Your average revenue per customer is meaningless
data.heapanalytics.com
data.heapanalytics.com
If, however, you are kind of self service, but do an Enterprise sales deal ... you have two funnels with two vastly different customer profiles. Not only is ARPU bad in this scenario, most numbers are.
The trick is narrowing the scope of a given metric enough that you can use it to make good decisions, and but not so much that it ignores important parts of a business.
Segmenting is great and can actually move the averages wildly, but that doesn't mean the averages are pointless. It means that averages are just one view of the data.
If I want a really high level view of the data, an average is great. If I want a super low level view of the data, looking at each customer or small groups of customers is great too. It depends on what you want to know and what problem you are trying to solve.
Use the right tool for the job.
On the technical side, when trying to boil down scads of metric datapoints into a single number, the statistics i usually end up using are the median and the 95th centile (or some higher centile). The median gives some sort of rough idea of where the middle of the main mass distribution is, ignoring the extremes, and the high centile gives an indication of where the top edge of the main mass is, ignoring freak outliers.
Would those be any use for revenue per customer?
Would we be better off with the median and some number that tries to capture the shape of the power law? Something that says "for every dollar you go up in revenue, the number of users drops by X%", or something like that?
For example, maybe you have a few different pricing tiers, and instead of optimizing for ARPU, you optimize for the number of people in a given tier that you can up-sell into a higher pricing tier. Now you're optimizing for something meaningful. Note that even if you did a good job in accomplishing this goal, your ARPU might actually go down, if you simultaneously saw a lot of growth in your lower pricing tiers from new signups.
10% comes from the top 0.1% who spend at least X$,
10% comes from 0.1%-0.9% who spend at least Y$
etc.
The value is to know what types of users you should focus on just be careful as you need a lot of data to make some of this stuff meaningful.
A lot of people do not understand the substance behind the numbers and this leads to garbage in, garbage out. That's what this article is about: Understanding what's behind the numbers and not being fooled by them. The previous piece was also about understanding the substance and knowing how to work the numbers to make other people happy with the proposed deal.
A good read on similar topics: How to Lie with Statistics.
First, I don't see how it's true that data with a relatively large amount of variance will tend to be power law distributed. Defining what a "large amount" of variance is is tough (it depends on your intuition and choice of variance metric) but there are lots of distributions with considerable variance that are, for example, normally distributed (many more than are power law distributed, as far as I can tell).
Second, if you find that this is misleading your projections, why not just use a different kind of average? For example, if you just want to know, "How much is the next customer likely to spend?", you might use the mode. Or, if you want a more robust average (i.e., less likely to be seriously thrown off by outliers), why not use the median? You can even complement these with confidence intervals if you want to get a sense of their precision.
Like twic already said, you need some indicator to understand what's going on with your business. I think that in many cases, this will be the mean. But if you want something more robust or more practical, perhaps the median or mode might suit you better.
For your second point – there are legitimate uses for other summary statistics (mode, median), but they can still be very misleading. For example, you mention using the mode of the distribution as a predictor of what the next customer will pay – this definitely wouldn't work for a company with a metered pricing plan, for example. Distributions are often not well characterized by singular summary statistics, Anscombe's Quartet (mentioned elsewhere in this comments section) is a good example of this: http://en.wikipedia.org/wiki/Anscombe's_quartet
Re: appropriate averages. Your case (metered billing) is an interesting one. I don't see why the mode is necessarily wrong - the next customer is most likely to spend the amount that is currently your most popular amount). However, in order to calculate a mode, you likely would want to bin your amounts into ranges so that $10.11 isn't treated as distinct from $10.12, etc.
You're definitely right about one summary statistic not being sufficient. I would advocate for visualization and summary statistics, with some estimate of your confidence in the estimate displayed visually.
Assume you know only the mean revenue and the maximum revenue (but forgot to measure variance). You could make an extreme scenario with the maximum possible variance to generate a "worst case" distribution. In this scenario, all customers either provide zero revenue or the maximum. This distribution has the maximum possible variance for a given mean and maximum.
Will you be profitable this year? Your chances will be better than the worst case scenario described above! If higher moments are known (variance, skew, etc.), more accurate bounds can be found.
In conclusion, the mean can be very useful, especially if higher moments are known.
If you run analytics on bad or incomplete information or information that doesn't realistically relate to something important to your bottom line, or if you try to interpret useful data without knowing what you should actually be looking at in the context of the service you're providing, it'll obviously be meaningless.