how to measure the fluctuation of data used for fitting a curve?

Question

Given a series of data points that are used to fit a smooth curve, how to measure the fluctuation of the data?

Answer 1

I guess that you mean how you measure the accuracy of the fitted model? One of the basic measure is the MSE which measures the empirical mean squared error of the prediction. I.e., let $\hat{y}(x_i)$ be the fitted value for the $x_i$-th data point and $y_i$ its actual $y$ value, hence the MSE is defined as $$ MSE(\text{model}) = \frac{1}{n}\sum_{i=1}^n(\hat{y}_i - y_i)^2\, . $$ Note that the more you "smooth" your data, the lower the computed MSE. This does not necessarily mean that you got the best model as too smooth model will be an over-fitting and hence your model will probably predict poorly out of sample data.