Aristocrats in flyovers

Worldly success, measured by the accumulation of money, is no doubt a very dazzling thing; and all men are naturally more or less the admirers of worldly success.

While the Scottish reformist, Samuel Smiles, authored this self-help quote in the mid-19^th century, the subject of the overemphasis society places on having excess money has been around for millenniums.  Jesus often refers to it in the Bible.  To this day, we are uniquely interested in who makes up the extraordinarily wealthy in our society.  Now, we’re not just referring to the popular expression “top 1%”, but rather “the top 1% of the top 1%” (or the top 0.01%).  A century ago, an American would need a wealth of a million dollars to achieve this status.  While inflation has grown by a multiple of mid-single digits since then, the wealth threshold required to be in the top 0.01% has grown by a multiple of a couple hundred. 

Of course this narrow sliver of the top 1% still accounts for tens of thousands of people, but to put further perspective on this, for an American to be included in the even more exclusive Forbes 400, a wealth of nearly two billion dollars is needed.  Looking at this list of Americans, we see (or want to assume that there are) clusters of people residing in geographic locations where it is more expensive and more exciting to live.  For example, southern New York and northern California.  And quite humbly, some of these billionaires take the time to read this statistics web log. 

Though people have their own geographic preferences as well; some of the country's uber-wealthy prefer to live in the serenity of less-trafficked locations.  Some of these states are those that Americans only see by air; they are sometimes referred to in a derogatory sense as flyover states.  What if we wanted to sharpen our saw by only focusing our attention on the richest person in each geography, curious (as Samuel Smiles suggested we might) about who makes up the well-to-do and wealthiest in each of the 50 states?  This article goes about this analysis.  Creating a mathematical model to learn from the aristocrats, and provide insights for those who want to enjoy the same monetary comfort of the top-end of uber-wealth.  We see here that the more populated states have typically higher average wealth, though there is a probability advantage we mistakenly discount in less populated states to being able to be become the very wealthy.

Take a look at this recent list, which identifies the richest person in each state.  Sure it’s fun to see familiar names, and to hypothesize about what patterns might exist in the data.  It also seems intuitive that larger populations would have a greater opportunity for the richest person to be further along the extreme end of the wealth distribution.  But how do we mathematically prove that? 

Probability models can help identify how this works.  We start this article with some mathematical theory, in how we can represent the very tail-end of a probability distribution.  This is a unique portion of the distribution that we rarely need to model other than to explore the freakish curiosity of some event, such as modeling the wealth of a state’s richest person. 

Naturally we know that in each state, all else equal, there will be a mound-shaped probability distribution of wealth among people.  Bounded by zero on the low-end, and perhaps right skewed to a limit on the high-end.  Also, more populated states would have –minimally- their typical value a little higher versus the typical value from less populated states.  Maybe the entire distribution fits something similar to a lognormal model, as does many economic time series data.  Except we know there are upper limits on how much wealth one can accrue in a limited lifetime.  And with a large enough standard deviation away from the middle of the distribution, the distribution would have a wider than normal tail distribution, in order to better allow for more extreme outcomes. 

These are just some initial starting points to think broadly about the overall parameters that shape the basic distribution.  Also note that the excess kurtosis (degree of fat-tails) associated with a lognormal distribution can be shown through complex moment functions: 

Focusing our attention now on just the extreme portion of the distribution that we are interested in, we also know that we might be better served there by using a more exotic distribution.  Such as a power function or a Pareto or a variation of the Weibull, etc.  Not all of these would have easily identifiable excess kurtosis.  And it turns out that it doesn’t matter the actual distribution upfront here.  Because given that there are millions of people in each state (and tens of millions in larger states), our only focus is to assess the patterns we observe at the very tip-end of the distribution.  See the illustration below, as well as in other applications we can linearly approximate (see here for example).

Take a moment to appreciate the range of combinations (curves shown in grey) that can be valid for this tail distribution we are modeling.  The illustration on the right, which we will ultimately not need, shows what we call the conditional or expected distribution at the extreme tails (so that the probability model sums to 100% or 1).  We don’t need it in this case since we are only focused on modeling the probability statistics with the maximum value only.

Let’s also take a chance to note some of required characteristics, which can help narrow the types of extreme value models we can use.  Of course we would want the slope of the distribution function to continue to be negative as we move towards the infinite end of the tail distribution.  Concurrently we would want the second derivative (or the change in the slope itself) to be positive, so that the curve takes the convex form that we generally see in the illustration above. 

This latter point is important to be able to isolate a non-parametric curve that best fits the tail end of the distribution.  It is also worth solving before we move ahead, via the probability calculus below, that for the popular normal distribution that implies our tail event is within the other 16% (per tail) probability of occurrence.  This of course means a greater than one standard deviation distance away from the average, which we clearly will satisfy for these extremely wealthy individuals.  Note that even for the normal distribution, the tails of which follow an exponential path as can be seen in the f(x) formula below.  And rules of thumbs still generally dominate the academic literature in working with the expected maximum value depending on sample size.

Here is the calculus formulae to understand when the second derivative [f’’(x)] turns positive.  For simplicity we assume the distribution is symmetrically centered about zero, and we could translate to another value for expanded analysis, if we needed.

So for the last equation above, an equation representing the second derivative of variable x, it can only be zero when x=σ, or equal to a standard deviation that cuts off 16% of the distribution in each tail.  Since a lognormal distribution –related to a normal random variable x- is equal to e^x, we can combine that with our undertanding of the expanding scale of the lognormal distribution at the upper end of it (e.g., the distance from the first quartile to the median is less then the distance from the third quartile to the median).  So this means that the positive convexity of a lognormal distribution is attained at just less than the mean-adjusted value of e^σ.

Now using the probability mathematics just developed, let’s now look at our theoretical probability distribution associated with the highest wealth in each state.  Note that on the vertical axis we are capturing one minus the cumulative probability distribution functions (cdf), and not the probability distribution function (pdf) itself.  Each actual state value is in red (except for two in blue).  And the pdf that is shown in the illustration above maps directly to these inverse cdfs in the chart below, and hence they are unnecessary for our proof of what is the model of the maximum wealth event along the probability distribution.





Again we amalgamated our 50 richest list, by taking the richest person in each state.  This technique does not mean that everyone would be a billionaire though, and 10 of these richest people on the list are not.  However, we can see something on the far right of the illustration, at the greatest wealth of about 80 billion dollars.  Bill Gates is not only the richest person for this one data point represnting Washington state, but he is therefore also richer than the richest people in all the other states.  The maximum of the maximums means he is also the richest person in the United States. 

This logic is also how the Miss America Pageant competition works.  In fact we will strengthen our case here, by showing below the past nearly century-worth of state winners of the Miss America scholarship competition.  Albeit this is a subjective competition, it is a clear data set with a rich history.

We see the pattern looks similar (and if one is a young woman the the probability of winning Miss America in one's state is better than being the richest person in that same state).  For example, we see with the gold theoretical model line, the same unmistakable path of how there are many states (particularly the less populated ones) which fail to get a high number of Miss America winners in the pageant’s history.  Among the five most winning states, include the very-well populated states of: New York, Illinois, and Ohio.  And we can see the flip-side of understanding how 2 of the most populated states not pulling their weight recently.  Neither California nor Texas has won in more than 30 years- if we assumed just a 1/50 (2%) probability for each state winning per year, then this probability of winning at least once over this time is nearly 70%. 

Returning to the illustration, these gold dashed lines are similar to the grey curved ranges in the top illustration above.  And instead in the probability model chart further above, for the richest person per state, we notice what may be oddities, colored in blue on the scatter plot.  As noted earlier, we show the more populated states tend to have that slim chance of seeing even more extreme outcomes.

What’s important here to understand is we are able to properly articulate the maximum expected wealth that one would normally see, based solely as a function of the state’s population size.  This is of statistical importance and we see this against the richest people per state on the chart below.  People can -of course- relocate once they achieve their wealth, though for many of the important examples in this list this didn’t happen and also the business domicile address was used to add additional stability to the results.  We might have expected for example, that many billionaires would relocate to Hawaii or Florida, but that wasn’t much true.  We should note that most of the people on this list (35 of 50) were first-generation entrepreneurs who started enterprises in those states in order to create their wealth.  America's most rich people have comfortably held their relative ranks over many years and have expanded their interests in life to beyond wealth creation.  And as with any aristocracy, in the list we see familiar names; people who have been glamourized corporate magnates and leading thought leaders.  This is likely true throughout the list and not just at the top of the chart below.

Let’s look now at the pattern in the actual top wealth, which similar to the probability model further above is shown in red.  There are a couple noticeable trend clusters in relating the state’s population size to the wealth of the richest person in that state.  The core trend moves along (though less than) the theoretical probability model computed and shown in gold.  There is also a smaller cluster that runs parallel to the theoretical model (though at higher values).  And we highlight a couple interesting individual state people in blue.

We are able to measure the fit of the actual wealth to the theoretical wealth of the richest person per state, based solely on the opportunities presented by the state’s population size.  And from this exercise we see that half of actual wealth for these richest can be explained by the state’s population size.  People such as Larry Ellison and Michael Dell were able to set-up their technology businesses anywhere, and the warmer climates that they enjoy should have been reflected in a higher than theoretical level of wealth.  But instead it is less.  So there is something here to our mathematical connection between maximum wealth, and the size of the underlying population.  Though we see some finite upper bounds in the most populated and well desired states.

We also see that there are a number of “anomalies” as well.  The most striking are Warren Buffett (consistently among the country's top few in wealth) and David Koch, revealed in the chart above though their blue data.  Of what can’t be explained by the state’s population, a third is attributed to the hyper success of these two powerful individuals.  They are also clustered together in two small neighboring states in the middle of the agricultural Heartland.  The flyovers, which probably shouldn’t be flyovers, of Nebraska and Kansas.  This is a reflection that these less populated states, away from the mega crowds of New York and California, have a greater than theoretically expected wealth residing in their richest people.  For these guys specifically, they are worth seven times more than would be expected just on population size alone (and there wealth rivals those from the most populated states, but with a lot less competition to get there).  These blue data are not outliers, but a passive reminder teaching us something important: that extreme wealth is sometimes where one doesn’t look. 

Also, we also wouldn’t expect in those states to see others of similar high wealth along that most extreme level of the distribution.  We use the reference of ultra-high net worth (UHNW) for those individuals with a net worth in an excess of 30 million dollars.  There are hundreds of UHNW people in Nebraska and Kansas, though judging based soley on the extreme wealth of Buffett and Koch using the left chart below, we’d expect many thousands assuming their extreme wealth implied a proportional wealth level across their domiciled states.  Instead, as we see on the right chart below, the two blue states have UHNW levels instead much more proportional to their state’s population.  In other words, outside of the most extreme wealth found in each of the less populated states, the rest of the population’s wealth very quickly falls back into the trend line.

For completeness sake, we recast the right chart above, through the lens of the familiar bar chart style.  We can enjoy the patterns in the number of actual versus theoretical UHNW per state.  This includes what we see in Nebraska and Kansas, where those states no longer stand out as much (in fact few states actually stand out anymore using this metric).  What we see suggesting a definite small wealth enhancement chance for those living in the most populated states, but that’s it. There are two countertrends in this data, from the hedge fund capital of Connecticut (notice the chief of Bridgewater Associates doing well on the chart further above), a small state just north east of New York, and from Wyoming where the UHNW data has been biased by a large influx of wealthy people migrating there for particular state tax breaks.  The richest person in Wyoming is not biased for this reason however since the source of business wealth was used more accurately in that source survey. 

There are proud film song words: “if I can make it there, I'm gonna make it anywhere”.  Obviously this idea is in reference to the city of New York, with an island jewel at the center and a house for half of the state’s population.  While this is a feel-good theme if you live there, the comments are not necessarily proven.  Though New York is, by all accounts, a cut-throat corporate jungle making it a competitive training ground for anyone.  If that’s not clear enough, spending time there will sap your money quickly and prove it.  The state’s recent, infamous gubernatorial candidate reminds us of this situation with a political party name that speaks for itself.  The Rent is Too Damn High.  Still, a disproportionate number of people (including very bright and inevitably successful people) from around the world flock towards New York, and also spots such as San Jose and San Francisco, for a shot at making it very rich and spectacularly famous.  But if the thought of “making it” is simply to do well relative to your most successful societal peers, then one would be probabilistically better off working in a less populated state.  This was the logic Bill Gates himself used to start Microsoft, away from the action (and drama) of New York.

The most famous physicist of the 20^th century quipped with disdain the value society places on pursuing excess money, for any sake including money itself.  Albert Einstein offered this rhetorical question below.

The example of great and pure characters is the only thing that can produce fine ideas and noble deeds … Can anyone imagine Moses, Jesus, or Gandhi armed with the money-bags of Carnegie?

Sympathetically the fascination about the lives, of the richest among us, isn’t usually the most important or productive thing to focus on.  But we can learn a little about the science of extreme statistics, and what it says about our ability to simply employ in our life both reasonable chance and success.