Sec. 1.1 HISTORY 3 



units, and methods for summarizing groups of data. Their efforts to 

 apply mathematical analysis to such problems helped to lay the 

 foundation for the statistical methods now in use. 



A third step in the evolution of statistical analysis and reasoning 

 came in the development of the mathematical theory of probability, 

 without which statistical reasoning could never have attained its 

 present reliability and usefulness. Games of chance were especially 

 popular among the well-to-do of the sixteenth and seventeenth cen- 

 turies; and many problems involving probability were presented to 

 the mathematicians of the day for solution. For example, an Italian 

 nobleman asked Galileo to explain the following facts: If three dice 

 are thrown, the numbers 9 and 10 can each be obtained from six 

 different combinations of the numbers on the faces of the dice; but 

 it has been found from experience that a sum of 10 appears more 

 frequently than a sum of 9. Why so? By an enumeration of all the 

 physically different ways that three dice can produce sums of 9 or 

 10, Galileo was able to answer this question clearly and convinc- 

 ingly. His answer appears to be the first published application of 

 the theory of probability.* Other prominent mathematicians such 

 as Pascal, Fermat, James and Daniel Bernoulli, de Moivre, Laplace, 

 Gauss, Simpson, Lagrange, Hermite, and Legendre developed many 

 important theorems and methods of attacking problems involving 

 chance events, and they passed this information on for later use by 

 mathematical statisticians. 



During the last quarter of the nineteenth century, Sir Francis 

 Galton took the lead in the development of the ideas of regression 

 and correlation when two (or more) measurements are made simul- 

 taneously on each member of a group of objects. He appears to have 

 built his ideas around problems in genetics. Karl Pearson and C. 

 Spearman extended this theory and applied it to studies in the social 

 sciences, especially psychology. Karl Pearson and others also had 



* The nature of Galileo's solution is as follows. A sum of 9 can be obtained 

 from any of the following combinations of numbers on three dice: 1, 2, 6; 1, 3, 

 5; 1, 4, 4; 2, 2, 5; 2, 3, 4; or 3, 3, 3. A sum of 10 is obtained from any of these: 

 1, 3, 6; 1, 4, 5; 2, 2, 6; 2, 3, 5; 2, 4, 4; or 3, 3, 4. There are six different com- 

 binations giving each of the sums 9 and 10; but the different combinations do 

 not occur equally frequently. For example, the combination 3, 3, 3 can be 

 thrown but one way whereas the combination 3, 3, 4 can occur on any of three 

 different throws, and hence would tend to appear three times as often as 3, 3, 3. 

 As a matter of fact, a 9 can be thrown in twenty-five different ways, a 10 in 

 twenty-seven different ways, which is the reason that the 10 appears more 

 frequently in games than the 9. S^C\Cl 



