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I'm looking for a reasonable way to measure how unlikely a data point is assuming it's generated by a random variable that follows log-normal. Do we have something like Z-value for normal distribution that can be applied to lognormal distribution?

To get the parameters of the distribution, I'm following a method similar to answers provided to this question:

shape, location, scale = scipy.stats.lognorm.fit(listofdata)
mu, sigma = np.log(scale), shape
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    $\begingroup$ Once you take logarithms, then you can work with z values on that scale. $\endgroup$
    – Nick Cox
    Commented Jul 30, 2020 at 21:20
  • $\begingroup$ The series I'm working with has the features of a lognormal distribution by its nature. I'm not taking a log of a normal data series. Does that change anything? Datapoints are heavily focused around 1 with long-tail toward infinity, with a maximum lower than 3. $\endgroup$
    – Mehdi Zare
    Commented Jul 30, 2020 at 21:23
  • $\begingroup$ @NickCox What would you say about taking the $0.025$ and $0.975$ quantiles like we would get from $z=\pm2$? (Or perhaps calculating the empirical quantile and walking it back to the z-value?) $\endgroup$
    – Dave
    Commented Jul 30, 2020 at 21:23
  • $\begingroup$ A long tail toward infinity and a maximum lower than 3: I have difficulty envisaging that. No matter: in my view the best check of lognormality is a normal quantile plot of the logarithms. @Dave's suggestions indicate the kind of numerical calculations you can add to that. $\endgroup$
    – Nick Cox
    Commented Jul 30, 2020 at 21:35

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To elaborate on my comment, if you want to give something that is like a z-value equivalent, you could calculate the quantile of the point that you have and then give the z-value for that quantile in a normal distribution.

If you get that your point is at quantile $0.975$, you know that to be be $z_{equivalent} = 1.96$, since quantile $0.975$ of a normal distribution is equal to a z-value of $1.96$.

I'm not sure how much I like this, but if you're trying to communicate with an audience that understands z-scores but not quantiles, this might communicate what you're trying to say.

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  • $\begingroup$ I don't have much experience with lognormal, but have been using z-value as a measure of how unlikely is a data point compared to the sample it's originated from. Do you imply that I can use the same concept on a lognormal distribution? I was doubtful of using z-value because the lognarm is not symmetrical around mean. $\endgroup$
    – Mehdi Zare
    Commented Jul 30, 2020 at 22:11
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    $\begingroup$ For a normal distribution, z-score is a proxy for the quantile, so if you say that you have a z-score of $1.96$, what you mean is that you're in the top $2.5\%$ of the distribution. What I propose is that you calculate the quantile directly. For lognormal, you won't have something familiar like $z=1.96 \implies$ quantile $0.975$, but if you get that you're at the $0.975$ quantile of your distribution, you could report that as being a z-score equivalent of $z=1.96$. This might be useful if your audience does not understand quantiles but is comfortable with z-scores. $\endgroup$
    – Dave
    Commented Jul 30, 2020 at 22:15
  • $\begingroup$ So if you get an observation that has a z-score equivalent of $5$, you could report that the point is as unlikely as a z-score of $5$ in a normal distribution, so pretty unlikely. $\endgroup$
    – Dave
    Commented Jul 30, 2020 at 22:15
  • $\begingroup$ That makes sense, thank you! $\endgroup$
    – Mehdi Zare
    Commented Jul 30, 2020 at 22:20

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