Two standard methods measure the secular trend hidden inside a time series. The method of moving averages smooths the series by replacing each value with the average of a block of neighbouring years; it is simple and objective but loses trend values at both ends of the series and cannot be used to forecast beyond the data. The method of least squares instead fits an algebraic straight line $Y = a+bX$ to the whole series by solving the normal equations $\sum Y = na+b\sum X$ and $\sum XY = a\sum X + b\sum X^2$ — coding the years around the middle year makes $\sum X = 0$ and simplifies $a = \sum Y / n$, $b = …