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Note that we usually use ln (log to the base e) which is preferred to log10 (see [[http://stats.stackexchange.com/questions/27682/what-is-the-reason-why-we-use-natural-logarithm-ln-rather-than-log-to-base-10 | here]] for a rationale. |
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 s2 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) |
$$\mbox{F}-1$$ (m) |
$$\mbox{Variance } \mbox{F}^text{-1}(\mbox{m})$$ |
||
Ln(m) |
$$em$$ |
($$e2m s2 ) / n $$ |
||
1/m |
1/m |
$$s2 / (m4 n)$$ |
||
$$\sqrt{\mbox{m}}$$ |
$$\mbox{m}2$$ |
[(2ms)2 ]/n |
||
$$2\mbox{ arcsine } \sqrt{\mbox{m}}$$ |
$$(\mbox{sin(m/2}))2$$ |
( $$(cos(m/2)sin(m/2))2 s2 ) /n $$ |
||
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 the Exploratory Data Analysis Graduate Statistics talk here).
Note that we usually use ln (log to the base e) which is preferred to log10 (see here for a rationale.