DISTRIBUTION OF THE SAMPLE MEAN 



157 



30 34 38 42 



50 54 



78 



58 62 

 Xorx 



Figure 6.21. Frequency distribution curve for a normal variate, X, and also 

 for the sampling mean, x , for samples with n = 10. 



is very nearly equal to cr/vn = 10/ v 10 = 3.1G, to two decimals, 

 where the symbol n is used to denote the number of observations 

 taken in the sample. 



The preceding discussion has suggested three features which are 

 exhibited by a large number of sample means obtained from a normal 

 population of numerical measurements. These features are: 



(a) Although it is impossible to predict the actual content of a 

 particular future sample it may be possible to predict the type of fre- 

 quency distribution which the sample means will follow, for example, 

 a normal distribution. 



(b) The average sample mean will be of essentially the same magni- 

 tude as the mean of the population sampled. 



(c) The sample means, X{, will display less variability than the X's 

 of the population. It is logical that the variability of the sample 

 means — from sample to sample — should decrease as the size of the 

 samples increases. It was suggested that a factor l/-\/n is involved 

 here. 



The following theorem is given without proof because that proof is 

 inappropriate to this book. The theorem is stated here for the purpose 

 of replacing the indefiniteness of statements (a), (b), and (c) above 

 with precise information which can be used in practice. 



Theorem. If a population of numerical measurements, X, con- 

 forms to a normal frequency distribution with mean, n, and stand- 



