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I'm asking for a pointer for a source of definitive descriptions of what types of data are best summarized by the arithmetic, geometric, and harmonic means.

I see regulators apply the geometric mean to water chemical concentrations rather than using the arithmetic mean. I want to know whether the geometric mean of a set of chemical concentrations (e.g.,in mg/L) is an appropriate representation of the expected value. If not, I want to explain this to non-technical decision-makers; if so, I want to understand why my assumption is wrong

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    $\begingroup$ "Appropriate" for what purpose? Often the GM is suitable for summarizing data or distributions of chemical concentrations, but for inputs to, say, a human health risk assessment the GM might be serving as a biased proxy for the AM. Thus, any reasonable answer to your question requires consideration of the context and purpose. $\endgroup$
    – whuber
    Commented Jan 22 at 17:15
  • $\begingroup$ In many US regulatory cases, the testing methodology for chemical concentrations is governed by an ASTM standard. Those ASTM standards often have a discussion about the test statistic for combining multiple samples and they explain the reasoning. Do you have access to those standards? I would link to examples, but they are generally behind a pay wall. $\endgroup$
    – R Carnell
    Commented Jan 22 at 17:43

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I can't really answer your question, but can provide some information to help you clarify it.

Many variables follow lognormal, not normal, distributions. This is often the case for the concentrations of chemicals.

For an ideal lognormal distribution, the geometric mean is the same as the median. The arithmetic mean is larger.

You asked about the expected value. That is another name for the arithmetic mean. The geometric mean of a sample is the most likely value for the median, so is the typical value.

For samples from a lognormal distribution, the arithmetic mean and geometic mean are distinct. In this question, I asked when to use each -- I included some background material and citations, and there are useful answers:

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

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  • $\begingroup$ My thanks to all of you: whuber, R. Carnell, and Harvey Motulski. I did not sufficiently explain the context for my question -- water quality discharge permit compliance -- and I really appreciate your comments. They help me resolve my reason for asking the question. I'm good now. $\endgroup$ Commented Jan 23 at 18:39
  • $\begingroup$ For permit compliance, it is usual for the permit to specify what property of the distribution is subject to compliance (and, usually, to indicate what statistic will be used to assess compliance). If the permit isn't explicit about that then it wasn't properly written. $\endgroup$
    – whuber
    Commented Jan 24 at 15:05

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