 

 as the dividing line between a warm day and a cool one. I might have 

 chosen some other temperature and, in fact, I did try other levels. I 

 then counted the occasions above 10° and below, and the number of 

 records of spawning that fell into these two classifications. This gave 

 the actual numbers in each class. Now, if there had been no connexion 

 between spawning and temperature at this level, these classes should 

 be in proportion. To fmd the expected number on this hypothesis, all 

 that has to be done is to divide the total number of observations by one 

 of the marginal totals, say that in the bottom left-hand corner, and 

 multiply by another, say that in the top right-hand comer, and one 

 obtains the expected number for the class in the top left-hand cell of 

 the table. This is then written down in brackets. The other expected 

 numbers follow by subtraction, for the totals of the expected numbers 

 must agree with the actual numbers. Sometimes it is so obvious that 

 the results are merely those that can be ascribed to chance, because the 

 actual and the expected numbers agree so closely, that it is really 

 superfluous to go on, and there are conclusions in the hterature that 

 would hardly have been published if the authors had even constructed 

 such a table, let alone worked it out as in the next few sentences. The 

 next step is to subtract each of the expected numbers from the corres- 

 ponding actual numbers, and square the results. Each square is then 

 divided by the expected number, and the results added. This number 



