Sec. 2.1 ARITHMETIC MEAN AND STANDARD DEVIATION 17 



this population were perfectly normal, that percentage would be 

 95.4. Also the interval /i±3(j includes 99.8 per cent of the ACE 

 scores, whereas a normal population would have 99.7 per cent of its 

 members in that interval. The reader can determine how closely the 

 population of Table 2.01 conforms to the normal requirement that 

 38.3 per cent of the measurements shall lie not more than one-half a 

 standard deviation above or below the mean, p. 



More discussion of normality and of population distributions will 

 come later; the point of the above discussion is that knowledge about 

 the mean and the standard deviation is useful in the study of one of 

 the most important types of populations of data. 



PROBLEMS 



1. Calculate the arithmetic mean and the standard deviation of the following 

 numbers: 2, 3, 9, 7, 5, 4, 10, 6, 3, 1, and 5. 



2. Make up three sets of numbers, each of which has fi = 7. 



3. Compute the x i for problem 1 and verify that 'Lx i = 0. 



4. Given the numbers 0, 8, 0, 1, 1, 1, 10, 2, 1, 1, 2, 3, 0, and 1, compute n. 

 Does fj. seem to you to be a good average for these numbers? Why? Ans. 2.21. 



5. Suppose that the mathematics grades for a certain class were 54, 95, 68, 71, 

 87, 75, 84, 63, 76, 81, 70, 90, 73, 77, and 61. Calculate /i and a, using the indi- 

 vidual x i first and then using formula 2.13 for <x. 



6. The following percentages of protein in samples of pasture grasses were 

 made available by Dr. George Wise, formerly of the Department of Dairy 

 Husbandry at Kansas State College. Compute /jl, <r 2 , and <r, given that SX 

 = 1423.33 and that 2X2 _ 21,924.2025. 



Ans. 14.23, 16.66, 4.08. 

 7. The following are bearings taken with a radio direction finder on a signal 

 sent repeatedly from a fixed location. Compute their arithmetic mean and 

 standard deviation as though these data constituted a population. 



X: 9, 8, 6, 4, -10, 6, 7, 10, 8, 9, 7, 6, 8, 8, 10, 10, 8, 8, 10, 9, 10, 7, 7, 3, and 8 

 (degrees from north). 



