STATISTICAL TREATMENTS 201 



population includes all the people. There would be an infinite number 

 of possible repetitions of a certain measurement, and these together 

 would constitute a population. A population has certain characteristics, 

 including variability, that can be described in detail. Obviously experi- 

 mental work with populations is impractical; usually we work with 

 samples of populations. A sample of the human population might include 

 all the people in a city or all the people in a room. The smallest sample 

 is one person. The features of a sample are similar to the features of a 

 population but obviously cannot be identical. If a sample is large, it 

 more truly represents the population. Statistical procedures have been 

 developed for dealing with populations and with large and small samples. 



The normal curve 



One of the characteristics of a population, or even of a sample, is a 

 natural variability. In general, the individual values in the population 

 of numbers tend to clump around a certain average value, but some sep- 

 arate values will be very much larger or very much smaller. There are 

 several ways of describing patterns of variation, but when dealing with 

 populations, all these methods approach the bell-shaped "normal curve." 

 A population of numbers is likely to be normally distributed, that is, to 

 vary according to the normal curve. 



Several "parameters" can be used to describe this normal distribution 

 within populations, but two of these are most important. The value 

 around which the other values seem to be centered is the average or 

 mean. When speaking of populations it is given the symbol m. The 

 mean is found by adding all the values and dividing by the number (N) 

 of values. This process is expressed mathematically by calling each value 

 or variate xi, X2, xs, . . . , x\. The symbol Xi stands for any value of x: xi, 

 then X2, then xs, and so on. The capital sigma (2) is used to indicate 



A' 



summation, so S Xi means "the sum of all the x values from xi to xn" The 



1 

 arithmetic mean then is 



N 



Zj Xi 

 1 



■m = 



N 



In the normal curve, the total of all deviations above the mean is equal 

 to the total of all deviations below the mean. If there is likely to be 



