STATISTICAL TREATMENTS 203 



mated from the characteristics of a sample or a series of samples. The bet- 

 ter the sampling procedure, the better the estimate of m and or'~. Each 

 sample, even if it includes only one variate, has a sample mean, which 

 might be nearly the same as the population mean or quite different. If 

 a large number of samples is taken, the average of all the sample means 

 provides a good estimate of the population mean. 



The sample mean is calculated in the same way as the population 

 mean but is given the symbol x. 



Xx 

 X = — 

 n 



where n is the number in the sample. 



If n is large, the variance of the sample (s") can be calculated in the 

 same way as the population variance. The sample variance only approxi- 

 mates the variance of the population, however, and tends to underesti- 

 mate the population variance by an amount equal to (w — 1)/m. For 

 this reason, sample variance is calculated by 



, s (x — xy 



s- = -. — 



n — 1 



The standard deviation (s) of the sample is the square root of this value. 

 A working formula which makes computation on an electric calculator 

 easier is 



2,X^ — 



2 W 



n — 1 



The standard error of the mean is a commonly used figure. It is 

 defined as the standard deviation of a distribution of means. If a large 

 number of samples is taken from a population, the means of the samples 

 are distributed in a normal curve. The variance of the distribution of 

 means, cr^^^ can be shown to equal crVw. The standard error of the mean, 

 then, is o-/\/n. There are other ways of arriving at a value for the stand- 

 ard error; some of these will be discussed later with some other special 

 statistics. 



Tests of significance 



The aim of statistical analysis is an estimate of the significance or 

 meaning of the data. The analysis reveals the probability that the ob- 



