When we measure things over time, they are no longer comparable. Apples become oranges. Because of this, a great deal of what we claim to know about the world just ain’t so. This leads to poor decision making. These claims make up what I call “the tyranny of the denominator”.

The idea is simple, but it has very uncomfortable implications. And once you see it once, you just can’t stop seeing it everywhere else.

Here are some quick examples, before we explore the implications for investors…

If stocks go up 22%, are investors wealthier?

It depends on inflation.

If our nation’s GDP grows by 2%, and population grows by 2% too, has our standard of living gone up?

As you can see from these examples, what we’re trying to measure – GDP and investor returns – is obfuscated by the fact that something else is also changing. In those examples, the value of money and the population amongst whom the GDP is divided.

But what does this have to do with the denominator – the bottom half of a fraction? Well, it’s just a way of expressing the fact that, when we try to measure something, the ground it’s standing on can shift too.

Usually, when we measure how things change, we focus on the numerator – the top half of a fraction. For example, GDP measures the economy. When GDP goes up, we have a higher standard of living. When it goes down, our standard of living falls.

But what about the denominator – the size of the population amongst which that GDP is divided?

A significant enough change in the population can render the GDP growth figure misleading. If the population grew more than GDP, our standard of living actually fell.

If the news reader reports a huge jump in GDP without mentioning how migration and demographics have changed, do we have any idea what’s really going on?

Consider another example, before we do some more explaining about how the tyranny of the denominator works.

Does raising taxes collect more tax revenue? It depends how many wealthy people leave the country, hide their earnings, or reduce their future earnings. The tax base changes once you change the tax rate.

And it depends on how raising taxes will affect economic growth. If you accept that lower taxes grow the economy more, then it’s only a question of when future income growth ends up producing more tax revenue at lower tax rates. A faster growing economy eventually pays more in tax at lower rates of taxation because the tax base is so much larger.

Civilisation-type computer game designers understand this problem very well. They make setting taxes a trade-off between short-term revenue and long-term revenue by way of a larger economy and tax base. The denominator – the size of the tax base – changes.

My point is that, throughout most of our lives, we seem to focus on the numerator and are blind to the changes in the denominator. We simply presume that the denominator is constant – that the population doesn’t change, the economy doesn’t grow, rich people submit to being taxed, and the value of money is constant. But they do change, especially over long periods of time.

Perhaps more importantly, I suspect that it’s changes in the denominator that can matter more to the data you’re analysing and basing your conclusions on. In other words, we just don’t know what we claim our historical data tells us.

Changes in the denominator can make comparisons over time extremely misleading for our decision making too. The comparisons can be so misleading that you’re better off not having an Office for National Statistics which gives politicians bad ideas to meddle.

The example that triggered this idea comes from inequality data. We’ve all heard about how inequality has soared in recent years as the rich get richer and the poor stay poor.

But there is a long list of issues with the way this analysis occurs. The list includes issues that are a prime example of the tyranny of the denominator in action.

Consider, for example, that most inequality data is based on a household level. One household’s income and wealth gets compared to another’s. And, over time, the level of inequality has indeed grown by this measure.

But what if it’s the households that are changing over time, not income inequality? What if the growing inequality as we measure it actually comes from who and how many earners and non-earners are in each home, rather than what they earn or own?

For example, we have the drop in fertility and the breakdown of the traditional family over time. The number of single-parent households has grown, as have childless two-income households. And we have changes in how many retirees and students there are: these are people who don’t earn much of an income, but for a good reason. This is a dramatic divergence in the income of households which reflects the changing household, not inequality of people’s income.

All of this biases inequality calculations by comparing apples to oranges because… times have changed. Comparing today to the past gives deeply misleading conclusions. The denominator – the household – looks very different today.

My point isn’t that inequality hasn’t grown. My point is that we don’t actually know. And our measures don’t tell us what we think they do. They may be measuring changes in households, not inequality.

Our ability to obtain data is another denominator that can change. It could be that we’re better at collecting income and wealth data from extremely low-income or high-income households, which has changed the picture. People on very low incomes might be forming more households – a good thing, but one that shows up as rising inequality in the data by growing the number of poor households within the dataset.

Bloomberg’s headline gives another example of the tyranny of the denominator in action: “Japan’s Record Trade Deficit Shows Growing Pain of Feeble Yen.” Those of you scratching your heads have every right to. For decades, we’ve all been taught that a weaker currency is the magical cure for a trade deficit, not the cause.

But consider the denominator – Japan’s actual trade. Japan imports and exports a great deal. A weaker currency makes their imports more expensive, worsening the balance. The same goes for Germany too, which also saw its legendary trade surplus evaporate this year.

The effectiveness of vaccinations is another example. Those who were not vaccinated are unlikely to be a representative sample of the population. If they were to die of Covid at a higher or lower rate than those vaccinated, this would be seen as evidence about the efficacy of the vaccines.

But it could reflect the tendency of the unvaccinated to be more or less healthy to begin with. Or their tendency to be more or less careful about Covid generally. Or their tendency to belong to certain groups. Or, more likely, something I can’t think of that they share, which sets them apart from the vaccinated, and which may explain their higher or lower case and fatality rate.

We all intuitively know that the very high death rate of those placed on ventilators doesn’t reflect the fact that ventilators are dangerous, but the fact that they were used for the most severe cases. It is the denominator – those on ventilators – that changed, not just the numerator – how many died.

Perhaps the most famous example of the tyranny of the denominator is a story about World War II. A statistician was asked to analyse where to add armour to bombers to prevent them from being shot down. His data consisted of the bullet holes in the bombers which had come back.

The Air Force wanted to add armour where the bullet holes tended to be concentrated, because this would protect the planes where they tended to get hit. The statistician wanted to add armour where the bullet holes were not concentrated.

The error was the tyranny of the denominator. The data came from the flights which had made it back. The planes shot down never showed up in the data set. I wrote about the story here.

But why does the tyranny of the denominator matter to investors? Well, financial markets are especially susceptible to the problem. Especially at the moment. And so you need to be aware of it in order to avoid falling prey.

Find out how, tomorrow. Because, right now, you need to see this.

Until tomorrow, as Mark Twain is credited with saying, “It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.”

Nick Hubble
Editor, Fortune & Freedom

PS Back in 2018, I wrote a series of reports on Italy’s upcoming elections and the impact they would have on markets. Over time, I turned those reports into a book. Today’s Fortune & Freedom will become a book one day. It’s only the seed of one for now. But I do need your help to write the rest of it.

I’m collecting the countless other examples of the tyranny of the denominator from our society and economics. Can you help me with your own examples? [email protected]