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Reading the numbers

Most technical arguments are not really about facts. They are about a handful of concepts that everyone uses and almost nobody pictures: a rate versus an amount, a ratio versus a difference, a curve that bends up versus one that bends up and keeps going.

This is a short guide to those concepts. Ten chapters, each one built around a figure you can drag. The aim is narrow and practical: when somebody says terawatt-hours or it doubles your risk or that's a log scale in a conversation, you should be able to see the thing in your head, and know which question to ask next.

Nothing here needs maths beyond multiplication. Every figure is drawn in your browser from a few lines of arithmetic — pull the sliders around and the ideas tend to arrive on their own.

The ten

Chapter 1 · Rates and amounts

Why terawatt-hours, not terawatts

A watt is a speed. A watt-hour is a distance. Mixing them up is the single most common slip in any conversation about energy — and the same trap is hiding in half a dozen other units.

Picture a tap filling a bucket.

• Power — watts (W, kW, MW, GW, TW) — is how fast the water is coming out right now. Litres per second. It is a rate, and rates only make sense at an instant.
• Energy — watt-hours (Wh, kWh, MWh, GWh, TWh) — is how much water ended up in the bucket. Litres. It is an amount, and amounts only make sense over a period.

The bridge between them is time: energy = power × time. A 2 kW kettle running for six minutes (0.1 hours) uses 0.2 kWh. That is the whole idea; everything else is bookkeeping about prefixes.

Why this matters for the energy debate

You will constantly hear a power station described as a 1 gigawatt plant. That is its maximum rate, not its output. Run flat out for a year — 8,760 hours — it would make 8.76 TWh. But nothing runs flat out. The fraction it actually achieves is the capacity factor:

• Nuclear: roughly 90% → about 7.9 TWh a year
• Offshore wind: roughly 40–50% → about 3.5–4.4 TWh
• Solar in the Netherlands: roughly 11% → about 1 TWh

Three things all honestly labelled 1 GW, producing amounts that differ by a factor of eight. This is why anyone being careful compares energy systems in TWh per year, and why a headline counting installed gigawatts is telling you about ambition rather than electricity.

The same trap, other units

• Megabits per second (rate) versus gigabytes (amount). Your broadband speed does not tell you your monthly usage without a duration.
• Euros per month (rate) versus euros (amount). Subscription pricing exists because these feel different.
• Deaths per year (rate) versus deaths (amount). A cumulative total and an annual rate can tell opposite stories about the same country — see chapter 9.
• Litres per 100 km (rate) versus litres in the tank (amount).

The tell is the word per, or a unit with a time in it. If a time unit appears in the denominator, it is a speed. If it appears multiplied — watt-hour, person-year, passenger-kilometre — it is an amount.

Chapter 2 · Scale

Orders of magnitude, and the bent axis

An order of magnitude is a factor of ten. Once the numbers in one conversation span a billion-to-one, an ordinary axis stops working — so we bend it, and a straight line starts to mean something new.

Human intuition is built for addition. We are good at ten more than and quite bad at ten times. Nearly every quantity in science — energy, money, distance, population, probability, compute — lives across many factors of ten, so the first move is always to stop counting and start counting zeros.

Two orders of magnitude cheaper means 100× cheaper. Within an order of magnitude means somewhere between a tenth and ten times, which sounds sloppy and is often the most honest thing anyone in the room can say.

On the linear axis, seven of the eight bars are invisible. They are not small — a Dutch household's yearly electricity is a real, expensive quantity — they are just crushed against zero by a number 10 billion times bigger. This is not a flaw in the chart. It is what spanning many orders of magnitude actually looks like.

What a log axis does

A logarithmic axis gives every ×10 the same amount of space. Equal steps across the page are equal multiplications, not equal additions. Which gives you one clean rule for reading it:

You already use log scales without calling them that. Each is a ×10 (or ×2) per step:

• Decibels — +10 dB is ten times the sound power. A 90 dB street is not a bit louder than a 60 dB office; it carries a thousand times the energy.
• pH — one step is a tenfold change in acidity, and the scale runs backwards.
• The Richter scale — one step is about 32× the energy released.
• Musical octaves — each octave is a doubling of frequency, which is why a piano keyboard is a log axis you can play.
• Stellar magnitude, f-stops on a camera, font size ratios. Same idea.

What it costs you

A log axis flatters everything. Explosive growth becomes a gentle slope; a tenfold gap becomes one modest step. Charts of a pandemic on a log axis were accurate and reassuring at the same time, which is precisely the problem. Two more limitations worth knowing:

• Zero does not exist on a log axis and neither do negative numbers. If a chart has a log axis, something has been dropped or floored.
• The axis labels are the whole story. Always read them before reading the shape. A log axis with unlabelled gridlines is either a mistake or a trick.

Chapter 3 · Curves

The shapes of growth

Logarithmic, linear, polynomial, exponential, S-shaped. These words describe how the output changes when you make the input bigger, and each one has a silhouette worth memorising.

Here is the whole family. Each panel shows one shape; the faint grey line in each is plain linear growth, so you can see whether the curve bows above it or below.

What each one feels like

• Constant, O(1) — the input does not matter at all. Looking something up in a hash table. The dream.
• Logarithmic, log n — doubling the input adds a fixed amount. Every step halves the remaining problem: binary search, a balanced tree, guessing a number. Feels essentially free, and gets more free the bigger things get.
• Polylogarithmic — (log n)² or (log n)³. Still slower-growing than any polynomial, however small its exponent. When somebody says polylog they mean basically free, but I am being honest that it isn't constant.
• Square root, √n — grows, but ever more slowly. The shape of diminishing returns, and of how measurement error shrinks with sample size (chapter 8).
• Linear, n — twice the input, twice the output. The honest default, and the shape people wrongly assume for everything.
• n log n — barely worse than linear, and the price of a good sort. On a chart you can hardly tell the two apart, which is the point.
• Polynomial, n² and n³ — twice the input, four or eight times the work. Perfectly fine when n is small, quietly ruinous when it isn't. Most it worked in testing disasters are quadratic.
• Exponential, 2ⁿ — each extra unit of input multiplies the output. Adding one item doubles the bill. Nothing survives this for long: at n = 60 you are past the number of grains of rice in the old chessboard story.
• Logistic, the S-curve — exponential growth that runs into a limit and bends over. Epidemics, technology adoption, bacteria in a dish, a species in a valley.

Bending the axes until the shape names itself

Here is the practical trick, and the reason people reach for log axes so often. Watch what happens to these three curves as you change what the axes measure.

Two rules fall out of that, and between them they cover most charts you will ever be shown:

• Straight on a log-y chart ⇒ exponential. Constant percentage growth. The steeper the line, the shorter the doubling time. This is why every chart of a virus, a startup's revenue or Moore's law ends up with a log y-axis.
• Straight on a log–log chart ⇒ a power law (output ∝ inputᵏ), and the slope is k. City sizes, earthquake energies, word frequencies and wealth all do this.

Doubling time, in your head

The one piece of exponential arithmetic worth carrying around: something growing at r percent per period doubles in roughly 70 ÷ r periods. (It is really 69.3, from the natural log of 2, but 70 divides nicely.)

It runs backwards too. Something shrinking 7% a year halves in about a decade, and that is what a half-life is: the same maths with the sign flipped.

Chapter 4 · Risk

Absolute versus relative risk

"Doubles your risk" is a ratio. It tells you nothing at all about how many people that is. Nearly every alarming health headline lives in this gap.

There are two ways to describe the same change in risk, and they are both true:

• Relative risk — the new risk divided by the old one. 18% higher, twice as likely, a third less. A ratio, with the size scaled out of it.
• Absolute risk — how many people out of a hundred, or a thousand. Two extra cases per 1,000 people. A difference, in units you can act on.

Doubling a risk of 1-in-100,000 gives you 2-in-100,000. Doubling a risk of 1-in-10 gives you 1-in-5. Identical headline, completely different lives. Press the presets below and watch how little the relative number constrains the picture.

The famous case is processed meat. In 2015 the WHO's cancer agency classified it as a definite carcinogen, and reported that 50 g a day — a couple of rashers — raises bowel cancer risk by about 18%. The headlines put bacon next to smoking, because both sit in the same category, which grades the strength of the evidence, not the size of the risk. In absolute terms: a lifetime bowel cancer risk of roughly 6 in 100 becomes roughly 7 in 100. Real, worth knowing, not remotely comparable to a habit that kills one smoker in two.

Number needed to treat

The cleanest way to state an absolute effect is to invert it. If a drug lowers your five-year risk from 50 in 1,000 to 40 in 1,000, the absolute reduction is 10 in 1,000, so 1 ÷ 0.01 = 100: a hundred people take the drug for five years, and one of them avoids a heart attack. That number — the number needed to treat — is what a doctor needs and what a press release almost never gives. Its mirror image for side effects is the number needed to harm.

Chapter 5 · Tests and evidence

Why a 99%-accurate test is usually wrong

If a condition is rare, most of the people who test positive for it do not have it. This is not a paradox and not a bad test — it is arithmetic, and it governs medicine, fraud detection, spam filters and every AI classifier you will ever deploy.

Start by throwing away the phrase 99% accurate. It is not a number; it is a mood. A test has two separate accuracies, and they trade off against each other:

• Sensitivity — of the people who do have the condition, what share does the test catch? Missing them is a false negative.
• Specificity — of the people who do not have it, what share does the test correctly clear? Wrongly flagging them is a false positive.

But neither of those is what you want to know when your own result comes back positive. You want the reverse: given a positive result, what is the chance I actually have it? That is the positive predictive value, and it depends on a third number the test cannot know — how common the condition is in the group being tested. Its base rate.

The mechanism is simple once you see the two blocks. The true positives are drawn from the tiny sick group; the false positives are drawn from the enormous healthy one. A 1% error rate applied to 990 healthy people produces about 10 false alarms — and if only 10 people in the thousand are actually ill, there are only about 10 true ones to compete with. Coin flip.

This is the shape of a hundred other problems

• Mass screening. Screening a whole healthy population for a rare cancer generates enormous numbers of false alarms, follow-up scans and biopsies. This is the real argument behind screening guidelines, and it is why they are usually narrower than people expect.
• Security and surveillance. A terrorist-detection system with a 0.1% false positive rate, run over a million travellers, produces a thousand innocent suspects.
• Fraud, spam and content moderation. The rarer the thing you are hunting, the more of your flagged pile is noise — which is why these systems drown their reviewers.
• AI classifiers. A model with a stellar benchmark score, pointed at a rare real-world event, will still deliver mostly false positives. The benchmark measured sensitivity and specificity; deployment cares about predictive value.

The fix is always the same: raise the base rate before you test. Test people with symptoms rather than everyone. Screen the age group where the disease is actually common. Run the cheap filter first and the expensive test only on what it flags. A positive result in a high-prevalence group means something; the same result from a mass screen often means very little.

Chapter 6 · Distributions

Averages, spread, and the long tail

The mean and the median agree only when a distribution is symmetric — and most of the interesting ones are not. Knowing which one you have been handed is often the whole argument.

• Mean — add everything up, divide by the count. Sensitive to every extreme value.
• Median — line everyone up and take the person in the middle. Ignores how extreme the extremes are.
• Mode — the most common value. Useful for lumpy things, rarely quoted.

Drag the skew below. On the left it is a symmetric bell and the two measures sit on top of each other. Push right and a long tail grows: a handful of enormous values drag the mean away while the median stays where most people actually live.

This is why average income rose and most people got poorer can both be true in the same year. Income, house prices, city sizes, book sales, startup returns, wildfire damage and citations per paper are all right-skewed. For any of them, the mean describes a person who may not exist.

Spread: standard deviation and sigma

An average with no spread attached is half a fact. Standard deviation (σ) is roughly the typical distance from the mean. For a genuine bell curve, the numbers to remember are:

Fat tails

The bell curve has an underrated property: extremes are not merely rare, they are effectively impossible. A 6σ human height would be about 2.6 m; a 6σ daily stock market move is supposed to happen once every few million years, and happens roughly every decade.

Distributions where extremes are rare but expected are called fat-tailed, and they cover most of the things that hurt: earthquakes, floods, market crashes, pandemics, wars, outages, security breaches. Two consequences worth internalising:

• The average is not the risk. For fat-tailed damage, the total over a century is dominated by two or three events, so planning around an average year plans for a year that never happens.
• No sample is ever enough. With a bell curve, a few hundred observations pin things down. With a fat tail, the biggest event so far is mostly a statement about how long you have been watching.

The best-known fat tail is the power law, the 80/20 or Pareto shape. Its defining feature is that it is scale-free: if the top 20% hold 80%, then inside that top 20% the same split holds again, all the way up. There is no typical size — which is exactly what makes "average" meaningless for it.

Chapter 7 · Relationships

What a correlation actually looks like

Almost everyone overestimates what a given correlation coefficient looks like on a page. Drag the slider until your intuition is recalibrated — it takes about ten seconds and it never wears off.

The correlation coefficient r runs from −1 to +1. Zero is no straight-line relationship; +1 is a perfect rising line; −1 a perfect falling one. That much everyone knows. What almost nobody has is a picture of what r = 0.4 looks like.

The number that keeps you honest is r²: the share of the variation in one variable that is accounted for by the other. A moderate correlation of 0.3 — the sort that carries a press release — explains 9% of what is going on. The other 91% is something else.

Two ways r lies to you

• r only sees straight lines. A perfect, deterministic U-shape has a correlation of about zero. Press a perfect curve above: total dependence, no correlation. Always look at the scatter plot, never just the coefficient.
• A handful of points can create or destroy it. One extreme outlier in a small sample can drag r from 0 to 0.6. This is what Anscombe's quartet — four datasets with identical statistics and wildly different shapes — was built to demonstrate.

And then the famous one

Correlation is not causation, which everyone can recite and few people apply. The useful version is the list of what else could be producing the pattern:

• A common cause. Ice cream sales and drownings both track the weather. The classic confounder.
• Reverse causation. Do sports build health, or do healthy people take up sport?
• Selection. If your sample was filtered on something related to both variables, a correlation appears from nothing. Almost all successful people do X findings are this.
• Coincidence. Compare enough pairs of series and striking correlations are guaranteed — the arithmetic of chapter 8's multiple comparisons, applied to data mining.

What does upgrade a correlation towards causation: randomised assignment (the gold standard — it severs the link to every confounder at once), a natural experiment where something arbitrary split the groups for you, a dose-response relationship, a plausible mechanism, consistency across very different populations, and replication.

Chapter 8 · Uncertainty

Error bars, confidence and p-values

An estimate with no interval attached is not a measurement, it is a guess with a tidy haircut. Here is how to read the intervals — and what 95% confident really promises.

Every measurement drawn from a sample is a bit wrong, and the useful question is how wrong. Three quantities get drawn as error bars and they are not interchangeable:

• Standard deviation — how spread out the underlying data is. Does not shrink with more data; it is a property of the world.
• Standard error — how precisely you have pinned down the average. Shrinks as you collect more.
• Confidence interval — a range, usually 95%, built from the standard error. Roughly twice the standard error either side.

On the same data these can differ by a factor of several, so a chart whose error bars are unlabelled is not telling you anything. Always ask which one you are looking at.

Precision improves with the square root of effort

The margin of error shrinks like 1/√n. To halve it you must quadruple the sample. That single fact explains why a well-run poll of 1,000 people is about ±3 points, why going to 4,000 only gets you to ±1.5, and why nobody bothers going further.

That picture is the correct definition of confidence. 95% confidence is a property of the procedure, not of your particular interval. It means: if you repeated this study forever, 95% of the intervals you construct would contain the true value. Your one interval either contains it or it doesn't — and you never find out which.

Notice also that the misses are not gentle. A study that misses is usually one that got an unusually extreme sample, which means the most striking results are disproportionately the wrong ones. Combine that with the fact that striking results are the ones that get published, and you have most of the replication crisis in two sentences.

What p < 0.05 actually says

A p-value answers exactly one question: if there were really no effect at all, how often would I see data at least this extreme by chance? p = 0.03 means three times in a hundred. That is all. In particular it is not:

• the probability that the effect is real (that requires a prior — chapter 5's problem again);
• the probability the result was a fluke;
• any statement whatsoever about how big the effect is.

The 0.05 threshold is a convention from the 1920s, chosen because it was convenient. There is no scientific meaning to the gap between p = 0.049 and p = 0.051.

Multiple comparisons

Test twenty independent hypotheses at p < 0.05 and you should expect one to come back significant from pure noise. Test a hundred and you get five. This is the engine behind p-hacking: run enough subgroups, outcomes and cut-off points, and a publishable result is guaranteed. The defences are pre-registration (say what you will test before you look), correcting the threshold for the number of tests, and replication in fresh data.

Chapter 9 · Denominators

Totals, rates and per-capita

Give me the same data and let me choose the denominator, and I can hand you either side of most arguments without stating a single false number.

Here are annual carbon dioxide emissions for six countries, and the only thing changing is what you divide by.

Neither view is a trick. They answer different questions. Totals answer how much is going into the atmosphere, and where would a cut make the biggest difference? Per-capita answers how are people in this country living, and what is fair? Notice that a country's diplomats will reliably prefer whichever chart is kinder to it, and that both delegations can be entirely truthful all week.

Other denominators are available, and each reframes the argument again: emissions per unit of GDP (are we efficient?), per kilowatt-hour (how clean is the grid?), cumulative since 1850 (who caused the existing warming?), or consumption-based rather than territorial (who gets the goods made by the factories?). Whenever a comparison feels unfair, look at the denominator first.

Percent versus percentage point

Unemployment goes from 4% to 5%. That is a rise of one percentage point, and also a rise of 25 percent. Both are correct, and the second is what you write if you want it to sound like a crisis. The rule:

• A percentage point is the arithmetic difference between two percentages.
• A percent change is the relative difference — the same ratio trap as chapter 4.
• Interest rates, tax rates, vote shares and unemployment should almost always be discussed in points. If a headline uses percent for one of them, it is choosing the bigger number.

The related sleight of hand: a fall of 50% followed by a rise of 50% does not get you back. It leaves you at 75%. Percentages do not commute, because the base moved.

Three more denominators worth checking

• Nominal versus real. Any money figure compared across years must be inflation-adjusted or it is meaningless. Record box office takings is almost always an artefact of nominal pricing.
• Counts versus rates. Cases doubled means nothing without knowing whether testing doubled. More deaths than last year means little without population and age structure — an older country has more deaths at identical health.
• Cumulative versus annual. China leads on annual emissions; the United States and Europe lead comfortably on the cumulative total since 1850. Both facts are used, by different people, in the same negotiation.

Chapter 10 · Phrasebook

The rest of the phrasebook

Shorter entries for the other phrases that turn up constantly and are rarely explained. Roughly grouped, all one-paragraph.

Numbers and units

Order of magnitude

A factor of ten. Three orders of magnitude is 1,000×. Used honestly, it signals that you know the answer's size but not its digits.

Significant figures

How many digits are actually meaningful. Writing 3.14159 when your measurement was ±10% invents precision that isn't there. A number's length is a claim.

ppm / ppb

Parts per million and per billion. CO₂ at 425 ppm is 0.0425% of the atmosphere — a reminder that a small share of something huge is still huge.

Back-of-the-envelope

A deliberately rough calculation aiming for the right order of magnitude. Often the most valuable thing in a meeting, and a good defence against a precise number that is wildly wrong.

Per capita

Per person. See chapter 9 — the most consequential division in public argument.

Half-life / doubling time / e-folding

Three ways to state the speed of compounding: the time to halve, to double, or to change by a factor of e (2.718). Interchangeable with a constant factor.

Capacity factor

Actual output divided by theoretical maximum output. The number that converts a power rating into an energy amount (chapter 1).

Studies and evidence

Randomised controlled trial

Participants are assigned to treatment or control by chance. The randomisation is what severs the link to confounders — that, not the size, is why it's the gold standard.

Double-blind

Neither the participant nor the person assessing them knows who got the treatment. Guards against both the placebo effect and the researcher's own thumb on the scale.

Observational study

Nobody assigned anything; the researchers watched what people did anyway. Cheaper, bigger, more realistic, and permanently vulnerable to confounding.

Cohort vs case-control

A cohort follows a group forwards in time. A case-control starts from people who already have the outcome and looks backwards. The second is much faster and much easier to bias.

Meta-analysis

Pools many studies into one estimate. Powerful, but it inherits every bias in the underlying literature — including publication bias.

Publication bias

Positive, surprising results get published; null results sit in a drawer. So the published literature is systematically more exciting than reality.

Preprint

A paper posted publicly before peer review. Fast and often excellent, but nobody independent has checked it yet. Peer review is a light filter, not a guarantee.

Replication

An independent team runs the study again and gets the same answer. The only real test, and it fails far more often than anyone expected.

Effect size

How big the difference actually is, in real units or standardised ones (Cohen's d). The number that should be quoted instead of the p-value.

Statistical power

The chance a study would detect the effect if it exists. Underpowered studies both miss real effects and, when they do find something, overstate it.

Ways to be fooled

Selection bias

Your sample was not drawn fairly from the population you're talking about. The single most common fatal flaw, and usually invisible in the summary.

Survivorship bias

You only see the things that made it. The war planes that came back, the funds still open, the founders who kept going. Reinforce the parts of the plane with no bullet holes.

Regression to the mean

An extreme measurement tends to be followed by a more ordinary one, for purely statistical reasons. Punish the worst performers and they improve; praise the best and they decline. Neither response did anything.

Simpson's paradox

A trend that holds in every subgroup can reverse when the groups are combined, because the groups differ in size. The reason controlling for X can flip a conclusion.

Goodhart's law

When a measure becomes a target, it stops being a good measure. Applies to KPIs, school league tables and AI reward functions alike.

Ecological fallacy

Concluding something about individuals from group averages. Richer countries vote differently than poorer ones; that says nothing about how a rich individual votes.

Base-rate neglect

Ignoring how common something is when interpreting evidence about it. Chapter 5, in three words.

Confounder

A third thing causing both of the two things you're comparing. Controlling for it means statistically removing its influence — which only works for the confounders you thought of.

Two more shapes

Signal-to-noise ratio

How much of what you're measuring is the thing versus the fuzz. Usually quoted in decibels, which is to say on a log scale (chapter 2).

Marginal

The next one, not the average one. Marginal cost is what the next unit costs, and it can be wildly different from the average — most arguments about pricing, energy and taxes are really about which of the two is relevant.

The eight questions

If you keep nothing else, keep these. In rough order of how often they crack something open:

1. Percent of what? — the absolute baseline behind any relative claim.
2. Compared to what? — a number with no comparison is decoration.
3. How many people? — the sample size, and how they were chosen.
4. Is that a rate or an amount? — watts or watt-hours.
5. What's the denominator? — and who chose it.
6. Mean or median? — and is the distribution skewed.
7. How wide is the interval? — not whether it cleared a threshold.
8. What else could produce this pattern? — confounding, selection, coincidence.

None of these require you to know more science than the person you are talking to. They just require you to picture the thing before agreeing with it.