A THEORY OF STATISTICS. 
No inference is possible from positive associations of 4B and BC. 
An inference is only possible from negative associations if a lie between 
‘183 . . . .and ‘215... . Note that cannot exceed 1. 
(3) y<3(6x— 222-1) 
> 3(3x + 222), 
subject to the conditions y<:0, <5z-1, pa. 
As in (2), no inference is possible from positive associations of 4C and BC 3 
an inference is possible from negative associations if z lie between ‘177 . . . . 
and 224 , . , , Note that 2 cannot exceed 3. 
CHAPTER V. 
1. 4,068. B, 0°36. 
CHAPTER VI. 
1..1200; 200. 2. 100; 20. 3. 146-25, 4. 2165. 
CHAPTER VII. 
2. Mean, 15673 1b. Median, 15467 Ib. Mode (approx.) 150'6 1b. (Note 
that the mean and the median should be taken to a place of decimals further 
than is desired for the mode: the true mode, found by fitting a theoretical 
frequency curve, is 151°1 1b.) 
3. Mean, 0'6330. Median, 0'6391. Mode (approx.), 0'651. (True mode 
is 0653.) 
4. £35'6 approximately. 
5. (1) 116°0. (2) Means 77°4, 89°0, ratio 114°9. (3) Geometrical means 772, 
88-9, ratio 1152. (4) 115-2. 
6. (1) 921,507. (2) 916,963. 
7. 1st qual. 10s. 63d. 2nd qual. 9s. 21d. 
8. n.p. Ifthe terms of the given binomial series are multiplied by 0, 1, 2, 3 
. . . , note that the resulting series is also a binomial when a common factor 
is removed. [The full proof is given in Chapter XV. § 6.] 
CHAPTER VIII. 
2. Standard deviation 21'3 1b. Mean deviation 16:4 1b. Lower quartile 
142°5, upper quartile 1684 ; whence @=12-95. Ratios: m.d./s.d. =0°77, 
@/s.d. =0'61. Skewness, 0:29. : 
3. Approximately lower quartile=£26'1, upper quartile=£54'6, ninth 
decile = £94. 
5. (1) M=732, ¢=17'3. (2) M=732, ¢=17"5. (3) M=73"2, &=180. 
(Note that while the mean is unaffected in the second place of decimals, the 
standard deviation is the higher the coarser the grouping.) 
6. ~/n.pg. The proof is given in Chapter XV. § 6. wr 
7. The assumption that observations are evenly distributed over the 
+0) 2