the following properties of exponentially decreasing weights are useful in making a suitable choice: at step , the weight approximately equals , the tail area the value , the head area . The tail area at step is . Where primarily the closest observations matter and the effect of the remaining observations can be ignored safely, then choose such that the tail area is sufficiently small. Weighted averages of functions The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus. Correcting for over- or under-dispersion Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that is too large. The correction that must be made is where is the reduced chi-squared: The square root can be called the standard error of the weighted mean (variance weights, scale corrected). When all data variances are equal, , they cancel out in the weighted mean variance, , which again reduces to the standard error of the mean (squared), , formulated in terms of the sample standard deviation (squared), See also Average Central tendency Mean Standard deviation Summary statistics Weight function Weighted average cost of capital Weighted geometric mean Weighted harmonic mean Weighted least squares Weighted median Weighted moving average Weighted sum of variables Weighting Standard error of a proportion estimation when using weighted data Ratio estimator References Further reading External links Tool to calculate Weighted Average Means Mathematical analysis Summary statistics