# Is there something like z-value for log-normal distributions?

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

• Once you take logarithms, then you can work with z values on that scale. Jul 30 '20 at 21:20
• 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. Jul 30 '20 at 21:23
• @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?)
– Dave
Jul 30 '20 at 21:23
• 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. Jul 30 '20 at 21:35

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.

• 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. Jul 30 '20 at 22:11
• 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.
– Dave
Jul 30 '20 at 22:15
• 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.
– Dave
Jul 30 '20 at 22:15
• That makes sense, thank you! Jul 30 '20 at 22:20