When summarizing data from a lognormal distribution, when does it make sense to report the arithmetic mean vs. geometric mean

Question

When data are sampled from a lognormal distribution with a reasonably large geometric standard deviation, the distribution is asymmetrical and the arithmetic mean will be distinct from (and larger than) the geometric mean.

• The arithmetic mean can be thought of as the center of gravity of the distribution. Parkin and Robinson suggest it makes sense to use the arithmetic mean "for situations where an estimate of the total mass
  of a variable (either a chemical constituent or a process) is required" (1).

• The geometric mean is equivalent to the median (for the entire distribution; not for any particular sample). It is the center point, in the sense that half the values are higher, and half are lower.

I've always thought that the geometric mean is a more sensible (and standard) way to describe the distribution.

Of course, there is no need to choose, and reporting both means makes sense in some situations. But if you want to report only one mean, what is the best way to choose (and explain your choice)?

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Addendum. Here is the key section from page 222 of Parkin and Robinson (using N not for population size, but for the element nitrogen):

1.Parkin, T. B. & Robinson, J. A. Analysis of lognormal data. Adv Soil Sci 20, 193–235 (1992).

Answer 1

If data comes from a lognormal distribution, then yes, the geometric mean is the single best summary statistic.

The key is the hypothesis. If the post had asked about choosing between normal and lognormal models, then the arithmetic mean might be more useful. Instead, the post asked about "data sampled from a lognormal distribution" -- i.e. a distribution already assumed or chosen to be lognormal. So whatever reasoning went into that assumption or choice probably would also suggest using the geometric mean.

Here are two examples:

1. A lognormal distribution might model weather or disasters such as water flows, rainfall, and hurricanes, or earthquakes, fire damages, and insurance claims. Then the sampled events can differ by orders of magnitude. So the arithmetic mean will be roughly $1/n$ times the largest case, which is not an especially useful number. By contrast, the geometric mean will estimate the median of the population, and be a reasonable guide to a common size for the events.

2. A lognormal distribution might model repeated financial processes, such as changes in stock prices, currency rates, or crypto values. Then the arithmetic mean does not scale nicely, so it is not an especially useful number. By contrast, reporting that the geometric mean is an increase of (e.g.) $1\%$ in a month or year or day is useful, because the powers $1.01^t$ will estimate the geometric means of samples over other time scales.

Answer 2

As the definition of the lognormal distribution suggests, if $Z\sim\mathcal{N}(\mu,\sigma^2)$ then $X=\exp(Z)$ follows the $\mathcal{logN}(\mu,\sigma^2)$ distribution. I think that the answer to "which mean to report?" first needs to go through "which part of the data are you interested in?". That is, are you interested in the measurements themselves ($X$) or in the underlying normal variable ($Z$)?

If you have some measurements and you are interested in the underlying variable (for example, your measure some power ratio and would like to learn something regarding this data in dB), then I'd report the geometric mean. Not only you inform the reader about $X$, but it also follows that $\sqrt[n]{\prod_{i=1}^{n}X_i}=\exp(\frac{1}{n}\sum_{i=1}^{n}Z_i)$. That is, one quantity tells two tales.

However, If you are interested in the measurements themselves - that is, you have some size which traditionally follows the lognormal distribution, such as the concentration of elements and their radioactivity in the Earth’s crust (example taken from here), then I'd report the arithmetic mean as it is the most common central metric.

PS. As far as I know, for a unimodal distribution the center of gravity lies in the mode, not the mean. While for $Z\sim\mathcal{N}(\mu,\sigma^2)$ we know that $\mu$ is not only the mean but also the median and the mode, this isn't the case for $X=\exp(Z)$. Its mean is defined as $\exp(\mu+0.5\sigma^2)$, its median is $\exp(\mu)$ and its mode is $\exp(\mu-\sigma^2)$.

Answer 3

I think that the standard deviation (or standard error) will help you decide. The variation within a log-normal distribution is asymmetric and so the conventionally calculated standard deviation will overstate the variation on the low end and understate it on the high end. That leads to confidence intervals that do not behave well.

In an example that will be familiar to you the confidence interval for a drug EC50 might go down to near or, sometimes, below zero. As that would imply the statistical possibility of super-duper potency, or more than infinite, that is misleading. (Unless you are a homeopathy practitioner!) The confidence interval for the logEC50 behaves correctly.

Given that, I prefer by far the geometric mean, and would call it the mean log EC50.

Answer 4

[I'm going to post an answer to my own question, in hopes of getting comments and a better answer. ]

Imagine your goal is to compare a variable like pollutant concentration or rainfall in various locations. At each general location, you collect measurements and find that over several time intervals. The distribution among those values is lognormal. You don't care about the typical value (median) in each location. What you really want to compare is the total amount of rain or pollutant in the locations. You can't know that, but the arithmetic mean is proportional to the total. Therefore comparing the arithmetic means is more sensible than comparing the geometric means.

Answer 5

If you only report one mean, you should choose the one with the most practical relevance. In situations where an estimate of the total mass matters, it is the arithmetic mean.

The mean is dominated by extreme values which is a desirable quality.

Situations where an estimate of the total mass matters (and not most common values) are situations where the total mass has most relevance, e.g. influence on the actions.

You are interested to know how extreme magnitudes of an earthquake or changes in a stock price, or rainfall can be because you have to prepare for that and before you prepare for most common magnitudes.

EDIT: So this answer can be seen as an extension to Spätzle's answer who says that it depends on the question you are trying to answer.

Matt F. Is right that the geometric mean is a good summary for the lognormal distribution because it fits the multplicative nature of the data generating process.

However practical implications might sometimes be more relevant which is why you want to show the arithmetic mean.