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Quite often in published research we see researchers apply log transformation to their data, and some claim that this makes the data closer to normal distribution. My questions are:

  1. Mathematically, why this might be true? In particular, it would be great if you could illustrate how log transformation brings the characteristics of the sample or data (such as: dispersion, skewness, etc. ) closer to those of a normal.

  2. Does it always bring data closer to normal (Or, are there situations in which it fails)?

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    $\begingroup$ 1. If the data is roughly log-normally distributed, then taking the logarithm transforms it to nearly normally distributed. 2. No - why would taking the log of a triangular RV make it normal? Or - any RV that can take negative values won't allow for a logarithmic transformation. $\endgroup$
    – corey979
    Commented Jun 24, 2020 at 22:12
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    $\begingroup$ A plot of log x versus x is the pivot here. Taking the logarithm of a right skewed distribution for which all values are positive will pull in high values relative to low values and make the distribution more nearly symmetric. Approach to normality is a more delicate question. Taking logarithms can't help otherwise -- with skewness. But there could be other good reasons for taking logarithms. $\endgroup$
    – Nick Cox
    Commented Jun 25, 2020 at 0:09
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    $\begingroup$ The above statement is too strong. For example, a slightly right skewed distribution may be made more strongly left skewed by taking logarithms: we say that the transformation is too strong and consider a weaker transformation, or indeed leaving the data as they arrived. . Away from the definition that the logarithm of a lognormal distribution is a normal distribution, it is difficulty to give universal rules. $\endgroup$
    – Nick Cox
    Commented Jun 25, 2020 at 0:30
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    $\begingroup$ Does this answer your question? Does a log transform always bring a distribution closer to normal? $\endgroup$
    – juod
    Commented Jun 25, 2020 at 0:47
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    $\begingroup$ In another aspect, exponential growth is quite common in nature. And taking a log will offset the growth and the residuals would be noise which would be normal. $\endgroup$
    – hbadger19042
    Commented Jun 25, 2020 at 2:56

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