4.1 The problem

Until now we have been considering how to describe or summarise a set of considered simply as an object in its own right. Very often we want to do more than this: we wish to use a collection of observed values to make about a larger set of values; we would like to consider a particular set of we have as representing a larger class. It turns out that to accomplish this is by no means . What is more, an exhaustive treatment of the difficulties is beyond the of this book. In this we can only provide the reader with a general outline of the problem of making from observed values. A full understanding of this exposition will depend to some degree on familiarity with the content of later . For this reason we suggest that this is first read to a general grasp of the problem, and returned to later for re-reading in the light of subsequent .

We will the problem of by introducing some of the cases which we will in greater detail in the to come. One, for example, in 8, concerns the size of the comprehension vocabulary of British children between 6 and 7 years of age. It is not possible, for practical reasons, to test all British children of this age. We simply will not have the . We can only test a sample of children. We have learned, in 2 and 3, how to make an description of an observed group, by, for example, a histogram or calculating the mean and standard of the vocabulary sizes of the subset of children . But our interest is often broader than this; we would like to know the mean and standard which would have been obtained by testing all children of the age. How close would these have been to the mean and standard actually observed? This will depend on the relationship we expect to hold between the group we have to measure and the larger group of children from which it has been . How far can we the characteristics of this latter group to be to those of the smaller group which has been observed? This is the problem of : how to from the properties of a part the likely properties of the whole. It will turn up repeatedly from now on. It is worth at the outset that because of the way in which samples are in many studies in linguistics and applied linguistics, it is often simply not possible to generalise beyond the samples. We will return to this difficulty.