18 SUMMARIZATION OF DATA Ch. 2 



8. The following data are like those of problem 7, but taken on a different 

 direction finder. Obtain the variance and the a for these data given that 2X 2 

 = 116,830, 2X = 1708. 



X: 66, 68, 69, 68, 71, 70, 66, 70, 68, 67, 68, 73, 68, 65, 72, 73, 68, 67, 69, 65, 64, 

 66, 67, 70, and 70 (degrees from north). Arts. 5.58, 2.36. 



9. Work problem 8, after subtracting 60 degrees from each bearing. How 

 much were fi and <r changed? How much were the x i changed? What if only 

 50 degrees had been subtracted? 



10. Use one-sixth of the range in problem 6 as an estimate of the standard 

 deviation, and compare this estimate with the true standard deviation. 



Ann. 3.38. 



11. Given that for Table 2.01 SX = 123,445, and 2X2 _ 12,693,988. Calculate 

 the arithmetic mean and the variance about the mean, /x. 



12. Given the following six yields of Ponca wheat at Manhattan, Kansas, 

 compute their mean after first subtracting 27 from each number. (Data 

 provided by Department of Agronomy, Kansas State College.) Yield 

 (bushels/acre) : 27.2, 40.9, 46.0, 38.1, 43.8, 46.3. 



Ans. 13.2 bushels per acre; therefore, true mean = 40.2. 



13. The test weights corresponding to the bushel yields of problem 12 were 

 as follows (data from same source) : 59.3, 60.7, 60.6, 60.2, 61.9, 58.1. Calculate 

 the midrange, the arithmetic mean, and the variance. 



14. In problems 12 and 13, which of the types of measurement, yield or test 

 weight, gives the more consistent results according to this evidence? Give 

 reasons. 



15. Write down every fiftieth score in Table 2.01, starting in the upper left- 

 hand corner of the table and working from left to right. Compute the arith- 

 metic mean of the sample thus obtained and compute the percentage error 

 relative to the true mean, 95.7. 



2.2 THE AVERAGE (OR MEAN) DEVIATION 



A measure of the variation about the arithmetic mean based 

 on the numerical values (signs are ignored) of the Xi was mentioned 

 in the preceding section. If we were to set out to devise a simple 

 and logical way to measure the dispersion of a group of numbers 

 about some point, such as the arithmetic mean, we might well decide 

 to use what is called the average (or mean) deviation. It is the 

 arithmetic mean of the Xi each taken as a positive number regardless 

 of its actual sign. For example, consider the weights of problem 

 2.11. The numerical deviations from the mean, jx, were found to be: 

 29, 16, 11, 7, 4, 10, 20, and 29 pounds. On the average — that is, con- 

 sidering the arithmetic mean as the average — the weights of those 



