The plot shows the true density function , which is normal with mean 0 and variance 1, and its nonparametric estimate obtained using a kernel smoothing with parameter . The initial value for of 0.95 corresponds to the kernel smoother that minimizes the integrated mean-square error, IMSE. Graphically, IMSE is the area between the two curves. Smaller values correspond to less smoothing and larger to more smoothing. With less smoothing, the red curve wobbles more around the true value, but there is less systematic bias.

Although this Demonstration only deals explicitly with an example nonparametric density estimation, the situation is a paradigm for all empirical statistical model building.

The variance-bias tradeoff is most simply explained mathematically in terms of estimating a single parameter with an estimator . Then the mean-square error of estimation for provides an estimate of the accuracy of the estimator and is defined by , where denotes mathematical expectation. The bias is defined by and the variance is ; hence . Thus there are two components to the error of estimation—one due to bias and the other variance.

This paradigm is very general and includes all statistical modelling problems involving smoothing or parameter estimation. For a more general discussion of this aspect, see §2.9 of T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd ed., New York: Springer, 2009.