Standard error of the log of a mean

Question

You estimate a positive mean \mu by the sample average \bar X, with \operatorname{Var}(\bar X) = \sigma^2 / n. You report results on the log scale. Use the delta method to find the approximate variance and standard error of \ln \bar X.

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The delta method comes from a first-order Taylor expansion of g around the true value \mu, which turns a smooth transform into a local rescaling by the slope g'(\mu): \operatorname{Var}\big(g(\bar X)\big) \approx \big[g'(\mu)\big]^2 \operatorname{Var}(\bar X). For g(\mu) = \ln \mu we have g'(\mu) = 1/\mu, so \operatorname{Var}(\ln \bar X) \approx \left(\frac{1}{\mu}ight)^2 \cdot \frac{\sigma^2}{n} = \frac{1}{n}\left(\frac{\sigma}{\mu}ight)^2. The standard error is therefore \operatorname{SE}(\ln \bar X) \approx \frac{1}{\sqrt n}\cdot\frac{\sigma}{\mu} = \frac{	ext{CV}}{\sqrt n}, where 	…

Standard error of the log of a mean

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