Percentages, Fractions, and Ratios

Opening Hook

In September 2021, the UK government announced that National Insurance contributions would rise by 1.25 percentage points. Across ten pages of government websites, this was described instead as a rise of 1.25%. These are not the same thing. The rate was already around 12% for most employees. A 1.25% increase on a rate of 12% is an increase of about 0.15 percentage points, taking the rate to roughly 12.15%. What was actually happening was a 1.25 percentage point increase, taking the rate to 13.25%. In proportional terms, this was a rise of more than ten percent in what employees paid, not one percent.

Consumer journalist Martin Lewis spotted it and publicly challenged the government to correct its own website, pointing out that the wording was misleading millions of workers about the scale of what was about to hit their payslips. Full Fact investigated and found the error on ten separate government pages.

The government eventually corrected the wording. But most people had already read the wrong version. Those who noticed anything unusual were left wondering whether 1.25% meant something was trivial or not. It was not trivial. It was a ten percent increase in their National Insurance rate. The language of percentages made the same change sound much smaller than it was.

This is not an unusual story. It is an ordinary one.

The Concept

A percentage is a way of expressing a proportion. “Per cent” means “per hundred,” so 25 percent simply means 25 out of every 100. You can write the same proportion as a fraction (25/100, or simplified, 1/4) or as a decimal (0.25) or as a ratio (1 to 3, meaning one part in every four). These are all different ways of writing the same thing. Knowing how to move between them is the foundation of reading almost any statistical claim.

The arithmetic is simpler than it might appear. To convert a fraction to a percentage, divide the top number (the numerator) by the bottom number (the denominator) and multiply by 100. So 3 out of 8 is 3 divided by 8, which is 0.375, which is 37.5 percent. To go back, divide by 100: 60 percent is 60 divided by 100, which is 0.6, or 3 out of every 5.

A ratio expresses how two quantities relate to each other. A ratio of 2 to 1 means for every two of one thing there is one of another. Odds, which you encounter in gambling and medical statistics, are a form of ratio: odds of 3 to 1 against something happening means it fails to happen three times for every one time it does, which is a probability of 1 in 4, or 25 percent.

So far, so mechanical. Here is where it gets less mechanical, and where most of the manipulation lives.

The percentage point versus the percentage change distinction. This is the single most reliably confusing distinction in all of everyday statistics, and it is confused constantly, sometimes accidentally and sometimes not. When a quantity expressed as a percentage changes, you can describe that change in two different ways, and they produce numbers that look completely different.

If the unemployment rate rises from 4% to 6%, you can say it rose by 2 percentage points. That is an arithmetic statement: 6 minus 4 equals 2. You can also say it rose by 50%. That is a proportional statement: 2 divided by 4, multiplied by 100. Both descriptions are technically accurate. One sounds like a moderate increase. The other sounds like a crisis. The choice of which to use is not neutral.

A government that wants to emphasise stability will say unemployment “rose by only 2 percentage points.” A commentator trying to convey alarm will say it “rose by 50 percent.” A reader who encounters only one of those figures and does not know which framing is being used cannot evaluate what they are reading.

The National Insurance case above is this confusion made practical. When someone says a rate “rises by 1.25%,” most readers hear “a small amount.” When the same rate was already sitting at 12%, a 1.25 percentage point rise represents a much larger proportional shift than the phrasing suggests.

The test is simple: when you see a percentage change, ask “a percentage of what?” A change described as “a percentage” is expressing a proportion of the original number. A change described as “percentage points” is expressing an arithmetic difference. If the source does not tell you which they mean, that is already a sign something might be off.

What “X times more likely” actually means. This phrase is used constantly in health journalism and advertising, and it is used in a way that is technically ambiguous at best and deliberately misleading at worst.

“Twice as likely” means your probability of the outcome is two times the baseline probability. If the baseline is 10%, twice as likely means 20%. Clear enough.

“Twice more likely” is different. In strict English, this means your probability is the original plus twice the original, which would be 30%. But almost nobody uses it this way. Almost everybody who writes “twice more likely” means “twice as likely.” The inconsistency is a quiet source of confusion.

Where it gets genuinely problematic is with larger multipliers and small baselines. “Five times more likely to develop a rare heart condition” sounds alarming. If the baseline risk of that condition is 1 in 100,000, five times more likely means 5 in 100,000. In absolute terms, your risk has gone from 0.001% to 0.005%. The multiplication is real. The alarm is manufactured by not telling you what you started from.

This is the denominator problem.

The denominator problem. A percentage is meaningless without knowing what it is a percentage of. This sounds obvious. In practice, it is violated constantly.

“A 100% increase” in anything sounds dramatic. If it refers to a rise from 1 case to 2 cases, it is describing a single additional occurrence. “Risk has doubled” is one of the most effective phrases in statistical manipulation, because it suppresses the denominator entirely. If your risk of something was 1 in a million, and it doubles, it is now 2 in a million. Your life should proceed essentially unchanged.

Every percentage needs a denominator. Who is in the group being counted? What is the time period? What is the population? A treatment that “reduces your risk by 50%” is saying: compared to a group that did not take this treatment, your risk is half what theirs was. But half of what? If the untreated group had a 2% risk of the outcome, reducing it by half takes your risk to 1%. That is a 1 percentage point reduction, which is different from halving your risk of dying tomorrow. One is a description of relative performance in a controlled comparison; the other is an existential claim. The language “reduces your risk by 50%” invites the second reading while only delivering the first.

Why It Matters

People making claims with numbers nearly always choose the framing that suits them. This is not a conspiracy. It is a structural incentive. A pharmaceutical company reporting drug trial results is not lying when it says the drug “reduces the risk of a heart attack by 36%.” That number may be accurate. What it is not doing is volunteering the absolute risk reduction, which might be 1.5 percentage points, or telling you how many people would need to take the drug for one person to benefit. Those numbers are available, but they are not in the advertisement.

The same structural pressure operates in political communication. When UK unemployment fell from 8.1% to 4.7% between 2011 and 2016, the government could reasonably describe this as unemployment “almost halving” (a relative description) or as falling by 3.4 percentage points (an absolute description). The relative description sounds more dramatic. It was used more often.

In any domain where someone has a financial, political, or reputational stake in how the numbers land, you should expect them to have chosen the framing that best serves their position. The question is not whether you trust them as a person. The question is whether you know enough to evaluate the claim you have been given.

The good news is that the test is the same in every case. Ask: is this a percentage point change or a percentage change? Ask: what is the denominator? Ask: what is the baseline, and what does this look like in absolute terms?

These three questions, applied consistently, disarm the majority of numerical manipulation you will encounter in ordinary public life.

How to Spot It

In 2021, the UK government described a 1.25 percentage point increase in National Insurance as a “1.25% rise.” Full Fact identified this error on ten separate government web pages. The government’s official guidance, press releases, and explainer pages all used the same phrasing.

This is a documented, verified instance of percentage point confusion published by one of Britain’s leading independent fact-checking organisations. The correction was eventually made. But ten pages means many people had already seen the misleading version, and many of those people probably assumed the rise was small, because 1.25% sounds small. The actual change was a roughly ten percent increase in what employees paid, because a 1.25 percentage point increase on a base rate of 12% is a proportional change of about 10.4%.

The tell is the missing baseline. When you see a percentage applied to something that is already a percentage, always ask what the original percentage was before forming any impression of the scale of change. A 1.25% rise sounds trivial. A rise from 12% to 13.25% is something you would notice in your take-home pay.

The second tell is the absence of absolute numbers. When a claim reports only a relative change (“doubled,” “50% more likely,” “one in three times the risk”), and does not volunteer the underlying rates, treat the omission as a signal. The absolute numbers are almost certainly less impressive than the relative ones, and the communicator has made a choice.

Your Challenge

A pharmaceutical company publishes the following claim in an advertisement: “In a clinical trial, our medication reduced the risk of a cardiovascular event by 35%.”

Before this number tells you anything meaningful, you need at least two more pieces of information. What are they? Once you have them, what calculation would you do to convert the “35% reduction” into something you could use to have an honest conversation with a doctor about whether the drug is worth taking?

There is no answer on this page. That is the point.

References

UK government National Insurance percentage point error and Full Fact investigation: Full Fact, “Government website makes percentage error on National Insurance rise” (2021). URL: https://fullfact.org/news/martin-lewis-national-insurance-government/. Martin Lewis public challenge to the government on Twitter, cited in the Full Fact article and in Rhyl Journal, “Martin Lewis clears confusion over National Insurance tax hike in the UK.” URL: https://www.rhyljournal.co.uk/news/19881463.martin-lewis-clears-confusion-national-insurance-tax-hike-uk/

Percentage point versus percentage change: definition and examples: Eurostat Statistics Explained, “Statistical concept: Percentage change and percentage points.” URL: https://ec.europa.eu/eurostat/statistics-explained/index.php/Beginners:Statistical_concept_-_Percentage_change_and_percentage_points. Wikipedia, “Percentage point.” URL: https://en.wikipedia.org/wiki/Percentage_point

Journalists Resource, “Percent change and percentage-point change: 4 tips to avoid math errors.” URL: https://journalistsresource.org/home/percent-change-math-for-journalists/

Absolute versus relative risk in pharmaceutical claims: Thomas J. et al., “Outcome Reporting Bias in COVID-19 mRNA Vaccine Clinical Trials,” Medicina (2021). PMC7996517. URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC7996517/. For absolute versus relative risk reduction in vaccine reporting generally: Olliaro P. et al., “COVID-19 vaccine efficacy and effectiveness — the elephant (not) in the room,” The Lancet Microbe (2021). URL: https://www.thelancet.com/journals/lanmic/article/PIIS2666-5247(21)00069-0/fulltext. The Pfizer–BioNTech vaccine relative risk reduction of 95.1% alongside an absolute risk reduction of approximately 0.7–0.84% is documented in: Brown R.B., “Outcome Reporting Bias in COVID-19 mRNA Vaccine Clinical Trials,” Medicina 57:2 (2021). URL: https://pmc.ncbi.nlm.nih.gov/articles/PMC7996517/