Plot twist: A quick history of charts, the elegant messengers of math

Charts have a starring role in humanity’s grand timeline.

The ancients tracked the dots in the night sky and plotted celestial patterns, creating their versions of cosmic PowerPoint slides, with clear calls to action for life on earth.

Through the millennia, our quest to understand the world has continued. The methods and metrics we use are now more specialised, more sophisticated. They provide new layers of insight and nuance.

During the Covid-19 lockdowns, for instance, seismic charts revealed a whole new way in which we had been impacting our planet. Put simply: vibrations in the earth had dropped by half. Between March and May 2020, with movement, transport and industrial operations paused, the earth itself was about 50% stiller.
In this way, quantum leaps in our techniques and technologies throw new light on our world. If our ancestors looked to the sky for clues, we have taken to the skies to learn more: Through satellite images, voyagers, the Mars landing missions, and rovers on the moon.
Here at home, India’s Economic Survey 2021-22 included a chapter titled Tracking Development through Satellite Images & Cartography, which identified a potential cause of a burning problem. Images taken by satellites above Punjab indicated that the kharif crop was being planted about three weeks later, causing its harvest to coincide with rabi sowing. Farmers may be burning stubble to save time, because of the shrinking gap between the two, contributing to Delhi’s unbreathable air in November and December, the findings suggest.
Each industry has charts that it uses extensively.
Consultants and MBAs swear by 2×2 frameworks, in which they encapsulate within four boxes on a grid, months’ worth of analysis and findings about products and behaviours. Data scientists refer to Area Under the Curve graphs to assess the performance of their machine-learning models.
To understand new technologies, investors and enterprises look to the research firm Gartner’s Hype Cycles, which visually represent the current stage and future adoption trajectory of emerging technology.
Industries have grown up around our intrinsic need for new ways of viewing and interpreting findings. The data-visualisation software market is now worth billions of dollars, and the pie is expanding.
For the general public, charts encapsulate important takeaways, without forcing them to navigate underlying complexity and nitty-gritty. But just as a map is not the territory, a chart is not the full story. It’s a point from which a deep dive may begin.
Take the bell curve, or the chart of normal distribution. It contains multitudes and may be the closest thing we have to a universal truth.
At the personal level, it has uncanny predictive power, acting almost as a prophecy. Where one falls on the bell curve largely determines your life’s trajectory.
At the extreme right is the narrow tail end that represents all the GOATs, greatests of all time. “Make something of yourself” is an exhortation to venture away from the wide middle.
Shift one’s attention to the left of the curve, and one finds the strugglers and stragglers. Are the people that inhabit the rest of this graph the keepers of those on the left? That in a nutshell is the social contract, and the answer to this question determines how countries and economies are run.
A good graph tells its tale simply, without any frills.
To paraphrase Michelangelo, a sculptor merely chisels away the superfluous material, to free the statue that already exists within the marble block.
A good chart, similarly, is a thing of beauty. It’s a union of elegance and insight, to create an aha moment.
Thousands of years after we scratched our first data onto rocks (click here for more on this), charts remain our most direct way of understanding our world.
Here are nine of the most impactful modern-day formats.
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The hockey stick graph
In 1999, climate scientist Michael Mann and his research partners created a graph of average northern hemisphere temperatures over six centuries (it was later extended to 2,000 years). The graph resembled a hockey stick, with relatively stable temperatures followed by a sharp rise in the 20th century.
The graph became a powerful form of evidence, and a key visual representation, of global warming and modern climate change, helping shape public opinion globally and influence climate policy.
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Keeling curve
More than 40 years before Mann’s hockey stick, climate-science pioneer Charles Keeling developed a new technique to accurately measure the concentration of carbon-dioxide in the air. In doing so, he also created the Keeling curve, in 1958, which is the longest uninterrupted record of atmospheric CO2.
The sharp upward swing of this curve provided the first significant evidence of the rising levels of CO2 in our air. In a single frame, it provided a snapshot of the impact of human activity on the planet, acting as an early indicator that we were entering the Anthropocene: the geological age we now find ourselves in, defined largely by human activity.
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Lorenz curve
Developed by economist Max Lorenz in 1905, this curve depicts income or wealth inequality in a country, by telling us what percentage of total wealth is in the hands of a given percentage of the population.
In a world where everyone had the same wealth, the graph would be a straight diagonal line. In the real world, the curve is shaped like a bow, and the more curved it is, the more unequal the wealth distribution.
The Gini coefficient, which is a statistical measure of economic inequality in a society, is based on the Lorenz curve.
Comparing Lorenz curves across time is a good way to measure whether levels of inequality are rising or falling in a society. Incidentally, the richest 1% of the Indian population currently own 40.1% of the country’s wealth — which is more than at any point since 1961.
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Laffer curve
Developed by economist Arthur Laffer in 1974, this is a theoretical representation of the relationship between tax rates and government tax revenues.
What’s interesting here is that a too-low tax rate is obviously not sustainable for a government. But a too-high one, it turns out, isn’t sustainable either.
Raising taxes beyond a certain point (the point varies, of course, based on the state of the economy and the phase it’s in) stifles entrepreneurship, commerce, spending — and compliance. And so, hypothetically, a government that imposed a tax rate of 0% would earn 0 in revenue; but so would a government that imposed a rate of 100%.
The trick, and the purpose of the chart, is to find that optimal rate that allows a government to maximise tax revenue without crossing that point at which the choking effect on spending and growth kicks in.
There are theoretical and empirical criticisms of the Laffer curve’s applicability, but it did influence Reaganomics (an era of tax cuts that did, at least temporarily, see interest rates, inflation and unemployment fall, boosting the US economy in the 1980s).
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Bell curve
The bell curve (or the curve of normal distribution) is a graph that shows how data is spread out around the average value. For example, if one measures the height of a large group of adults, most people will be close to the average height, with only a limited number being either very short or very tall.
The bell curve is the natural pattern of most of the physical world (most rocks are hard; most pillows are soft; most corn grows within a certain timeframe, to a certain height).
Most things, in other words, follow the 68-95-99.7 rule, which states that 68% of data points lie within 1 standard deviation (SD) of the mean, 95% lie within 2 SDs and 99.7% within 3 SDs of the average.
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Fat-tailed distribution
In a normal distribution, there are very few data points at the extreme ends of the curve. These extreme values, or outliers, are critical in risk assessment, because one has to account for the likelihood of extreme events.
The fat-tailed distribution has a larger proportion of extreme values at either end. One typically sees this graph when the natural order of things is disturbed.
The thing about the fat tail is that the unlikely event is still rare; but when it does occur, because systems were not built to accommodate it, the impact can be dramatic, unpredictable and long-lasting.
For two recent examples, think of the pandemic, and the global economic downturn of 2008.
Extreme weather events, which were once uncommon, are becoming more frequent, which is another fat tail we urgently need to engage with.
For more on this, there is the former options trader Nassim Nicholas Taleb’s engrossing book, The Black Swan: The Impact of the Highly Improbable (2007; no relation to the 2010 psychological thriller about a ballerina).
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Exponential curves
Legend has it that the man who invented the game of chess was asked, by his king, to pick a reward. He said he wanted grains of rice, starting with one grain in the first square of his chessboard, with the number doubling in every square thereafter.
Why didn’t he ask for land or gold? Seems like a lost opportunity, people said.
Well, then as now, the human brain struggled to understand exponentials. We are not used to thinking of growth in geometric proportions.
Do the math and the chess reward would have amounted to about 18.5 quintillion grains, which is 52,000 times the rice reserves in all of the Food Corporation of India’s warehouses.
How can that be? Think about the million-billion comparison that is now virtually an adage: a million seconds is about 11.5 days; a billion seconds is about 31.7 years.
(Incidentally, after billion comes trillion, quadrillion, then quintillion.)
Exponential graphs are characterised by rapid growth (or decline). The rate of change is not constant, but it increases (or decreases) multiplicatively. The values grow slowly at first, then skyrocket.
Think of the jump in the number of Covid-19 cases in 2020. The investment mantra of “start early and benefit from compounding effect” is another illustration.
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The Kaplan-Meier plot
This chart is used in clinical trials, to analyse chances of survival and other outcomes over time. It has a stepped shape, with each step representing an event (such as a relapse or a side-effect), making it easier to identify at what intervals certain events occur.
It allows researchers to compare how well different treatments work, through a visual inspection. With survival rates, for instance, if the drop is steep, events are occurring sooner and more frequently for that group.
The Kaplan-Meier plot is often used in conjunction with other metrics, such as the hazard ratio (relative risk measure of different treatments).
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Heat maps
A heat map is a visualisation technique in which colour is used to represent data in a grid, with each cell’s colour or intensity of shade corresponding to a value.
A heat map provides a quick visual summary, and is intuitive. They are a staple of business reporting and are used, to great effect, to represent price changes of stock baskets, sales and distribution patterns, even employee performance.