Variance of a transformed mean
It is sometimes necessary to transform data to, for example, downweight the influence of outliers, prior to performing any analysis. The reciprocal of reaction times is used for this purpose.
A transformed mean, m, with variance $$\mbox{s}^text{2}$$ on a sample of size, n, has a backtransformed variance (ie on the original scale) given below obtained using the delta method.
Note: Please ignore the '^' signs in the second column of the below table. These appear to be needed, for some reason, to format the table below.
F(m) |
$$\mbox{F}text{-1}(\mbox{m})$$ |
$$\mbox{Variance } \mbox{F}^text{-1}(\mbox{m})$$ |
||
Ln(m) |
$$etext{m}$$ |
$$\frac{\mbox{e}text{2m}\mbox{s}text{2}}{\mbox{n}}$$ |
||
1/m |
1/m |
$$\frac{\mbox{s}text{2}}{\mbox{m}text{4} \mbox{n}}$$ |
||
$$\sqrt{\mbox{m}}$$ |
$$\mbox{m}text{2}$$ |
$$\frac{\mbox{2ms}^text{2}}{\mbox{n}}$$ |
||
$$2\mbox{ arcsine } \sqrt{\mbox{m}}$$ |
$$(\mbox{sin(m/2}))text{2}$$ |
$$\frac{\mbox{sin(m/2)}text{4}\mbox{s}text{2}}{\mbox{n}}$$ |
||
- [:FAQ/FVars/Fab: Variances of simple combinations of estimates such as means]
Ordinarily when using power transforms we transform before taking the mean e.g. taking logs of raw data and then taking means of these logged values rather than averaging the raw data first and logging the resultant mean (See exploratory analysis graduate Statistics Talk [:StatsCourse2011:here)].