So...
1) You admit that decay is not linear.
Yes. But that is the point.
2) You admit that we can't see the whole pattern.
We
can see the whole pattern. If there were deviations from our pattern we would see it quickly on a residual plot
You state correctly that a short-term plotting of findings can instantly tell us if the decay is not linear. However, just because it is linear for the short-term past tells us nothing about its long-term action.
Of
course it is not linear. What the hell gave you that impression? We are continuously taking multiplying by 1/2. That is not a linear eqation. We can
make it a linear equation by modifying the units. The equation y=2^x can be turned into a linear function by transforming the units into a logarithim. It seems that I need to explain this entire relationship to you.
1. We take data plotting remaining percentage of radioactive elements against time.
2. We plot this daata on a graph.
Now this data will
not form a line or curve. There is no reason to expect it too due to human error and a variety of other factors, namely our equipment.
3. We note that the data is non-linear, so we transform the graph so it becomes linear, or we just leave it be if the equation is not complicated. Linear equations simplify predictions though.
4. We calculate the least squares regression line of this data. The general equation of this line is
(predicted Y) = A*X+B (in statistics this changes but is besides the point)
5. We calculate the
for each of our data points. We take our actual value and subtract the expected value. This gives us a value for each data point called a residual.
6. Now we plot the residuals against X. If points are randomly scattered above and below a line, there is no bigger pattern. If there is a clear pattern, such as a curve, then there is a bigger pattern.
The ~17% error is due to (surprise surprise) known environmental factors that caused varying degrees of carbon buildup. Which, of course, begs the question... how can we know that we are accounting for all factors? Are we even safe to assume that we have likely accounted for all factors?
7. Let's trust your 17% margin of error figure. that means we must have an r^2 (correlation squared) value of .83. R^2 tells us how Y responds to X. An r^2 value of .83 means that 83% of the Y(isoptope concetration) is explained by change in X(time). Working backwards gives us an r (correlation) of +/- .911, meaning that there is a very strong positive of negative linear relationship between x and y. While the actual relationship is not linear, we have manipulated the data so that it is. Calculation of r is a bit long, but ask and I will explain how.