is due to measurement errors. If we knew that the first three measurements (in 1975, 1978, & 1980) constituted random scatter about the same well-defined trend of 1983-1989, then surprisingly it would be more accurate to predict values for these three years from the trend than to use the actual measurements.
An extreme example of the difference between extrapolation and interpolation for time series is world population (Figure 13). The validity of interpolated population within the last 2000 years depends on how much one trusts the simple pattern of Figure 13. The prolonged gap between 1 A.D. and 1650 conceivably could mask excursions as large as that of 1650-present, yet we know independently from history that such swings have not occurred. The combination of qualitative historical knowledge and the pattern of Figure 13 suggests that even the Black Death, which killed a large proportion of the population, caused less total change than is now occurring per decade. For purposes of defining the trend and for interpolation, then, both the distance between bracketing data points and the rate of change are important. Thus the great increase in sampling density at the right margin of Figure 13 is entirely appropriate, although a single datum at about 1000 A.D. would have lent considerable improvement to trend definition.
Extrapolation of world population beyond the limits of Figure 13 is both instructive and a matter of world concern. Predicting populations prior to 1 A.D. would be based on very scanty data, yet it appears that values would have been greater than zero and less than the 1 A.D. value of 0.2 billion. In contrast, extrapolation of the pattern to future populations suggests that the world population soon will be infinite. Reality intervenes to tell us that it is impossible for the pattern of Figure 13 to continue for much longer.
The three examples above are atypical in that they all are time series -- measurements of temporal changes of a variable. Interpolation, extrapolation, and indeed any interpretation of a time series is ambiguous, because time is an acausal variable. Often one can hypothesize a relationship between two variables that lends confidence to one’s interpretation. In contrast, the source of variations within a time series may be unmeasured and possibly even unidentified.
The challenge of avoiding the confounding effect of time is present in all sciences. It is particularly acute within the social sciences, because some variables that might affect human behavior are difficult to hold constant throughout an experiment. For example, consider the relationship between height and weight of boys, shown in Figure 14a. The relationship is nonlinear, and we might be tempted to extrapolate that a 180-cm-high boy could be as much as twice as heavy as a 160-cmhigh boy. Clearly neither height nor weight is normally distributed, and in fact it would be absurd to speak of the average height or weight of boys, unless one specified the boys’ age. Figure 14a is actually based on a tabulation for boys of different ages. Age is the causal variable that controls both height and weight and leads to a correlation between the two. Both change systematically but
