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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)?

ExcitedSnail
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  • 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.
    • – corey979 Jun 24 '20 at 22:12
  • Thanks! But it doesn't fully address my question. My question is about bringing data closer to normal(not nearly normal), it could be still far from normal after transformation. For example, does data that look like coming from triangular RV get closer to normal after log transformation? – ExcitedSnail Jun 24 '20 at 23:43
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    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. – Nick Cox Jun 25 '20 at 00:09
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    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. – Nick Cox Jun 25 '20 at 00:30
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    Does this answer your question? Does a log transform always bring a distribution closer to normal? – juod Jun 25 '20 at 00:47
  • @juod Thanks! This is helpful. It answered my second question, but not much for the first one. – ExcitedSnail Jun 25 '20 at 02:18
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    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. – hbadger19042 Jun 25 '20 at 02:56
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    Do these help? 1. https://stats.stackexchange.com/questions/93082/if-x-is-normally-distributed-can-logx-also-be-normally-distributed/93097#93097
    2. https://stats.stackexchange.com/questions/67437/confusion-related-to-which-transformation-to-use/
    3. https://stats.stackexchange.com/questions/418316/what-if-we-take-the-logarithm-of-x-how-does-skewness-and-kurtosis-change/     – Glen_b Jun 25 '20 at 03:09
  • @Glen_b Yes, they are helpful. Thanks! – ExcitedSnail Jun 29 '20 at 18:37
  • @kevin012 Thanks! It's helpful. – ExcitedSnail Jun 29 '20 at 18:38