ON BIOLOGICAL MEASUREMENTS. 291 



Example : Length of cuckoo's egg (after 0. H. Latter). 

 Length class, mm. . . 19-0 19-5 20-0 20-5 21-0 21-5 22-0 22-5 23-0 



Frequency . . .1 3 33 39 156 152 392 288 286 



Length class, mm. . . 23-5 24-0 24-5 25-0 25-5 26-0 26-5 Total 



Frequency ... 100 86 21 12 2 1 1572 



A series of numbers arranged in this way form what is called the frequency distribu- 

 tion of the sample. 



The total of 1572 eggs is distributed in 16 length classes, each with a range of 

 half a millimetre, each class being specified by its central length. Thus the entry 

 under 21-5 mm. indicates that 152 of the eggs measured were judged to lie between 

 the precise limits 21-25 and 21-75 mm. The class range need not be equal to the 

 unit of measurement, but should be (either one unit or) an integral number of such 

 units ; the table above was condensed from a record giving the length to 0-1 mm. 



A fruitful source of bias is avoided, at the time the measurements are actually 

 taken, by using length classes bounded by the divisions marked on the measuring 

 instrument used, instead of the more common practice of using length classes centred 

 on visible divisions, and bounded by imaginary ones. The effect of the latter pro- 

 cedure appears to be especially noticeable in micro-measurements. If working with 

 length classes of 1 mm. adopt class boundaries of 0-1, 1-2, 2-3 mm., &c, with class 

 centres at 0-5, 1-5, 2-5 mm., &c. If working with length classes of 0-5 mm. adopt 

 class boundaries of 0-0-5, O'5-l-O mm. &c, with class centres at 0-25, 0-75, 1-25 mm., 

 &c. 



Measurement groups free from bias, bounded by divisions which can be accurately 



visualised. 

 Units Half units 



Groups usually emplo3 T ed, centred on divisions which can be accurately visualised/ 

 but bounded by imaginary divisions. 

 The use of small units is less important than accuracy of the class boundaries, and 

 it is above all essential that these boundaries should be clearly indicated. For 

 example, headings such as these are ambiguous : 



Age . . .6 years 7 vears 8 years 



Frequency . . 15 '38 62 



It is impossible to tell whether the 38 individuals were between 7*0 and 8-0, 

 or between 6-5 and 7-5 ; the former interpretation adhering to the popular convention 

 of age, the latter to the scientific convention of specifying the central measurement 

 of each class. 



In the choice of the class interval, which should be uniform throughout, little 

 additional information is supplied by a very fine classification ; for material which is 

 apparently homogeneous a class interval equal to a quarter of the Standard Deviation 

 is sufficiently small ; this will usually be provided for by dividing the material into 

 about 20-25 classes. Coarser groupings are by no means valueless. To bring out 

 the peculiarities of heterogeneous material finer grouping will sometimes be required. 

 Small samples shoidd not be grouped more coarsely than large samples. Extreme 

 measurements shoidd never be pooled as, e.g., 'more than 25 mm.'; since in the statis- 

 tical treatment the precise determination of these is of particidar importance. 



3. SUMMARY OF SINGLE MEASUREMENT DATA. 



If space does not allow a complete specification of the observations, these may be 

 summarised by means of a few quantities calculated from them ; each of these quanti- 

 ties is technically termed a statistic. If this course be taken, great care and some 

 additional knowledge will be needed to make the summary adequate. For instance, 

 the mean and range of the lengths of the individuals of a sample contain only a small 



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