QUANTITY OF VARIETY 7/13 



of cars, then the three components would be said to be "'indepen- 

 dent", or to "vary independently", within this defined set. 



It should be noticed that such a statement refers essentially to 

 what is observed to occur within the set; it need contain no reference 

 to any supposed cause for the independence (or for the constraint). 



Ex. 1 : When Pantagruel and his circle debated whether or not the time had 

 come for Panurge to marry, they took advisers, who were introduced thus: 

 ". . . Rondibilis, is married now, who before was not — Hippothadeus was 

 not before, nor is yet — Bridlegoose was married once, but is not now — and 

 Trouillogan is married now, who wedded was to another wife before." 

 Does this set of vectors show constraint? 



Ex. 2: If each component can be Head (H) or Tail {T), does the set of four 

 vectors {H,H,H)AT,T,H), (H,T,T), {T,H,T) show constraint in relation to the 

 set showing independence ? 



7/13. Degrees of freedom. When a set of vectors does not show 

 the full range of possibilities made available by the components 

 (S.7/1 1), the range that remains can sometimes usefully be measured 

 by saying how many components with independence would give 

 the same variety. This number of components is called the degrees 

 of freedom of the set of vectors. Thus the traffic lights (S.7/8) show 

 a variety of four. If the components continued to have two states 

 apiece, two components with independence could give the same 

 variety (of four). So the constraint on the lights can be expressed 

 by saying that the three components, not independent, give the same 

 variety as two would if independent; i.e. the three lights have two 

 degrees of freedom. 



If all combinations are possible, then the number of degrees of 

 freedom is equal to the number of components. If only one combin- 

 ation is possible, the degrees of freedom are zero. 



It will be appreciated that this way of measuring what is left 

 free of constraint is applicable only in certain favourable cases. 

 Thus, were the traffic lights to show three, or five combinations, the 

 equivalence would no longer be representable by a simple, whole, 

 number. The concept is of importance chiefly when the components 

 vary continuously, so that each has available an infinite number of 

 values. A reckoning by degrees of freedom may then still be 

 possible, though the states cannot be counted. 



Ex. 1 : If a dealer in second-hand cars boasts that his stock covers a range of 

 10 ages, 8 horse powers, and 12 colours, in all combinations, how many 

 degrees of freedom has his stock? 



Ex. 2 : The angular positions of the two hands on a clock are the two compo- 

 nents of a vector. Has the set of vectors (in ordinary working round the 

 12 hours) a constraint if the angles are measured precisely? 



9 129 



