Suppose that as a result of observation or experience of some kind we have obtained a set of values of a variable u corresponding to equidistant values of its argument; let these be denoted by u
1, u
2, … un
If they have been derived from observations of some natural phenomenon, they will be affected by errors of observation; if they are statistical data derived from the examination of a comparatively small field, they will be affected by irregularities arising from the accidental peculiarities of the field; that is to say, if we examine another field and derive a set of values of u from it, the sets of values of u derived from the two fields will not in general agree with each other In any case, if we form a table of the differences δu
1 = u
2 – u
1, δu
2 = u
3 – u
2, …, δ2
u
1 = δu
2 − δu
1, etc., it will generally be found that these differences are so irregular that the difference-table cannot be used for the purposes to which a difference-table is usually put, viz., finding interpolated values of u, or differential coefficients of u with respect to its argument, or definite integrals involving u; before we can use the difference-tables we must perform a process of “smoothing,” that is to say, we must find another sequence u
1′, u
2′, u
3′, …, un
′, whose terms differ as little as possible from the terms of the sequence u
1, u
2, … un
, but which has regular differences. This smoothing process, leading to the formation of u
1′, u
2′ … un
′, is called the graduation or adjustment of the observations.