In many instances, it is impossible to evaluate a parameter precisely, for two reasons:
The population may be so large as to preclude measurement of every object, and meas-
urement may entail destruction of the object, as in finding the breaking strength of a ca-
ble. In these cases it is necessary to estimate the parameter by drawing a representative
sample and evaluating the corresponding statistic. The process of estimating a parameter
by means of a statistic is known as statistical inference.
Since a statistic is a function of the manner in which the sample is drawn and thus is
influenced by chance, the statistic is a random variable. The probability distribution of a
statistic is called the sampling distribution of that statistic. Consider that all possible sam-
ples of a given size have been drawn and the statistic S corresponding to each sample has
been calculated. A characteristic of this set of values of S, such as the arithmetic mean, is
referred to as a characteristic of the sampling distribution of S. In the subsequent material,
the term mean refers exclusively to the arithmetic mean.
Notational System
Table 26 is presented for ease of reference. Here N = number of objects in the population;
IJL and a = arithmetic mean and standard deviation, respectively, of the population; n =
number of objects in the sample; X and s = arithmetic mean and standard deviation, re-
spectively, of the sample; fjis and <rs = arithmetic mean and standard deviation, respec-
tively, of the sampling distribution of the statistic S.
Basic Equations
The mean and standard deviation of the sampling distribution of the mean are
Vx = V (14)/ N-n s, _,
°* = (7VW^7 <^15 >If the population is infinite, Eq. 15 reduces to
VX=-^ (15a)TABLE 26. Notation
Sampling distribution
Characteristic Sample Population of a statistic S
Mean X JJL JJLS
Standard deviation s a as
Number of items n N