Computing error bars for decay.

The decay error bars are based on the scatter of the points about the least squares fit plotted in the bampqc.ps file. The variance in decay is computed from the deviations about the line. In general, for any least squares solution of the form $y=mx+b$ , the slope, m, is given by,

$$m=\frac{\left(\sum x_{i}\right)\left(\sum y_{i}\right)-N\sum\left(x_{i}y_{i}\right)}{\left(\sum x_{i}\right)^{2}-N\sum\left(x_{i}^{2}\right)}.$$ (15)

Here, N is the number of (x,y) pairs. We can estimate the variance in y from

$$\sigma_{y}^{2}=\frac{\sum\left(y_{i}-\left(mx_{i}+b\right)\right)^{2}}{N-1}.$$ (16)

The variance in the slope, m, follows from the equation used to compute m, equation(15), and the variance in the y values (which will be taken to be a constant given by equation(16)). For uncorrelated errors, the variance in the slope is given by

$$\sigma_{m}^{2}=\sum\left(\frac{\partial m}{\partial y_{i}}\right)^{2}\sigma_{y_{i}}^{2},$$ (17)

which reduces to the following

$$\sigma_{m}^{2}=\frac{N\sigma_{y}^{2}}{\left[N\sum\left(x_{i}^{2}\right)-\left(\sum x_{i}\right)^{2}\right]}.$$ (18)

The slope, m, for our problem is of course the decay factor (dB/m). Conversion to 1/m units follows by dividing by 8.68589 in the usual way.